Maths can lead to a career in anything... including fashion
My passion for the fusion of science and design developed at University. Although I learned to knit and crochet as a child, it was while at university studying maths that my interest in knitting really developed, and I started to design and make unusual and interesting clothes. Being self-taught, I was not restricted by any boundaries and felt I could translate any idea into knitting by working out a logical way of doing it.
This approach clearly owed something to my mathematical background, and for me, there was a natural relationship between the two. I often put many ideas and techniques together to create complex designs, many of which are are non-repetitive, and combine colour, texture and form so that the result appears totally natural. Even now as a research professor, knitting continues to be, for me, the perfect blend of creativity, craft and technology, which my education seemed to want to separate.
Knitting used to be the poor relation of textiles but has now grown to be properly recognized, and has a vital part to play in fashion. I built an international business in fashion knitwear and yarns, with my designs and knitting kits selling in major retail stores worldwide such as Liberty and Harrods in London, Saks on 5th Avenue in New York and Takashimaya in Tokyo. Innovative and experimental crafted knitwear have recently made a big splash on the London Fashion Week catwalk, with Mark Fast’s intricate designs.
But knitting is not the only textile that has been influenced by science and mathematics. My current work as director for the Centre for Fashion Science at the London College of Fashion aims for collaboration between scientists and designers to create new concepts, products and processes which harness innovations in science and technology. This will break new ground in fashion–related research and hopefully merge desirability and fashion with sustainability and well-being.
The convergence of digital technologies and disciplines such as nanotechnology, with materials and cognitive sciences has opened up unprecedented possibilities for these technologies to create fashion and textiles that can respond to the user and have a function which enhances other aspects of our lifestyles.
For example, some of the centre’s researchers are exploring applications of body scanning for made-to-measure fashion and accessories and virtual try-on developments for key retailers. This could allow consumers to see if trousers fit their figure without having the hassle of the changing rooms.
Other areas of research include the development personalised design. This includes projects in to bespoke bags, which are ergonomically designed to fit the body; research examining how new forms of textiles can be developed to conform to the body providing clothes that truly are the ‘perfect fit’; and the study of seamless garment knitting for comfort and personalised fit utilising advanced knitting technology. A team at London College of Fashion is also developing an online style advisor system for ‘mass customization’, allowing the consumer to make clothing to their own specification. This is part of a European collaborative research project involving manufacturers and institutions from Germany, France, Greece, Italy and the UK.
Wonderland, a research project funded by the EPSRC, is collaboration between Professors Helen Storey (co-director Centre for Fashion Science at LCF) and Prof Tony Ryan (a polymer chemist from Sheffield University) which is enabling fashion to create the seemingly impossible. For instance, the disappearing dress, made from dissolving textiles which disintegrate in water, creating vibrant underwater fireworks; bringing a whole new possibility for sustainable clothes development. There are also incredible extensions of this, using the technologies to explore intelligent packaging, which once finished with, can be dissolved under hot water to form a gel in which seeds can be grown. This concept could revolutionise the packaging industry and resolve the age old problem of waste plastic.
These are just a few examples of how science and technology will increasingly link up with textiles, fashion and the creative industries to both push the boundaries of useful new technology and create design-led and innovative product ideas. These are exciting times for fashion and science, and I am excited to be at the cusp of the research which could change our lives.
Sandy Black is supporting the Science: [So what? So everything] campaign, which aims to highlight the science behind our everyday lives and the exciting careers in science.
Monday, November 2, 2009
Mental Arithmetic
Being able to do calculations in your head has some obvious uses, like checking your change in the supermarket, or figuring out the scores on the cricket ground. But is it useful for anything else in this age of computers and calculators?
The most important thing mental arithmetic does is to give you a feel for numbers, relationships between them and the patterns they make. This is obviously essential if you want to do well in maths at school, or go on to take a job that uses a lot of maths. But even if you plan to move into a completely different field after school, you will need a good grip on numbers and the ability to solve simple maths problems in your head. It’s a bit like learning a language: even though you can resort to a dictionary whenever you don’t know a word, you will never be able to speak fluently unless you know a lot of the vocabulary off the top of your head.
A safety net
Doctors and nurses, for example, need to do complicated calculations to work out the doses of the drugs they have to give to their patients. Even if they do have a calculator to hand, they have to be able to spot if they have made a mistake – punching in an extra zero or forgetting a decimal point can lead to the dose being ten times too high or low, with disastrous consequences for the patient. Being good at mental arithmetic is like having a safety-net that will guard you against mistakes.
Visual imagination
People whose work is very visual, like designers, architects and artists, for example, need to be able to imagine the way three-dimensional objects move in space. A look at a fashion designer’s sketch pad will show you that he or she has an intuitive feel for proportions and perspective. Such a feel for space and the objects in it can only come through practice and experience. The mental arithmetic you learn at school helps you exercise your visual imagination.
Maths in music
If you play a musical instrument, you will already have learnt how maths is important in music. Certain combinations of sounds are pleasing to us, because of the mathematical relationships between their frequencies. This is all about fractions, and musicians and sound engineers need an intuitive understanding of the relationships between numbers.
These are just three examples of how a good understanding of maths and numbers helps even in areas which, on the face of it, have nothing to do with maths. But also in your private life such an understanding is essential. When you open the paper or switch on the TV, you are bombarded by statistics, issued by politicians to get you to vote for them or by companies to get you to buy their products. When you open your mail, you will find bank statements and bills. You won’t always have the time to sit down and check them with a calculator and a maths book. But if you understand numbers, a quick glance can give you a good indication as to whether what is presented to you makes sense or not. If you’re good at mental arithmetic, you’ll glide much more smoothly through a world that is, after all, made up of numbers and patterns.
The most important thing mental arithmetic does is to give you a feel for numbers, relationships between them and the patterns they make. This is obviously essential if you want to do well in maths at school, or go on to take a job that uses a lot of maths. But even if you plan to move into a completely different field after school, you will need a good grip on numbers and the ability to solve simple maths problems in your head. It’s a bit like learning a language: even though you can resort to a dictionary whenever you don’t know a word, you will never be able to speak fluently unless you know a lot of the vocabulary off the top of your head.
A safety net
Doctors and nurses, for example, need to do complicated calculations to work out the doses of the drugs they have to give to their patients. Even if they do have a calculator to hand, they have to be able to spot if they have made a mistake – punching in an extra zero or forgetting a decimal point can lead to the dose being ten times too high or low, with disastrous consequences for the patient. Being good at mental arithmetic is like having a safety-net that will guard you against mistakes.
Visual imagination
People whose work is very visual, like designers, architects and artists, for example, need to be able to imagine the way three-dimensional objects move in space. A look at a fashion designer’s sketch pad will show you that he or she has an intuitive feel for proportions and perspective. Such a feel for space and the objects in it can only come through practice and experience. The mental arithmetic you learn at school helps you exercise your visual imagination.
Maths in music
If you play a musical instrument, you will already have learnt how maths is important in music. Certain combinations of sounds are pleasing to us, because of the mathematical relationships between their frequencies. This is all about fractions, and musicians and sound engineers need an intuitive understanding of the relationships between numbers.
These are just three examples of how a good understanding of maths and numbers helps even in areas which, on the face of it, have nothing to do with maths. But also in your private life such an understanding is essential. When you open the paper or switch on the TV, you are bombarded by statistics, issued by politicians to get you to vote for them or by companies to get you to buy their products. When you open your mail, you will find bank statements and bills. You won’t always have the time to sit down and check them with a calculator and a maths book. But if you understand numbers, a quick glance can give you a good indication as to whether what is presented to you makes sense or not. If you’re good at mental arithmetic, you’ll glide much more smoothly through a world that is, after all, made up of numbers and patterns.
The Universal Language
Mathematics is the only language shared by all human beings regardless of culture, religion, or gender. Pi is still approximately 3.14159 regardless of what country you are in. Adding up the cost of a basket full of groceries involves the same math process regardless of whether the total is expressed in dollars, rubles, or yen. With this universal language, all of us, no matter what our unit of exchange, are likely to arrive at math results the same way.
Very few people, if any, are literate in all the world's tongues—English, Chinese, Arabic, Bengali, and so on. But virtually all of us possess the ability to be "literate" in the shared language of math. This math literacy is called numeracy, and it is this shared language of numbers that connects us with people across continents and through time. It is what links ancient scholars and medieval merchants, astronauts and artists, peasants and presidents.
With this language we can explain the mysteries of the universe or the secrets of DNA. We can understand the forces of planetary motion, discover cures for catastrophic diseases, or calculate the distance from Boston to Bangkok. We can make chocolate chip cookies or save money for retirement. We can build computers and transfer information across the globe. Math is not just for calculus majors. It's for all of us. And it's not just about pondering imaginary numbers or calculating difficult equations. It's about making better daily decisions and, hopefully, leading richer, fuller lives.
Very few people, if any, are literate in all the world's tongues—English, Chinese, Arabic, Bengali, and so on. But virtually all of us possess the ability to be "literate" in the shared language of math. This math literacy is called numeracy, and it is this shared language of numbers that connects us with people across continents and through time. It is what links ancient scholars and medieval merchants, astronauts and artists, peasants and presidents.
With this language we can explain the mysteries of the universe or the secrets of DNA. We can understand the forces of planetary motion, discover cures for catastrophic diseases, or calculate the distance from Boston to Bangkok. We can make chocolate chip cookies or save money for retirement. We can build computers and transfer information across the globe. Math is not just for calculus majors. It's for all of us. And it's not just about pondering imaginary numbers or calculating difficult equations. It's about making better daily decisions and, hopefully, leading richer, fuller lives.
Population Growth
In the last few centuries, the number of people living on Earth has increased many times over. By the year 2000, there will be 10 times more people on Earth than there were 300 years ago.
How can population grow so fast? Think of a family tree. At the top are 2 parents, and beneath them the children they had. Listed beneath those children are the children they had, and so on and so on, down through each generation. As long as the family members continue to reproduce, the family tree continues to increase in size, getting larger with each passing generation. This same basic idea applies to the world's population.
Exponential growth
Population grows in the same way that money grows when it's left to compound interest in a bank. With money, growth comes through accumulating interest upon interest. The interest payments you accumulate eventually earn interest, increasing your money. With population growth, new members of the population eventually produce other new members of the population. The population increases exponentially as time passes.
--------------------------------------------------------------------------------
WORLD POPULATION
Year Population
1700 600,000,000
1800 900,000,000
1900 1,500,000,000
2000 6,000,000,000
--------------------------------------------------------------------------------
A crucial difference between money and population is that money can increase without limits while population can't. Any population of living creatures is constrained by the availability of food, water, land, or other important resources. Once those resources are depleted, a population won't continue to grow exponentially. It will plateau, or even decline, as a result of disease or malnutrition. Unlike calculating interest, calculating population growth is an imprecise business.
How fast will it grow?
To arrive at a reasonable estimate of how the world's population will grow in the next 50 years, you need to look at birth and death rates (the rates at which people are being born and dying in any given period). If birth and death rates stayed the same across the years in all parts of the world, population growth could be figured with a fairly simple formula much like the one used to figure compound interest. Birth and death rates aren't constant across countries and through time, however. Disease or disaster can cause death rates to increase for a certain period. A booming economy might mean higher birth rates for a given year.
The rate of Earth's population growth is slowing down. Throughout the 1960s, the world's population was growing at a rate of about 2% per year. By 1990, that rate was down to 1.5%, and by the year 2015, it's expected to drop to 1%. Family planning initiatives, an aging population, and the effects of diseases such as AIDS are some of the factors behind this rate decrease.
Even at these very low rates of population growth, the numbers are staggering. By 2015, despite a low expected 1% growth rate, experts estimate there will be 7 billion people on the planet. By 2050, there may be as many as 10 billion people living on Earth. Can the planet support this population? When will we reach the limit of our resources?
How can population grow so fast? Think of a family tree. At the top are 2 parents, and beneath them the children they had. Listed beneath those children are the children they had, and so on and so on, down through each generation. As long as the family members continue to reproduce, the family tree continues to increase in size, getting larger with each passing generation. This same basic idea applies to the world's population.
Exponential growth
Population grows in the same way that money grows when it's left to compound interest in a bank. With money, growth comes through accumulating interest upon interest. The interest payments you accumulate eventually earn interest, increasing your money. With population growth, new members of the population eventually produce other new members of the population. The population increases exponentially as time passes.
--------------------------------------------------------------------------------
WORLD POPULATION
Year Population
1700 600,000,000
1800 900,000,000
1900 1,500,000,000
2000 6,000,000,000
--------------------------------------------------------------------------------
A crucial difference between money and population is that money can increase without limits while population can't. Any population of living creatures is constrained by the availability of food, water, land, or other important resources. Once those resources are depleted, a population won't continue to grow exponentially. It will plateau, or even decline, as a result of disease or malnutrition. Unlike calculating interest, calculating population growth is an imprecise business.
How fast will it grow?
To arrive at a reasonable estimate of how the world's population will grow in the next 50 years, you need to look at birth and death rates (the rates at which people are being born and dying in any given period). If birth and death rates stayed the same across the years in all parts of the world, population growth could be figured with a fairly simple formula much like the one used to figure compound interest. Birth and death rates aren't constant across countries and through time, however. Disease or disaster can cause death rates to increase for a certain period. A booming economy might mean higher birth rates for a given year.
The rate of Earth's population growth is slowing down. Throughout the 1960s, the world's population was growing at a rate of about 2% per year. By 1990, that rate was down to 1.5%, and by the year 2015, it's expected to drop to 1%. Family planning initiatives, an aging population, and the effects of diseases such as AIDS are some of the factors behind this rate decrease.
Even at these very low rates of population growth, the numbers are staggering. By 2015, despite a low expected 1% growth rate, experts estimate there will be 7 billion people on the planet. By 2050, there may be as many as 10 billion people living on Earth. Can the planet support this population? When will we reach the limit of our resources?
Sunday, November 1, 2009
Maths in everyday life - 3
Market research
Have you ever been stopped by someone in the street wanting you to complete a questionnaire? If yes, have you thought about what happens to these answers?
All the data collected will be analysed by market researchers and statisticians, who then work out what people actually want. This could be something like finding out what people think about a new mobile phone, the look of a new MP3 player, anything really.
Market research is carried out by almost every business to make sure the products or services fulfil the consumer’s wants and needs. This helps the company stay ahead of the competition.
Our planet and us
At the end of a long day you finally go to bed. The Sun has gone down and you know for certain that it will rise again tomorrow, and so does every other living thing on Earth. The life of every animal and plant is governed by the movement of the Earth around the Sun, and every animal and plant is perfectly adapted to the environment it lives in.
We humans try to understand the natural world around us using sciences like physics and biology. These sciences are written in the language of maths. Physicists and astronomers use mathematical formulae to express the secrets of the universe and to send humans into space. Biologists believe that the secret of life lies in the genes and mathematical processes determine how genes get passed on and cause animals and plants to evolve so that they fit their environment perfectly. Our brains work because of the complicated interaction of neurons — so complicated that only maths has a chance to describe it. Even the way the patterns form on a Leopard’s coat is best understood using maths.
Just a day like any other. And like any other day, one where mathematics has played a major part.
Have you ever been stopped by someone in the street wanting you to complete a questionnaire? If yes, have you thought about what happens to these answers?
All the data collected will be analysed by market researchers and statisticians, who then work out what people actually want. This could be something like finding out what people think about a new mobile phone, the look of a new MP3 player, anything really.
Market research is carried out by almost every business to make sure the products or services fulfil the consumer’s wants and needs. This helps the company stay ahead of the competition.
Our planet and us
At the end of a long day you finally go to bed. The Sun has gone down and you know for certain that it will rise again tomorrow, and so does every other living thing on Earth. The life of every animal and plant is governed by the movement of the Earth around the Sun, and every animal and plant is perfectly adapted to the environment it lives in.
We humans try to understand the natural world around us using sciences like physics and biology. These sciences are written in the language of maths. Physicists and astronomers use mathematical formulae to express the secrets of the universe and to send humans into space. Biologists believe that the secret of life lies in the genes and mathematical processes determine how genes get passed on and cause animals and plants to evolve so that they fit their environment perfectly. Our brains work because of the complicated interaction of neurons — so complicated that only maths has a chance to describe it. Even the way the patterns form on a Leopard’s coat is best understood using maths.
Just a day like any other. And like any other day, one where mathematics has played a major part.
Maths in everyday life - 2
Transport
You wait for the bus. Maths is used to timetable buses, trains and planes, taking into account the limited stocks of vehicles, the cost of fuel and the number of people wanting to use them for various journeys across the day. It is used to decide the timing of traffic lights, and to understand traffic flow and congestion. This is another application of operational research (see planning).
Maths also affects the designs of these vehicles, particularly planes, cars and boats. Aerodynamics, another application of fluid dynamics, is obviously vital in designing a plane that can actually fly, but is also important in designing any sort of vehicle to move as smoothly through the air as possible so that it can go faster and take less fuel.
Understanding chance and risk
You get off the bus and someone is selling tickets in a raffle for a new car. Should you buy one? What are the chances of you winning in this, or in any other, lottery? All games of chance are governed by probabilities of events happening. If you understand probability, you have a much better chance of coming out on top, or in deciding that it's not worth spending your money at all.
The bus got stuck in traffic, and you arrived at your destination really late. Should you ride your bike next time, you wonder. But is that too risky? And, if you do, should you wear a helmet?
Your friend calls you to say she has had some negative test results from the doctor. Is she definitely sick, or what are the chances that the test is wrong and she is not sick at all? There are a number of treatment options available, but all have different rates of success and different side effects. To decide which one to take, she will need to weigh up the risks of side effects of each treatment against the chances of success, using the probabilities of each of these.
Business and industry also need to make decisions about risk and chance and need mathematicians - or operational researchers - to help them with this.
Communicating
Finally you get home and get a chance to check your email. Computers store all information as binary numbers, and use mathematical operations to manipulate this information, whether you are editing an email, resizing a photo or setting preferences in your web browser.
Computers communicate digitally, sending packets of information across networks or wires, fibre optic cables and phone lines. The way this information is encoded uses maths. Maths is also used to create better compression methods, so that you can download mp3s faster, or that movie makers can fit longer movies and more special features on DVDs.
You wait for the bus. Maths is used to timetable buses, trains and planes, taking into account the limited stocks of vehicles, the cost of fuel and the number of people wanting to use them for various journeys across the day. It is used to decide the timing of traffic lights, and to understand traffic flow and congestion. This is another application of operational research (see planning).
Maths also affects the designs of these vehicles, particularly planes, cars and boats. Aerodynamics, another application of fluid dynamics, is obviously vital in designing a plane that can actually fly, but is also important in designing any sort of vehicle to move as smoothly through the air as possible so that it can go faster and take less fuel.
Understanding chance and risk
You get off the bus and someone is selling tickets in a raffle for a new car. Should you buy one? What are the chances of you winning in this, or in any other, lottery? All games of chance are governed by probabilities of events happening. If you understand probability, you have a much better chance of coming out on top, or in deciding that it's not worth spending your money at all.
The bus got stuck in traffic, and you arrived at your destination really late. Should you ride your bike next time, you wonder. But is that too risky? And, if you do, should you wear a helmet?
Your friend calls you to say she has had some negative test results from the doctor. Is she definitely sick, or what are the chances that the test is wrong and she is not sick at all? There are a number of treatment options available, but all have different rates of success and different side effects. To decide which one to take, she will need to weigh up the risks of side effects of each treatment against the chances of success, using the probabilities of each of these.
Business and industry also need to make decisions about risk and chance and need mathematicians - or operational researchers - to help them with this.
Communicating
Finally you get home and get a chance to check your email. Computers store all information as binary numbers, and use mathematical operations to manipulate this information, whether you are editing an email, resizing a photo or setting preferences in your web browser.
Computers communicate digitally, sending packets of information across networks or wires, fibre optic cables and phone lines. The way this information is encoded uses maths. Maths is also used to create better compression methods, so that you can download mp3s faster, or that movie makers can fit longer movies and more special features on DVDs.
Maths in everyday life
Maths is everywhere. Without realizing, we use maths every day, and it plays a part in nearly all our daily activities. Every time we pick up the phone, use the internet, manage money, decide to take a risk, check the weather report, go to the doctors or travel anywhere, maths plays its part.
The weather
You wake up in the morning and get ready to go out. You check the weather report to decide what to wear and what to take - an umbrella? Sunglasses? A hat? The forecasts you read in the paper are a result of solving complicated equations involving the way air, clouds and water move around the planet - part of an area of mathematics called fluid dynamics.
Long-term weather trends can affect our whole environment. For instance, climate change might lead to more storms and more floods. These problems are part of the work of environmental statisticians who study data received from around the world and try to predict what might happen over the next several years. For more information related to this subject please visit the RSS guide to environmental statistics link below.
Buildings and construction
If it is rainy and cold outside, you will be happy to stay at home a while longer and have a nice hot cup of tea. But someone has built the house you are in, made sure it keeps the cold out and the warmth in, and provided you with running water for the tea. This someone is most likely an engineer. Engineers are responsible for just about everything we take for granted in the world around us, from tall buildings, tunnels and football stadiums, to access to clean drinking water. They also design and build vehicles, aircraft, boats and ships. What's more, engineers help to develop things which are important for the future, such as generating energy from the sun, wind or waves. Maths is involved in everything an engineer does, whether it is working out how much concrete is needed to build a bridge, or determining the amount of solar energy necessary to power a car.
Money
Once out the door you walk past a newsstand where the headline "house prices boom again" scream from the front pages. House prices are determined by supply and demand, and the one thing anyone buying a house needs is money. Most people get a loan from a bank to buy a house, and then have to pay back the money they borrowed plus some interest and fees. To choose which bank to borrow from, and to decide if you can afford to borrow at all, you need to understand compound interest.
The housing market is part of the bigger economic picture. How the economy is doing affects how much things cost, how much we are paid and how much the government spends - and maths is used to monitor the economy and predict how it will change. A large part of the world's economy is invested in the stock market, and highly skilled mathematicians are employed to try to understand, and even predict, movements in the stock market.
Planning
Whether you are managing money (trying to save for a gap-year holiday), resources (trying to make those Easter eggs last) or time (deciding how much time to spend on studying for each subject), you are doing calculations, sometimes automatically in your head, trying to optimise your outputs (money saved, grades achieved) given your inputs (pay from part-time job, hours in the day). In industry this is called operational research, and is used to improve the processes used in manufacturing, in how businesses are run, and in making the best use of resources such as beds in a hospital or police on the beat.
The weather
You wake up in the morning and get ready to go out. You check the weather report to decide what to wear and what to take - an umbrella? Sunglasses? A hat? The forecasts you read in the paper are a result of solving complicated equations involving the way air, clouds and water move around the planet - part of an area of mathematics called fluid dynamics.
Long-term weather trends can affect our whole environment. For instance, climate change might lead to more storms and more floods. These problems are part of the work of environmental statisticians who study data received from around the world and try to predict what might happen over the next several years. For more information related to this subject please visit the RSS guide to environmental statistics link below.
Buildings and construction
If it is rainy and cold outside, you will be happy to stay at home a while longer and have a nice hot cup of tea. But someone has built the house you are in, made sure it keeps the cold out and the warmth in, and provided you with running water for the tea. This someone is most likely an engineer. Engineers are responsible for just about everything we take for granted in the world around us, from tall buildings, tunnels and football stadiums, to access to clean drinking water. They also design and build vehicles, aircraft, boats and ships. What's more, engineers help to develop things which are important for the future, such as generating energy from the sun, wind or waves. Maths is involved in everything an engineer does, whether it is working out how much concrete is needed to build a bridge, or determining the amount of solar energy necessary to power a car.
Money
Once out the door you walk past a newsstand where the headline "house prices boom again" scream from the front pages. House prices are determined by supply and demand, and the one thing anyone buying a house needs is money. Most people get a loan from a bank to buy a house, and then have to pay back the money they borrowed plus some interest and fees. To choose which bank to borrow from, and to decide if you can afford to borrow at all, you need to understand compound interest.
The housing market is part of the bigger economic picture. How the economy is doing affects how much things cost, how much we are paid and how much the government spends - and maths is used to monitor the economy and predict how it will change. A large part of the world's economy is invested in the stock market, and highly skilled mathematicians are employed to try to understand, and even predict, movements in the stock market.
Planning
Whether you are managing money (trying to save for a gap-year holiday), resources (trying to make those Easter eggs last) or time (deciding how much time to spend on studying for each subject), you are doing calculations, sometimes automatically in your head, trying to optimise your outputs (money saved, grades achieved) given your inputs (pay from part-time job, hours in the day). In industry this is called operational research, and is used to improve the processes used in manufacturing, in how businesses are run, and in making the best use of resources such as beds in a hospital or police on the beat.
Why keep on studying maths?
Mathematics is one of the most general, and one of the most fundamental subjects that you can study. Although it may sometimes be difficult to see immediately the relevance of maths to a future career, and perhaps to subjects you intend to study later, maths in fact gives you an excellent grounding for any subject and is itself a prerequisite for many.
Why take A level maths?
Deciding which A levels to take can be difficult. Sometime it is a conflict of what you really like or are good at and what you want to do in order to be successful in your career or even something that your parents are doing. If you are thinking of going on to university, your first step should be to look at all the courses you might possibly want to take and see what the requirements are. Some courses specify particular subjects at A level and obviously this should be your first consideration.
For a degree in maths, statistics, physics, engineering or actuarial science, for example, you will almost certainly have to have a good maths A level and possible a further maths A level. However, there are many degrees which do not specify any particular subjects they wish you to have taken and, for many of these, maths will be an excellent choice.
If you are thinking of looking for a job straight after A levels, maths is a great "core" subject - in fact, it is one of the most important subjects you can take. This is because the ability to understand and manipulate numbers and mathematical concepts is extremely useful for almost any job.
There is always a demand for employees who can think logically and process information accurately - skills which an A level in maths will teach you.
Further maths?
If you enjoy doing maths, you might want to consider taking Further Maths at A/S or A level. The problems you are faced with in further maths are more challenging that the ordinary mathematics problems. They can also be more applicable in real case situations and have applications in physics, chemistry or biology. You can do this even if your school or college does not offer it.
Scientists need maths and stats!
Maths is the language of the physical sciences, and many of the quantitative parts of the social sciences as well. It also enables you to communicate complicated concepts. These more general skills that will be useful no matter what you want to do in the future.
Practically any course in the social or laboratory sciences will include a module on statistics, and students who stopped studying maths at GCSE often find this very difficult. If you have A level maths, even if this was not a required subject for admission, you will be ahead of the game!
Practically any course in the social or laboratory sciences will include a module on statistics, and students who stopped studying maths at GCSE often find this very difficult. If you have A level maths, even if this was not a required subject for admission, you will be ahead of the game
Engineering
If you study engineering you will use maths for most of your courses. Whether you are doing civil, mechanical, electrical or materials engineering, you will need to use geometry, calculus, and algebra to work with mathematical formulas for physical forces, electrical currents and other phenomena. For example, when studying civil engineering you will have to calculate the force distribution for different structures, such as truss bridges. This involves a combination of trigonometry and solving the equations of forces on the bridge. On other occasions you need to use integration to calculate the centre of mass of an object, or use differential equations to understand the flow of water through pipes.
Economics
Studying economics requires you not only to be able to handle data and work with figures, but to understand the concepts underlying economic questions as well. You will use maths to calculate compound interest, and arithmetic series to calculate growth of investments. You will need to use calculus to optimise cost-and-profit calculations, and to use maths to express and analyse supply and demand problems.
Medicine
Medical students have to take courses in statistics, which is not surprising given that much of today's medical knowledge is evidence-based. You will need to critically interpret data, such as the results of trials of new medical practise, judge the reliability of tests and assess the risks of treatments. Maths is also used in medical research for modeling tumour growth and the effects of therapy, planning treatment and understanding and interpreting medical scans.
Archaeology
Mathematics is even necessary in many of the social sciences, such as psychology and archaeology. Archaeologists use a variety of mathematical and statistical techniques to present the data from archaeological surveys and try to distinguish patterns in their results that shed light on past human behavior. Statistical measures are used during excavation to monitor which pits are most successful and decide on further excavation. Finds are analyzed using statistical and numerical methods to spot patterns in the way the archaeological record changes over time, and geographically within a site and across the country. Archaeologists also use statistics to test the reliability of their interpretations.
And best of all...
Obviously, if you want to go on to study mathematics or statistics, all the maths and stats you have done in the past will play its part. If you enjoy and even love these subjects, this is the greatest help in doing well. But even if this isn't exactly you, building up your mathematical toolbox will come in handy whatever you intend to study.
Why take A level maths?
Deciding which A levels to take can be difficult. Sometime it is a conflict of what you really like or are good at and what you want to do in order to be successful in your career or even something that your parents are doing. If you are thinking of going on to university, your first step should be to look at all the courses you might possibly want to take and see what the requirements are. Some courses specify particular subjects at A level and obviously this should be your first consideration.
For a degree in maths, statistics, physics, engineering or actuarial science, for example, you will almost certainly have to have a good maths A level and possible a further maths A level. However, there are many degrees which do not specify any particular subjects they wish you to have taken and, for many of these, maths will be an excellent choice.
If you are thinking of looking for a job straight after A levels, maths is a great "core" subject - in fact, it is one of the most important subjects you can take. This is because the ability to understand and manipulate numbers and mathematical concepts is extremely useful for almost any job.
There is always a demand for employees who can think logically and process information accurately - skills which an A level in maths will teach you.
Further maths?
If you enjoy doing maths, you might want to consider taking Further Maths at A/S or A level. The problems you are faced with in further maths are more challenging that the ordinary mathematics problems. They can also be more applicable in real case situations and have applications in physics, chemistry or biology. You can do this even if your school or college does not offer it.
Scientists need maths and stats!
Maths is the language of the physical sciences, and many of the quantitative parts of the social sciences as well. It also enables you to communicate complicated concepts. These more general skills that will be useful no matter what you want to do in the future.
Practically any course in the social or laboratory sciences will include a module on statistics, and students who stopped studying maths at GCSE often find this very difficult. If you have A level maths, even if this was not a required subject for admission, you will be ahead of the game!
Practically any course in the social or laboratory sciences will include a module on statistics, and students who stopped studying maths at GCSE often find this very difficult. If you have A level maths, even if this was not a required subject for admission, you will be ahead of the game
Engineering
If you study engineering you will use maths for most of your courses. Whether you are doing civil, mechanical, electrical or materials engineering, you will need to use geometry, calculus, and algebra to work with mathematical formulas for physical forces, electrical currents and other phenomena. For example, when studying civil engineering you will have to calculate the force distribution for different structures, such as truss bridges. This involves a combination of trigonometry and solving the equations of forces on the bridge. On other occasions you need to use integration to calculate the centre of mass of an object, or use differential equations to understand the flow of water through pipes.
Economics
Studying economics requires you not only to be able to handle data and work with figures, but to understand the concepts underlying economic questions as well. You will use maths to calculate compound interest, and arithmetic series to calculate growth of investments. You will need to use calculus to optimise cost-and-profit calculations, and to use maths to express and analyse supply and demand problems.
Medicine
Medical students have to take courses in statistics, which is not surprising given that much of today's medical knowledge is evidence-based. You will need to critically interpret data, such as the results of trials of new medical practise, judge the reliability of tests and assess the risks of treatments. Maths is also used in medical research for modeling tumour growth and the effects of therapy, planning treatment and understanding and interpreting medical scans.
Archaeology
Mathematics is even necessary in many of the social sciences, such as psychology and archaeology. Archaeologists use a variety of mathematical and statistical techniques to present the data from archaeological surveys and try to distinguish patterns in their results that shed light on past human behavior. Statistical measures are used during excavation to monitor which pits are most successful and decide on further excavation. Finds are analyzed using statistical and numerical methods to spot patterns in the way the archaeological record changes over time, and geographically within a site and across the country. Archaeologists also use statistics to test the reliability of their interpretations.
And best of all...
Obviously, if you want to go on to study mathematics or statistics, all the maths and stats you have done in the past will play its part. If you enjoy and even love these subjects, this is the greatest help in doing well. But even if this isn't exactly you, building up your mathematical toolbox will come in handy whatever you intend to study.
Mathematics education
Mathematics education is the practice of teaching and learning mathematics, as well as the field of scholarly research on this practice. Researchers in mathematics education are in the primarily concerned with the tools, methods and approaches that facilitate practice or the study of practice. However mathematics education research, known on the continent of Europe as the didactics of mathematics, has developed into a fully fledged field of study, with its own characteristic concepts, theories, methods, national and international organisations, conferences and literature. This article describes some of the history, influences and recent controversies concerning mathematics education as a practice.
History
Elementary mathematics was part of the education system in most ancient civilisations, including Ancient Greece, the Roman empire, Vedic society and ancient Egypt. In most cases, a formal education was only available to male children with a sufficiently high status, wealth or caste.
In Plato's division of the liberal arts into the trivium and the quadrivium, the quadrivium included the mathematical fields of arithmetic and geometry. This structure was continued in the structure of classical education that was developed in medieval Europe. Teaching of geometry was almost universally based on Euclid's Elements. Apprentices to trades such as masons, merchants and money-lenders could expect to learn such practical mathematics as was relevant to their profession.
The first mathematics textbooks to be written in English and French were published by Robert Recorde, beginning with The Grounde of Artes in 1540.
In the Renaissance the academic status of mathematics declined, because it was strongly associated with trade and commerce. Although it continued to be taught in European universities, it was seen as subservient to the study of Natural, Metaphysical and Moral Philosophy.
This trend was somewhat reversed in the seventeenth century, with the University of Aberdeen creating a Mathematics Chair in 1613, followed by the Chair in Geometry being set up in University of Oxford in 1619 and the Lucasian Chair of Mathematics being established by the University of Cambridge in 1662. However, it was uncommon for mathematics to be taught outside of the universities. Isaac Newton, for example, received no formal mathematics teaching until he joined Trinity College, Cambridge in 1661.
In the eighteenth and nineteenth centuries the industrial revolution led to an enormous increase in urban populations. Basic numeracy skills, such as the ability to tell the time, count money and carry out simple arithmetic, became essential in this new urban lifestyle. Within the new public education systems, mathematics became a central part of the curriculum from an early age.
By the twentieth century mathematics was part of the core curriculum in all developed countries.
During the twentieth century mathematics education was established as an independent field of research. Here are some of the main events in this development:
• In 1893 a Chair in mathematics education was created at the University of Göttingen, under the administration of Felix Klein
• The International Commission on Mathematical Instruction (ICMI) was founded in 1908, and Felix Klein became the first president of the organization
• A new interest in mathematics education emerged in the 1960s, and the commission was revitalized
• In 1968, the Shell Centre for Mathematical Education was established in Nottingham
• The first International Congress on Mathematical Education (ICME) was held in Lyon in 1969. The second congress was in Exeter in 1972, and after that it has been held every four years
In the 20th century, the cultural impact of the “electric age” (McLuhan) was also taken up by educational theory and the teaching of mathematics. While previous approach focused on "working with specialized 'problems' in arithmetic", the emerging structural approach to knowledge had "small children meditating about number theory and 'sets'."
History
Elementary mathematics was part of the education system in most ancient civilisations, including Ancient Greece, the Roman empire, Vedic society and ancient Egypt. In most cases, a formal education was only available to male children with a sufficiently high status, wealth or caste.
In Plato's division of the liberal arts into the trivium and the quadrivium, the quadrivium included the mathematical fields of arithmetic and geometry. This structure was continued in the structure of classical education that was developed in medieval Europe. Teaching of geometry was almost universally based on Euclid's Elements. Apprentices to trades such as masons, merchants and money-lenders could expect to learn such practical mathematics as was relevant to their profession.
The first mathematics textbooks to be written in English and French were published by Robert Recorde, beginning with The Grounde of Artes in 1540.
In the Renaissance the academic status of mathematics declined, because it was strongly associated with trade and commerce. Although it continued to be taught in European universities, it was seen as subservient to the study of Natural, Metaphysical and Moral Philosophy.
This trend was somewhat reversed in the seventeenth century, with the University of Aberdeen creating a Mathematics Chair in 1613, followed by the Chair in Geometry being set up in University of Oxford in 1619 and the Lucasian Chair of Mathematics being established by the University of Cambridge in 1662. However, it was uncommon for mathematics to be taught outside of the universities. Isaac Newton, for example, received no formal mathematics teaching until he joined Trinity College, Cambridge in 1661.
In the eighteenth and nineteenth centuries the industrial revolution led to an enormous increase in urban populations. Basic numeracy skills, such as the ability to tell the time, count money and carry out simple arithmetic, became essential in this new urban lifestyle. Within the new public education systems, mathematics became a central part of the curriculum from an early age.
By the twentieth century mathematics was part of the core curriculum in all developed countries.
During the twentieth century mathematics education was established as an independent field of research. Here are some of the main events in this development:
• In 1893 a Chair in mathematics education was created at the University of Göttingen, under the administration of Felix Klein
• The International Commission on Mathematical Instruction (ICMI) was founded in 1908, and Felix Klein became the first president of the organization
• A new interest in mathematics education emerged in the 1960s, and the commission was revitalized
• In 1968, the Shell Centre for Mathematical Education was established in Nottingham
• The first International Congress on Mathematical Education (ICME) was held in Lyon in 1969. The second congress was in Exeter in 1972, and after that it has been held every four years
In the 20th century, the cultural impact of the “electric age” (McLuhan) was also taken up by educational theory and the teaching of mathematics. While previous approach focused on "working with specialized 'problems' in arithmetic", the emerging structural approach to knowledge had "small children meditating about number theory and 'sets'."
Saturday, October 31, 2009
Banū Mūsā
The Banū Mūsā brothers (Arabic: بنو موسى, "Sons of Mūsā") were three 9th century Persian scholars, of Baghdad, active in the House of Wisdom:
• Abu Ja'far Muhammad ibn Mūsā ibn Shākir (before 803 – 873) (Arabic: محمد بن موسى بن شاكر) , who specialised in astronomy, engineering, geometry and physics.
• Ahmad ibn Mūsā ibn Shākir (803 – 873) (Arabic: أحمد بن موسى بن شاكر) , who specialised in engineering and mechanics.
• Al-Hasan ibn Mūsā ibn Shākir (810 – 873) (Arabic: الحسن بن موسى بن شاكر) , who specialised in engineering and geometry.
The Banu Musa were the sons of Mūsā ibn Shākir, who had been a highwayman and later an astrologer to the Caliph al-Ma'mūn. At his death, he left his young sons in the custody of the Caliph, who entrusted them to Ishaq bin Ibrahim al-Mus'abi, a former governor of Baghdad. The education of the three brothers was carried out by Yahya bin Abu Mansur who worked at the famous House of Wisdom library and translation centre in Baghdad.
Works
Book of Ingenious Devices
The Banu Musa brothers invented a number of automata (automatic machines) and mechanical devices, and they described a hundred such devices in their Book of Ingenious Devices. Some of these inventions include:
• Valve and plug valve
• Float valve
• Feedback controller
• Automatic flute player
• Programmable machine
• Mechanical trick devices
• Hurricane lamp
• Self-trimming lamp (Ahmad ibn Mūsā ibn Shākir)
• Self-feeding lamp
• Gas mask
• Grab
• Clamshell grab
• Fail-safe system
• Differential pressure
The Banu Musa also invented "the earliest known mechanical musical instrument", in this case a hydropowered organ which played interchangeable cylinders automatically. According to Charles B. Fowler, this "cylinder with raised pins on the surface remained the basic device to produce and reproduce music mechanically until the second half of the nineteenth century." The Banu Musa also invented an automatic flute player which appears to have been the first programmable machine.
Book on the motion of the orbs
In physics and astronomy, Muhammad ibn Musa was a pioneer of astrophysics and celestial mechanics. In the Book on the motion of the orbs, he was the first to discover that the heavenly bodies and celestial spheres were subject to the same laws of physics as Earth, unlike the ancients who believed that the celestial spheres followed their own set of physical laws different from that of Earth.
Astral Motion and The Force of Attraction
In mechanics and astronomy, Muhammad ibn Musa, in his Astral Motion and The Force of Attraction, discovered that there was a force of attraction between heavenly bodies, foreshadowing Newton's law of universal gravitation.
On mechanics
Ahmad (c. 805) specialised in mechanics and wrote a work on pneumatic devices called On mechanics.
Premises of the book of conics
The eldest brother, Ja'far Muḥammad, wrote a critical revision on Apollonius' Conics, called the Premises of the book of conics.
The Book of the Measurement of Plane and Spherical Figures
The Banu Musa's most famous mathematical treatise is The Book of the Measurement of Plane and Spherical Figures, which considered similar problems as Archimedes did in his On the Measurement of the Circle and On the Sphere and the Cylinder.
The elongated circular figure
The youngest brother, al-Hasan (c. 810), specialised in geometry and wrote a work on the ellipse called The elongated circular figure.
***Image :
Drawing of Self trimming lamp in Ahmad ibn Mūsā ibn Shākir's treatise on mechanical devices. The manuscript was written in Arabic.
Muslim Mathematicians
(1) Abd al-Hamīd ibn Turk
Abd al-Hamīd ibn Turk (fl. 830), known also as ‘Abd al-Hamīd ibn Wase ibn Turk Jili was a ninth century Muslim mathematician. Not much is known about his biography. The two records of him, one by the Persian Ibn Nadim and the other by al-Qifti are not identical. However al-Qifi mentions his name as Abd al-Hamīd ibn Wase ibn Turk Jili. Jili means from Gilan.
He wrote a work on algebra of which only a chapter called "Logical Necessities in Mixed Equations", on the solution of quadratic equations, has survived.
He authored a manuscript entitled Logical Necessities in Mixed Equations, which is very similar to al-Khwarzimi's Al-Jabr and was published at around the same time as, or even possibly earlier than, Al-Jabr. The manuscript gives the exact same geometric demonstration as is found in Al-Jabr, and in one case the same example as found in Al-Jabr, and even goes beyond Al-Jabr by giving a geometric proof that if the determinant is negative then the quadratic equation has no solution. The similarity between these two works has led some historians to conclude that algebra may have been well developed by the time of al-Khwarizmi and 'Abd al-Hamid.
(2) Abū al-Hasan ibn Alī al-Qalasādī
Abū al-Hasan ibn ‘Alī al-Qalaṣādī (1412 in Baza, Spain – 1486 in Beja, Tunisia) was an Arab Muslim mathematician and an Islamic scholar specializing in Islamic inheritance jurisprudence. He is known for being one of the most influential voices in algebraic notation since antiquity and for taking "the first steps toward the introduction of algebraic symbolism." He wrote numerous books on arithmetic and algebra, including al-Tabsira fi'lm al-hisab (Arabic: التبصير في علم الحساب "Clarification of the science of arithmetic").
Symbolic algebra
In Islamic mathematics, al-Qalasadi made the first attempt at creating an algebraic notation since Ibn al-Banna two centuries earlier, who was himself the first to make such an attempt since Diophantus and Brahmagupta in ancient times. The notations of his predecessors, however, lacked symbols for mathematical operations. Al-Qalasadi's algebraic notation was the first to have symbols for these functions and was thus "the first steps toward the introduction of algebraic symbolism." He represented mathematical symbols using characters from the Arabic alphabet, where:
• wa means "and" for addition (+)
• illa means "less" for subtraction (-)
• fi means "times" for multiplication (*)
• ala means "over" for division (/)
• j represents jadah meaning "root"
• sh represents shay meaning "thing" for a variable (x)
• m represents mal for a square (x2)
• k represents kab for a cube (x3)
• l represents yadilu for equality (=)
(3) Abū Kāmil Shujā ibn Aslam
Abū Kāmil Shujāʿ ibn Aslam ibn Muḥammad ibn Shujā (c. 850 – c. 930) Abu Kamil (Arabic: ابو كامل) for short, was an Egyptian Muslim mathematician during the Islamic Golden Age. He has also been called al-Hasib al-Misri—literally, "the Egyptian calculator."
Unlike the many polymaths of this era—notably al-Khwarizmi, al-Kindi, Ibn al-Haytham (Alhacen in the West), al-Biruni, Ibn Sina (Avicenna), and Ibn Rushd (Averroes)—Abu Kamil was a specialist. His field was algebra.
His mathematical techniques were later adopted by Fibonacci, thus allowing Abu Kamil an important role in introducing algebra to Europe.
Works
His Book on rare things in the art of calculation treated systems of equations whose solutions are whole numbers or fractions and also combinatorics. This work led to later research into the real numbers, solutions of polynomials, and finding roots by later scientists of the age such as al-Karaji and Ibn Yaḥyā al-Maghribī al-Samawʾal. His work The Book of Precious Things in the Art of Reckoning contains general methods for solving linear equations.
He was also the first to treat irrational numbers as algebraic objects. He was the first to accept irrational numbers (often in the form of a square root, cube root or fourth root) as solutions to quadratic equations or as coefficients in an equation. He was also the first to solve three non-linear simultaneous equations with three unknown variables.
Abd al-Hamīd ibn Turk (fl. 830), known also as ‘Abd al-Hamīd ibn Wase ibn Turk Jili was a ninth century Muslim mathematician. Not much is known about his biography. The two records of him, one by the Persian Ibn Nadim and the other by al-Qifti are not identical. However al-Qifi mentions his name as Abd al-Hamīd ibn Wase ibn Turk Jili. Jili means from Gilan.
He wrote a work on algebra of which only a chapter called "Logical Necessities in Mixed Equations", on the solution of quadratic equations, has survived.
He authored a manuscript entitled Logical Necessities in Mixed Equations, which is very similar to al-Khwarzimi's Al-Jabr and was published at around the same time as, or even possibly earlier than, Al-Jabr. The manuscript gives the exact same geometric demonstration as is found in Al-Jabr, and in one case the same example as found in Al-Jabr, and even goes beyond Al-Jabr by giving a geometric proof that if the determinant is negative then the quadratic equation has no solution. The similarity between these two works has led some historians to conclude that algebra may have been well developed by the time of al-Khwarizmi and 'Abd al-Hamid.
(2) Abū al-Hasan ibn Alī al-Qalasādī
Abū al-Hasan ibn ‘Alī al-Qalaṣādī (1412 in Baza, Spain – 1486 in Beja, Tunisia) was an Arab Muslim mathematician and an Islamic scholar specializing in Islamic inheritance jurisprudence. He is known for being one of the most influential voices in algebraic notation since antiquity and for taking "the first steps toward the introduction of algebraic symbolism." He wrote numerous books on arithmetic and algebra, including al-Tabsira fi'lm al-hisab (Arabic: التبصير في علم الحساب "Clarification of the science of arithmetic").
Symbolic algebra
In Islamic mathematics, al-Qalasadi made the first attempt at creating an algebraic notation since Ibn al-Banna two centuries earlier, who was himself the first to make such an attempt since Diophantus and Brahmagupta in ancient times. The notations of his predecessors, however, lacked symbols for mathematical operations. Al-Qalasadi's algebraic notation was the first to have symbols for these functions and was thus "the first steps toward the introduction of algebraic symbolism." He represented mathematical symbols using characters from the Arabic alphabet, where:
• wa means "and" for addition (+)
• illa means "less" for subtraction (-)
• fi means "times" for multiplication (*)
• ala means "over" for division (/)
• j represents jadah meaning "root"
• sh represents shay meaning "thing" for a variable (x)
• m represents mal for a square (x2)
• k represents kab for a cube (x3)
• l represents yadilu for equality (=)
(3) Abū Kāmil Shujā ibn Aslam
Abū Kāmil Shujāʿ ibn Aslam ibn Muḥammad ibn Shujā (c. 850 – c. 930) Abu Kamil (Arabic: ابو كامل) for short, was an Egyptian Muslim mathematician during the Islamic Golden Age. He has also been called al-Hasib al-Misri—literally, "the Egyptian calculator."
Unlike the many polymaths of this era—notably al-Khwarizmi, al-Kindi, Ibn al-Haytham (Alhacen in the West), al-Biruni, Ibn Sina (Avicenna), and Ibn Rushd (Averroes)—Abu Kamil was a specialist. His field was algebra.
His mathematical techniques were later adopted by Fibonacci, thus allowing Abu Kamil an important role in introducing algebra to Europe.
Works
His Book on rare things in the art of calculation treated systems of equations whose solutions are whole numbers or fractions and also combinatorics. This work led to later research into the real numbers, solutions of polynomials, and finding roots by later scientists of the age such as al-Karaji and Ibn Yaḥyā al-Maghribī al-Samawʾal. His work The Book of Precious Things in the Art of Reckoning contains general methods for solving linear equations.
He was also the first to treat irrational numbers as algebraic objects. He was the first to accept irrational numbers (often in the form of a square root, cube root or fourth root) as solutions to quadratic equations or as coefficients in an equation. He was also the first to solve three non-linear simultaneous equations with three unknown variables.
al-Khwārizmī's work in Algebra
Al-Kitāb al-mukhtasar fī hisāb al-jabr wa-l-muqābala (Arabic: الكتاب المختصر في حساب الجبر والمقابلة “The Compendious Book on Calculation by Completion and Balancing”) is a mathematical book written approximately 830 CE. The book was written with the encouragement of the Caliph Al-Ma'mun as a popular work on calculation and is replete with examples and applications to a wide range of problems in trade, surveying and legal inheritance. The term algebra is derived from the name of one of the basic operations with equations (al-jabr) described in this book. The book was translated in Latin as Liber algebrae et almucabala by Robert of Chester (Segovia, 1145) hence "algebra", and also by Gerard of Cremona. A unique Arabic copy is kept at Oxford and was translated in 1831 by F. Rosen. A Latin translation is kept in Cambridge.
The al-jabr is considered the foundational text of modern algebra. It provided an exhaustive account of solving polynomial equations up to the second degree, and introduced the fundamental methods of "reduction" and "balancing", referring to the transposition of subtracted terms to the other side of an equation, that is, the cancellation of like terms on opposite sides of the equation.
Al-Khwārizmī's method of solving linear and quadratic equations worked by first reducing the equation to one of six standard forms (where b and c are positive integers)
• squares equal roots (ax2 = bx)
• squares equal number (ax2 = c)
• roots equal number (bx = c)
• squares and roots equal number (ax2 + bx = c)
• squares and number equal roots (ax2 + c = bx)
• roots and number equal squares (bx + c = ax2)
by dividing out the coefficient of the square and using the two operations al-jabr (Arabic: الجبر “restoring” or “completion”) and al-muqābala ("balancing"). Al-jabr is the process of removing negative units, roots and squares from the equation by adding the same quantity to each side. For example, x2 = 40x − 4x2 is reduced to 5x2 = 40x. Al-muqābala is the process of bringing quantities of the same type to the same side of the equation. For example, x2 + 14 = x + 5 is reduced to x2 + 9 = x.
The above discussion uses modern mathematical notation for the types of problems which the book discusses. However, in Al-Khwārizmī's day, most of this notation had not yet been invented, so he had to use ordinary text to present problems and their solutions. For example, for one problem he writes, (from an 1831 translation)
"If some one say: "You divide ten into two parts: multiply the one by itself; it will be equal to the other taken eighty-one times." Computation: You say, ten less thing, multiplied by itself, is a hundred plus a square less twenty things, and this is equal to eighty-one things. Separate the twenty things from a hundred and a square, and add them to eighty-one. It will then be a hundred plus a square, which is equal to a hundred and one roots. Halve the roots; the moiety is fifty and a half. Multiply this by itself, it is two thousand five hundred and fifty and a quarter. Subtract from this one hundred; the remainder is two thousand four hundred and fifty and a quarter. Extract the root from this; it is forty-nine and a half. Subtract this from the moiety of the roots, which is fifty and a half. There remains one, and this is one of the two parts."
J. J. O'Conner and E. F. Robertson wrote in the MacTutor History of Mathematics archive:
"Perhaps one of the most significant advances made by Arabic mathematics began at this time with the work of al-Khwarizmi, namely the beginnings of algebra. It is important to understand just how significant this new idea was. It was a revolutionary move away from the Greek concept of mathematics which was essentially geometry. Algebra was a unifying theory which allowed rational numbers, irrational numbers, geometrical magnitudes, etc., to all be treated as "algebraic objects". It gave mathematics a whole new development path so much broader in concept to that which had existed before, and provided a vehicle for future development of the subject. Another important aspect of the introduction of algebraic ideas was that it allowed mathematics to be applied to itself in a way which had not happened before."
R. Rashed and Angela Armstrong write:
"Al-Khwarizmi's text can be seen to be distinct not only from the Babylonian tablets, but also from Diophantus' Arithmetica. It no longer concerns a series of problems to be resolved, but an exposition which starts with primitive terms in which the combinations must give all possible prototypes for equations, which henceforward explicitly constitute the true object of study. On the other hand, the idea of an equation for its own sake appears from the beginning and, one could say, in a generic manner, insofar as it does not simply emerge in the course of solving a problem, but is specifically called on to define an infinite class of problems."
(*** image :- Statue of Khwarizmi in front of the Faculty of Mathematics of Amirkabir University of Technology in Tehran, Iran.)
List of female mathematicians
The majority of mathematicians are male, but there have been some demographic changes since World War II. Women are still underrepresented in mathematics and neighboring branches of science such as physics. A number of prizes instituted by the AMS and other mathematical societies are aimed at changing this situation.
Prominent female mathematicians have included:
A
• Tatyana Afanasyeva (1876–1974)
• Maria Agnesi (1718–1799)
• Annie Dale Biddle Andrews (1885–1940)
• Hertha Marks Ayrton (1854–1923)
B
• Nina Bari (1901–1961)
• Ruth Aaronson Bari (1971–2005)
• Agnes Sime Baxter (1870–1917)
• Dorothy Lewis Bernstein (1914–1988)
• Vasanti N. Bhat-Nayak (1938–2009)
• Joan Birman (1927– )
• Gertrude Blanch (1897–1996)
• Lenore Blum
• Alicia Boole Stott (1860–1940)
• Mary Everest Boole (1832–1916)
• Marjorie Lee Browne (1914–1979)
C
• Mary Cartwright (1900–1998)
• Émilie du Châtelet (1706–1749)
• Fan Chung (1949– )
• Marianna Csörnyei (1975– )
D
Ingrid Daubechies is known for her Daubechies wavelets
• Ingrid Daubechies (1954– )
E
• Tatyana Ehrenfest (1905– 1984)
• Karin Erdmann (1948– )
G
• Sophie Germain (1776– 1831)
• Shafi Goldwasser (1958– )
H
• Christine Hamill (1923– 1956)
• Caroline Herschel (1750– 1848)
• Susan Howson (1973– )
• Hypatia of Alexandria (died 415)
K
Sofia Kovalevskaya is known for her contributions to partial differential equations
• Sofia Kovalevskaya (1850–1891)
• Krystyna Kuperberg (1944– )
L
• Olga Aleksandrovna Ladyzhenskaya (1922– 2004)
• Judith Q. Longyear (1938–1995)
• Ada Lovelace (1815–1852)
M
• Jessie MacWilliams (1917–1990)
• Mileva Marić (1875–1948)
• Dusa McDuff (1945– )
• Maryam Mirzakhani (1977– )
• Ruth Moufang (1905–1977)
N
• Hanna Neumann (1914–1971)
• Emmy Noether (1882–1935)
P
• Rózsa Péter (1905–1977)
• Elena Cornaro Piscopia (1646-1684)
• Cheryl Praeger (1948– )
R
• Marina Ratner (1938– )
• Idun Reiten (1942– )
• Julia Robinson (1919– 1985)
• Mary Ellen Rudin (1924– )
S
• Charlotte Scott (1858–1931)
• Nina Snaith
• Mary Somerville (1780–1872)
• Bhama Srinivasan (1935–)
• Irene Stegun (1919–2008)
• Esther Szekeres (1910–2005)
T
• Éva Tardos (1957– )
• Olga Taussky-Todd (1906–1995)
• Audrey Terras (1942– )
• Theano (6th century BC)
U
• Karen Uhlenbeck (1942– )
W
• Katrin Wehrheim
• Anna Johnson Pell Wheeler (1883– )
• Melanie Wood (1981– )
• Dorothy Maud Wrinch (1894–1976)
Y
• Grace Chisholm Young (1868–1944)
• Sofya Yanovskaya (1896–1966)
***This is a list of female mathematicians who are notable. Many have made significant contributions to mathematics. There are many many more (also from Pakistan), I will try to find, and then I will post the list here … keep intouch … (Nadeem Ahmad)
Prominent female mathematicians have included:
A
• Tatyana Afanasyeva (1876–1974)
• Maria Agnesi (1718–1799)
• Annie Dale Biddle Andrews (1885–1940)
• Hertha Marks Ayrton (1854–1923)
B
• Nina Bari (1901–1961)
• Ruth Aaronson Bari (1971–2005)
• Agnes Sime Baxter (1870–1917)
• Dorothy Lewis Bernstein (1914–1988)
• Vasanti N. Bhat-Nayak (1938–2009)
• Joan Birman (1927– )
• Gertrude Blanch (1897–1996)
• Lenore Blum
• Alicia Boole Stott (1860–1940)
• Mary Everest Boole (1832–1916)
• Marjorie Lee Browne (1914–1979)
C
• Mary Cartwright (1900–1998)
• Émilie du Châtelet (1706–1749)
• Fan Chung (1949– )
• Marianna Csörnyei (1975– )
D
Ingrid Daubechies is known for her Daubechies wavelets
• Ingrid Daubechies (1954– )
E
• Tatyana Ehrenfest (1905– 1984)
• Karin Erdmann (1948– )
G
• Sophie Germain (1776– 1831)
• Shafi Goldwasser (1958– )
H
• Christine Hamill (1923– 1956)
• Caroline Herschel (1750– 1848)
• Susan Howson (1973– )
• Hypatia of Alexandria (died 415)
K
Sofia Kovalevskaya is known for her contributions to partial differential equations
• Sofia Kovalevskaya (1850–1891)
• Krystyna Kuperberg (1944– )
L
• Olga Aleksandrovna Ladyzhenskaya (1922– 2004)
• Judith Q. Longyear (1938–1995)
• Ada Lovelace (1815–1852)
M
• Jessie MacWilliams (1917–1990)
• Mileva Marić (1875–1948)
• Dusa McDuff (1945– )
• Maryam Mirzakhani (1977– )
• Ruth Moufang (1905–1977)
N
• Hanna Neumann (1914–1971)
• Emmy Noether (1882–1935)
P
• Rózsa Péter (1905–1977)
• Elena Cornaro Piscopia (1646-1684)
• Cheryl Praeger (1948– )
R
• Marina Ratner (1938– )
• Idun Reiten (1942– )
• Julia Robinson (1919– 1985)
• Mary Ellen Rudin (1924– )
S
• Charlotte Scott (1858–1931)
• Nina Snaith
• Mary Somerville (1780–1872)
• Bhama Srinivasan (1935–)
• Irene Stegun (1919–2008)
• Esther Szekeres (1910–2005)
T
• Éva Tardos (1957– )
• Olga Taussky-Todd (1906–1995)
• Audrey Terras (1942– )
• Theano (6th century BC)
U
• Karen Uhlenbeck (1942– )
W
• Katrin Wehrheim
• Anna Johnson Pell Wheeler (1883– )
• Melanie Wood (1981– )
• Dorothy Maud Wrinch (1894–1976)
Y
• Grace Chisholm Young (1868–1944)
• Sofya Yanovskaya (1896–1966)
***This is a list of female mathematicians who are notable. Many have made significant contributions to mathematics. There are many many more (also from Pakistan), I will try to find, and then I will post the list here … keep intouch … (Nadeem Ahmad)
Math Tricks - 2
4. Multiplying together two numbers that differ by a small even number
This trick only works if you’ve memorized or can quickly calculate the squares of numbers. If you’re able to memorize some squares and use the tricks described later for some kinds of numbers you’ll be able to quickly multiply together many pairs of numbers that differ by 2, or 4, or 6.
Let’s say you want to calculate 12×14.
When two numbers differ by two their product is always the square of the number in between them minus 1.
12×14 = (13×13)-1 = 168.
16×18 = (17×17)-1 = 288.
99×101 = (100×100)-1 = 10000-1 = 9999
If two numbers differ by 4 then their product is the square of the number in the middle (the average of the two numbers) minus 4.
11×15 = (13×13)-4 = 169-4 = 165.
13×17 = (15×15)-4 = 225-4 = 221.
If the two numbers differ by 6 then their product is the square of their average minus 9.
12×18 = (15×15)-9 = 216.
17×23 = (20×20)-9 = 391.
5. Squaring 2-digit numbers that end in 5
If a number ends in 5 then its square always ends in 25. To get the rest of the product take the left digit and multiply it by one more than itself.
35×35 ends in 25. We get the rest of the product by multiplying 3 by one more than 3. So, 3×4 = 12 and that’s the rest of the product. Thus, 35×35 = 1225.
To calculate 65×65, notice that 6×7 = 42 and write down 4225 as the answer.
85×85: Calculate 8×9 = 72 and write down 7225.
6. Multiplying together 2-digit numbers where the first digits are the same and the last digits sum to 10
Let’s say you want to multiply 42 by 48. You notice that the first digit is 4 in both cases. You also notice that the other digits, 2 and 8, sum to 10. You can then use this trick: multiply the first digit by one more than itself to get the first part of the answer and multiply the last digits together to get the second (right) part of the answer.
An illustration is in order:
To calculate 42×48: Multiply 4 by 4+1. So, 4×5 = 20. Write down 20.
Multiply together the last digits: 2×8 = 16. Write down 16.
The product of 42 and 48 is thus 2016.
Notice that for this particular example you could also have noticed that 42 and 48 differ by 6 and have applied technique number 4.
Another example: 64×66. 6×7 = 42. 4×6 = 24. The product is 4224.
A final example: 86×84. 8×9 = 72. 6×4 = 24. The product is 7224
7. Squaring other 2-digit numbers
Let’s say you want to square 58. Square each digit and write a partial answer. 5×5 = 25. 8×8 = 64. Write down 2564 to start. Then, multiply the two digits of the number you’re squaring together, 5×8=40.
Double this product: 40×2=80, then add a 0 to it, getting 800.
Add 800 to 2564 to get 3364.
This is pretty complicated so let’s do more examples.
32×32. The first part of the answer comes from squaring 3 and 2.
3×3=9. 2×2 = 4. Write down 0904. Notice the extra zeros. It’s important that every square in the partial product have two digits.
Multiply the digits, 2 and 3, together and double the whole thing. 2×3x2 = 12.
Add a zero to get 120. Add 120 to the partial product, 0904, and we get 1024.
56×56. The partial product comes from 5×5 and 6×6. Write down 2536.
5×6x2 = 60. Add a zero to get 600.
56×56 = 2536+600 = 3136.
One more example: 67×67. Write down 3649 as the partial product.
6×7x2 = 42×2 = 84. Add a zero to get 840.
67×67=3649+840 = 4489.
8. Multiplying by doubling and halving
There are cases when you’re multiplying two numbers together and one of the numbers is even. In this case you can divide that number by two and multiply the other number by 2. You can do this over and over until you get to multiplication this is easy for you to do.
Let’s say you want to multiply 14 by 16. You can do this:
14×16 = 28×8 = 56×4 = 112×2 = 224.
Another example: 12×15 = 6×30 = 6×3 with a 0 at the end so it’s 180.
48×17 = 24×34 = 12×68 = 6×136 = 3×272 = 816. (Being able to calculate that 3×27 = 81 in your head is very helpful for this problem.)
9. Multiplying by a power of 2
To multiply a number by 2, 4, 8, 16, 32, or some other power of 2 just keep doubling the product as many times as necessary. If you want to multiply by 16 then double the number 4 times since 16 = 2×2x2×2.
15×16: 15×2 = 30. 30×2 = 60. 60×2 = 120. 120×2 = 240.
23×8: 23×2 = 46. 46×2 = 92. 92×2 = 184.
54×8: 54×2 = 108. 108×2 = 216. 216×2 = 432.
Practice these tricks and you’ll get good at solving many different kinds of arithmetic problems in your head, or at least quickly on paper. Half the fun is identifying which trick to use. Sometimes more than one trick will apply and you’ll get to choose which one is easiest for a particular problem.
This trick only works if you’ve memorized or can quickly calculate the squares of numbers. If you’re able to memorize some squares and use the tricks described later for some kinds of numbers you’ll be able to quickly multiply together many pairs of numbers that differ by 2, or 4, or 6.
Let’s say you want to calculate 12×14.
When two numbers differ by two their product is always the square of the number in between them minus 1.
12×14 = (13×13)-1 = 168.
16×18 = (17×17)-1 = 288.
99×101 = (100×100)-1 = 10000-1 = 9999
If two numbers differ by 4 then their product is the square of the number in the middle (the average of the two numbers) minus 4.
11×15 = (13×13)-4 = 169-4 = 165.
13×17 = (15×15)-4 = 225-4 = 221.
If the two numbers differ by 6 then their product is the square of their average minus 9.
12×18 = (15×15)-9 = 216.
17×23 = (20×20)-9 = 391.
5. Squaring 2-digit numbers that end in 5
If a number ends in 5 then its square always ends in 25. To get the rest of the product take the left digit and multiply it by one more than itself.
35×35 ends in 25. We get the rest of the product by multiplying 3 by one more than 3. So, 3×4 = 12 and that’s the rest of the product. Thus, 35×35 = 1225.
To calculate 65×65, notice that 6×7 = 42 and write down 4225 as the answer.
85×85: Calculate 8×9 = 72 and write down 7225.
6. Multiplying together 2-digit numbers where the first digits are the same and the last digits sum to 10
Let’s say you want to multiply 42 by 48. You notice that the first digit is 4 in both cases. You also notice that the other digits, 2 and 8, sum to 10. You can then use this trick: multiply the first digit by one more than itself to get the first part of the answer and multiply the last digits together to get the second (right) part of the answer.
An illustration is in order:
To calculate 42×48: Multiply 4 by 4+1. So, 4×5 = 20. Write down 20.
Multiply together the last digits: 2×8 = 16. Write down 16.
The product of 42 and 48 is thus 2016.
Notice that for this particular example you could also have noticed that 42 and 48 differ by 6 and have applied technique number 4.
Another example: 64×66. 6×7 = 42. 4×6 = 24. The product is 4224.
A final example: 86×84. 8×9 = 72. 6×4 = 24. The product is 7224
7. Squaring other 2-digit numbers
Let’s say you want to square 58. Square each digit and write a partial answer. 5×5 = 25. 8×8 = 64. Write down 2564 to start. Then, multiply the two digits of the number you’re squaring together, 5×8=40.
Double this product: 40×2=80, then add a 0 to it, getting 800.
Add 800 to 2564 to get 3364.
This is pretty complicated so let’s do more examples.
32×32. The first part of the answer comes from squaring 3 and 2.
3×3=9. 2×2 = 4. Write down 0904. Notice the extra zeros. It’s important that every square in the partial product have two digits.
Multiply the digits, 2 and 3, together and double the whole thing. 2×3x2 = 12.
Add a zero to get 120. Add 120 to the partial product, 0904, and we get 1024.
56×56. The partial product comes from 5×5 and 6×6. Write down 2536.
5×6x2 = 60. Add a zero to get 600.
56×56 = 2536+600 = 3136.
One more example: 67×67. Write down 3649 as the partial product.
6×7x2 = 42×2 = 84. Add a zero to get 840.
67×67=3649+840 = 4489.
8. Multiplying by doubling and halving
There are cases when you’re multiplying two numbers together and one of the numbers is even. In this case you can divide that number by two and multiply the other number by 2. You can do this over and over until you get to multiplication this is easy for you to do.
Let’s say you want to multiply 14 by 16. You can do this:
14×16 = 28×8 = 56×4 = 112×2 = 224.
Another example: 12×15 = 6×30 = 6×3 with a 0 at the end so it’s 180.
48×17 = 24×34 = 12×68 = 6×136 = 3×272 = 816. (Being able to calculate that 3×27 = 81 in your head is very helpful for this problem.)
9. Multiplying by a power of 2
To multiply a number by 2, 4, 8, 16, 32, or some other power of 2 just keep doubling the product as many times as necessary. If you want to multiply by 16 then double the number 4 times since 16 = 2×2x2×2.
15×16: 15×2 = 30. 30×2 = 60. 60×2 = 120. 120×2 = 240.
23×8: 23×2 = 46. 46×2 = 92. 92×2 = 184.
54×8: 54×2 = 108. 108×2 = 216. 216×2 = 432.
Practice these tricks and you’ll get good at solving many different kinds of arithmetic problems in your head, or at least quickly on paper. Half the fun is identifying which trick to use. Sometimes more than one trick will apply and you’ll get to choose which one is easiest for a particular problem.
Math Tricks - 1
Being able to perform arithmetic quickly and mentally can greatly boost your self-esteem, especially if you don’t consider yourself to be very good at Math. And, getting comfortable with arithmetic might just motivate you to dive deeper into other things mathematical.
This article presents nine ideas that will hopefully get you to look at arithmetic as a game, one in which you can see patterns among numbers and pick then apply the right trick to quickly doing the calculation.
The tricks in this article all involve multiplication.
Don’t be discouraged if the tricks seem difficult at first. Learn one trick at a time. Read the description, explanation, and examples several times for each technique you’re learning. Then make up some of your own examples and practice the technique.
As you learn and practice the tricks make sure you check your results by doing multiplication the way you’re used to, until the tricks start to become second nature. Checking your results is critically important: the last thing you want to do is learn the tricks incorrectly.
1. Multiplying by 9, or 99, or 999
Multiplying by 9 is really multiplying by 10-1.
So, 9×9 is just 9x(10-1) which is 9×10-9 which is 90-9 or 81.
Let’s try a harder example: 46×9 = 46×10-46 = 460-46 = 414.
One more example: 68×9 = 680-68 = 612.
To multiply by 99, you multiply by 100-1.
So, 46×99 = 46x(100-1) = 4600-46 = 4554.
Multiplying by 999 is similar to multiplying by 9 and by 99.
38×999 = 38x(1000-1) = 38000-38 = 37962.
2. Multiplying by 11
To multiply a number by 11 you add pairs of numbers next to each other, except for the numbers on the edges.
Let me illustrate:
To multiply 436 by 11 go from right to left.
First write down the 6 then add 6 to its neighbor on the left, 3, to get 9.
Write down 9 to the left of 6.
Then add 4 to 3 to get 7. Write down 7.
Then, write down the leftmost digit, 4.
So, 436×11 = is 4796.
Let’s do another example: 3254×11.
The answer comes from these sums and edge numbers: (3)(3+2)(2+5)(5+4)(4) = 35794.
One more example, this one involving carrying: 4657×11.
Write down the sums and edge numbers: (4)(4+6)(6+5)(5+7)(7).
Going from right to left we write down 7.
Then we notice that 5+7=12.
So we write down 2 and carry the 1.
6+5 = 11, plus the 1 we carried = 12.
So, we write down the 2 and carry the 1.
4+6 = 10, plus the 1 we carried = 11.
So, we write down the 1 and carry the 1.
To the leftmost digit, 4, we add the 1 we carried.
So, 4657×11 = 51227 .
3. Multiplying by 5, 25, or 125
Multiplying by 5 is just multiplying by 10 and then dividing by 2. Note: To multiply by 10 just add a 0 to the end of the number.
12×5 = (12×10)/2 = 120/2 = 60.
Another example: 64×5 = 640/2 = 320.
And, 4286×5 = 42860/2 = 21430.
To multiply by 25 you multiply by 100 (just add two 0’s to the end of the number) then divide by 4, since 100 = 25×4. Note: to divide by 4 your can just divide by 2 twice, since 2×2 = 4.
64×25 = 6400/4 = 3200/2 = 1600.
58×25 = 5800/4 = 2900/2 = 1450.
To multiply by 125, you multipy by 1000 then divide by 8 since 8×125 = 1000. Notice that 8 = 2×2x2. So, to divide by 1000 add three 0’s to the number and divide by 2 three times.
32×125 = 32000/8 = 16000/4 = 8000/2 = 4000.
48×125 = 48000/8 = 24000/4 = 12000/2 = 6000.
This article presents nine ideas that will hopefully get you to look at arithmetic as a game, one in which you can see patterns among numbers and pick then apply the right trick to quickly doing the calculation.
The tricks in this article all involve multiplication.
Don’t be discouraged if the tricks seem difficult at first. Learn one trick at a time. Read the description, explanation, and examples several times for each technique you’re learning. Then make up some of your own examples and practice the technique.
As you learn and practice the tricks make sure you check your results by doing multiplication the way you’re used to, until the tricks start to become second nature. Checking your results is critically important: the last thing you want to do is learn the tricks incorrectly.
1. Multiplying by 9, or 99, or 999
Multiplying by 9 is really multiplying by 10-1.
So, 9×9 is just 9x(10-1) which is 9×10-9 which is 90-9 or 81.
Let’s try a harder example: 46×9 = 46×10-46 = 460-46 = 414.
One more example: 68×9 = 680-68 = 612.
To multiply by 99, you multiply by 100-1.
So, 46×99 = 46x(100-1) = 4600-46 = 4554.
Multiplying by 999 is similar to multiplying by 9 and by 99.
38×999 = 38x(1000-1) = 38000-38 = 37962.
2. Multiplying by 11
To multiply a number by 11 you add pairs of numbers next to each other, except for the numbers on the edges.
Let me illustrate:
To multiply 436 by 11 go from right to left.
First write down the 6 then add 6 to its neighbor on the left, 3, to get 9.
Write down 9 to the left of 6.
Then add 4 to 3 to get 7. Write down 7.
Then, write down the leftmost digit, 4.
So, 436×11 = is 4796.
Let’s do another example: 3254×11.
The answer comes from these sums and edge numbers: (3)(3+2)(2+5)(5+4)(4) = 35794.
One more example, this one involving carrying: 4657×11.
Write down the sums and edge numbers: (4)(4+6)(6+5)(5+7)(7).
Going from right to left we write down 7.
Then we notice that 5+7=12.
So we write down 2 and carry the 1.
6+5 = 11, plus the 1 we carried = 12.
So, we write down the 2 and carry the 1.
4+6 = 10, plus the 1 we carried = 11.
So, we write down the 1 and carry the 1.
To the leftmost digit, 4, we add the 1 we carried.
So, 4657×11 = 51227 .
3. Multiplying by 5, 25, or 125
Multiplying by 5 is just multiplying by 10 and then dividing by 2. Note: To multiply by 10 just add a 0 to the end of the number.
12×5 = (12×10)/2 = 120/2 = 60.
Another example: 64×5 = 640/2 = 320.
And, 4286×5 = 42860/2 = 21430.
To multiply by 25 you multiply by 100 (just add two 0’s to the end of the number) then divide by 4, since 100 = 25×4. Note: to divide by 4 your can just divide by 2 twice, since 2×2 = 4.
64×25 = 6400/4 = 3200/2 = 1600.
58×25 = 5800/4 = 2900/2 = 1450.
To multiply by 125, you multipy by 1000 then divide by 8 since 8×125 = 1000. Notice that 8 = 2×2x2. So, to divide by 1000 add three 0’s to the number and divide by 2 three times.
32×125 = 32000/8 = 16000/4 = 8000/2 = 4000.
48×125 = 48000/8 = 24000/4 = 12000/2 = 6000.
Wednesday, October 28, 2009
OMAR AL-KHAYYAM
OMAR AL-KHAYYAM
Ghiyath al-Din Abul Fateh Omar Ibn Ibrahim al-Khayyam was born at Nishapur, the provincial capital of Khurasan around 1044 C.E. (c. 1038 to 1048). Persian mathematician, astronomer, philosopher, physician and poet, he is commonly known as Omar Khayyam. Khayyam means the tent-maker, and although generally considered as Persian, it has also been suggested that he could have belonged to the Khayyami tribe of Arab origin who might have settled in Persia. Little is known about his early life, except for the fact that he was educated at Nishapur and lived there and at Samarqand for most of his life. He was a contemporary of Nidham al-Mulk Tusi. Contrary to the available opportunities, he did not like to be employed at the King's court and led a calm life devoted to search for knowledge. He travelled to the great centres of learn- ing, Samarqand, Bukhara, Balkh and Isphahan in order to study further and exchange views with the scholars there. While at Samarqand he was patronised by a dignatory, Abu Tahir. He died at Nishapur in 1123-24.
Algebra would seem to rank first among the fields to which he contributed. He made an attempt to classify most algebraic equations, including the third degree equations and, in fact, offered solutions for a number of them. This includes geometric solutions of cubic equations and partial geometric solutions of most other equations. His book Maqalat fi al-Jabr wa al-Muqabila is a master- piece on algebra and has great importance in the development of algebra. His remarkable classification of equations is based on the complexity of the equations, as the higher the degree of an equation, the more terms, or combinations of terms, it will contain. Thus, Khayyam recognizes 13 different forms of cubic equatlon. His method of solving equations is largely geometrical and depends upon an ingenious selection of proper conics. He also developed the binomial expansion when the exponent is a positive integer. In fact, he has been considered to be the first to find the binomial theorem and determine binomial coefficients. In geometry, he studied generalities of Euclid and contributed to the theory of parallel lines.
The Saljuq Sultan, Malikshah Jalal al-Din, called him to the new observatory at Ray around 1074 and assigned him the task of determining a correct solar calendar. This had become necessary in view of the revenue collections and other administrative matters that were to be performed at different times of the year. Khayyam introduced a calendar that was remarkably accurate, and was named as Al-Tarikh-al-Jalali. It had an error of one day in 3770 years and was thus even superior to the Georgian calendar (error of 1 day in 3330 years).
His contributions to other fields of science include a study of generalities of Euclid, development of methods for the accurate determination of specific gravity, etc. In metaphysics, he wrote three books Risala Dar Wujud and the recently discovered Nauruz- namah. He was also a renowned astronomer and a physician.
Apart from being a scientist, Khayyam was also a well-known poet. In this capacity, he has become more popularly known in the Western world since 1839, when Edward Fitzgerald published an English translation of his Rubaiyat (quatrains). This has since become one of the most popular classics of world literature. It should be appreciated that it is practically impossible to exactly translate any literary work into another language, what to talk of poetry, especially when it involves mystical and philosophical messages of deep complexity. Despite this, the popularity of the translation of Rubaiyat would indicate the wealth of his rich thought.
Khayyam wrote a large number of books and monographs in the above areas. Out of these, 10 books and thirty monographs have been identified. Of these, four concern mathematics, three physics, three metaphysics, one algebra and one geometry.
His influence on the development of mathematics in general and analytical geometry, in particular, has been immense. His work remained ahead of others for centuries till the times of Descartes, who applied the same geometrical approach in solving cubics. His fame as a mathematician has been partially eclipsed by his popularity as a poet; nonetheless his contribution as a philosopher and scientist has been of significant value in furthering the frontiers of human knowledge.
YAQUB IBN ISHAQ AL-KINDI
YAQUB IBN ISHAQ AL-KINDI
Abu Yousuf Yaqub Ibn Ishaq al-Kindi was born at Kufa around 800 C.E. His father was an official of Haroon al-Rashid. Al-Kindi was a contemporary of al-Mamun, al-Mu'tasim and al-Mutawakkil and flourished largely at Baghdad. He vas formally employed by Mutawakkil as a calligrapher. On account of his philosophical views, Mutawakkil was annoyed with him and confiscated all his books. These were, however, returned later on. He died in 873 C.E. during the reign of al-M'utamid.
Al-Kindi was a philosopher, mathematician, physicist, astronomer, physician, geographer and even an expert in music. It is surprising that he made original contributions to all of these fields. On account of his work he became known as the philosopher of the Arabs.
In mathematics, he wrote four books on the number system and laid the foundation of a large part of modern arithmetic. No doubt the Arabic system of numerals was largely developed by al- Khawarizmi, but al-Kindi also made rich contributions to it. He also contributed to spherical geometry to assist him in astronomical studies.
In chemistry, he opposed the idea that base metals can be converted to precious metals. In contrast to prevailing alchemical views, he was emphatic that chemical reactions cannot bring about the transformation of elements. In physics, he made rich contributions to geometrical optics and wrote a book on it. This book later on provided guidance and inspiration to such eminent scientists as Roger Bacon.
In medicine, his chief contribution comprises the fact that he was the first to systematically determine the doses to be adminis- tered of all the drugs known at his time. This resolved the conflic- ting views prevailing among physicians on the dosage that caused difficulties in writing recipes.
Very little was known on the scientific aspects of music in his time. He pointed out that the various notes that combine to produce harmony, have a specific pitch each. Thus, notes with too low or too high a pitch are non-pleatant. The degree of harmony depends on the frequency of notes, etc. He also pointed out the fact that when a sound is produced, it generates waves in the air which strike the ear-drum. His work contains a notation on the determination of pitch.
He was a prolific writer, the total number of books written by him was 241, the prominent among which were divided as follows:
Astronomy 16, Arithmetic 11, Geometry 32, Medicine 22,
Physics 12, Philosophy 22, Logic 9, Psychology 5, arr Music 7.
In addition, various monographs written by him concern tides, astronomical instruments, rocks, precious stones, etc. He was also an early translator of Greek works into Arabic, but this fact has largely been over-shadowed by his numerous original writings. It is unfortunate that most of his books are no longer extant, but those existing speak very high of his standard of scholarship and contribution. He was known as Alkindus in Latin and a large number of his books were translated into Latin by Gherard of Cremona. His books that were translated into Latin during the Middle Ages comprise Risalah dar Tanjim, Ikhtiyarat al-Ayyam, Ilahyat-e-Aristu, al-Mosiqa, Mad-o-Jazr, and Aduiyah Murakkaba.
Al-Kindi's influence on development of science and philosophy was significant in the revival of sciences in that period. In the Middle Ages, Cardano considered him as one of the twelve greatest minds. His works, in fact, lead to further development of various subjects for centuries, notably physics, mathematics, medicine and music.
Thabit Ibn Qurra Ibn Marwan al-Sabi al-Harrani
Thabit Ibn Qurra Ibn Marwan al-Sabi al-Harrani
Thabit Marwanwas born in the year 836 C.E. at Harran (present Turkey). As the name indicates he was basically a member of the Sabian sect, but the great Muslim mathematician Muhammad Ibn Musa Ibn Shakir, impressed by his knowledge of languages, and realising his potential for a scientific career, selected him to join the scientific group at Baghdad that was being patronised by the Abbasid Caliphs. There, he studied under the famous Banu Musa brothers. It was in this setting that Thabit contributed to several branches of science, notably mathematics, astronomy and mechanics, in addition to translating a large number of works from Greek to Arabic. Later, he was patronized by the Abbasid Caliph al-M'utadid. After a long career of scholarship, Thabit died at Baghdad in 901 C.E.
Thabit's major contribution lies in mathematics and astronomy. He was instrumental in extending the concept of traditional geometry to geometrical algebra and proposed several theories that led to the development of non-Euclidean geometry, spherical trigonometry, integral calculus and real numbers. He criticized a number of theorems of Euclid's elements and proposed important improvements. He applied arithmetical terminology to geometrical quantities, and studied several aspects of conic sections, notably those of parabola and ellipse. A number of his computations aimed at determining the surfaces and volumes of different types of bodies and constitute, in fact, the processes of integral calculus, as developed later.
In astronomy he was one of the early reformers of Ptolemic views. He analysed several. Problems related to the movements of sun and moon and wrote treatises on sun-dials.
In the fields of mechanics and physics he may be recognised as the founder of static’s. He examined conditions of equilibrium of bodies, beams and levers.
In addition to translating a large number of books himself, he founded a school of translation and supervised the translation of a further large number of books from Greek to Arabic.
Among Thabit's writings a large number have survived, while several are not extant. Most of the books are on mathematics, followed by astronomy and medicine. The books have been written in Arabic but some are in Syriac. In the Middle Ages, some of his books were translated into Latin by Gherard of Cremona. In recent centuries, a number of his books have been translated into European languages and published.
He carried further the work of the Banu Musa brothers and later his son and grandson continued in this tradition, together with the other members of the group. His original books as well as his translations accomplished in the 9th century exerted a positive influence on the development of subsequent scientific research.
ABUL WAFA MUHAMMAD AL-BUZJANI
ABUL WAFA MUHAMMAD AL-BUZJANI
Abul Wafa Muhammad Ibn Muhammad Ibn Yahya Ibn Ismail al-Buzjani was born in Buzjan, Nishapur in 940 C.E. He flourished as a great mathematician and astronomer at Baghdad and died in 997/998 C.E. He learnt mathematics in Baghdad. In 959 C.E. he migrated to Iraq and lived there till his death.
Abul Wafa's main contribution lies in several branches of mathematics, especially geometry and trigonometry. In geometry his contribution comprises solution of geometrical problems with opening of the compass; construction of a square equivalent to other squares; regular polyhedra; construction of regular hectagon taking for its side half the side of the equilateral triangle inscribed in the same circle; constructions of parabola by points and geometrical solution of the equations:
x4 = a and x4 + ax3 = b
Abul Wafa's contribution to the development of trigonometry was extensive. He was the first to show the generality of the sine theorem relative to spherical triangles. He developed a new method of constructing sine tables, the value of sin 30' being correct to the eighth decimal place. He also developed relations for sine (a+b) and the formula:
2 sin2 (a/2) = 1 - cos a , and
sin a = 2 sin (a/2) cos (a/2)
In addition, he made a special study of the tangent and calculated a table of tangents. He introduced the secant and cosecant for the first time, knew the relations between the trigonometric lines, which are now used to define them, and undertook extensive studies on conics.
Apart from being a mathematician, Abul Wafa also contributed to astronomy. In this field he discussed different movernents of the moon, and discovered 'variation'. He was also one of the last Arabic translators and commentators of Greek works.
He wrote a large number of books on mathematics and other subjects, most of which have been lost or exist in modified forms. His contribution includes Kitab 'Ilm al-Hisab, a practical book of arithmetic, al-Kitab al-Kamil (the Complete Book), Kitab al-Handsa (Applied Geometry). Apart from this, he wrote rich commentaries on Euclid, Diophantos and al-Khawarizmi, but all of these have been lost. His books now extant include Kitab 'Ilm al-Hisab, Kitab al- Handsa and Kitab al-Kamil.
His astronomical knowledge on the movements of the moon has been criticized in that, in the case of 'variation' the third inequality of the moon as he discussed was the second part of the 'evection'. But, according to Sedat, what he discovered was the same that was discovered by Tycho Brache six centuries later. Nonetheless, his contribution to trigonometry was extremely significant in that he developed the knowledge on the tangent and introduced the secant and cosecant for the first time; in fact a sizeable part of today's trigonometry can be traced back to him.
MOHAMMAD BIN MUSA AL-KHAWARIZMI
One of the Greatest Mathematicians: MOHAMMAD BIN MUSA AL-KHAWARIZMI
Abu Abdullah Mohammad Ibn Musa al-Khawarizmi was born at Khawarizm (Kheva), south of Aral sea. Very little is known about his early life, except for the fact that his parents had migrated to a place south of Baghdad. The exact dates of his birth and death are also not known, but it is established that he flourished under Al- Mamun at Baghdad through 813-833 and probably died around 840 C.E.
Khawarizmi was a mathematician, astronomer and geographer. He was perhaps one of the greatest mathematicians who ever lived, as, in fact, he was the founder of several branches and basic concepts of mathematics. In the words of Phillip Hitti, he influenced mathematical thought to a greater extent than any other medieval writer. His work on algebra was outstanding, as he not only initiated the subject in a systematic form but he also developed it to the extent of giving analytical solutions of linear and quadratic equations, which established him as the founder of Algebra. The very name Algebra has been derived from his famous book Al-Jabr wa-al-Muqabilah. His arithmetic synthesized Greek and Hindu knowledge and also contained his own contribution of fundamental importance to mathematics and science. Thus, he explained the use of zero, a numeral of fundamental importance developed by the Arabs. Similarly, he developed the decimal system so that the overall system of numerals, 'algorithm' or 'algorizm' is named after him. In addition to introducing the Indian system of numerals (now generally known as Arabic numerals), he developed at length several arithmetical procedures, including operations on fractions. It was through his work that the system of numerals was first introduced to Arabs and later to Europe, through its translations in European languages. He developed in detail trigonometric tables containing the sine functions, which were probably extrapolated to tangent functions by Maslama. He also perfected the geometric representation of conic sections and developed the calculus of two errors, which practically led him to the concept of differentiation. He is also reported to have collaborated in the degree measurements ordered by Mamun al-Rashid were aimed at measuring of volume and circumference of the earth.
The development of astronomical tables by him was a significant contribution to the science of astronomy, on which he also wrote a book. The contribution of Khawarizmi to geography is also outstanding, in that not only did he revise Ptolemy's views on geography, but also corrected them in detail as well as his map of the world. His other contributions include original work related to clocks, sundials and astrolabes.
Several of his books were translated into Latin in the early 12th century. In fact, his book on arithmetic, Kitab al-Jam'a wal- Tafreeq bil Hisab al-Hindi, was lost in Arabic but survived in a Latin translation. His book on algebra, Al-Maqala fi Hisab-al Jabr wa-al- Muqabilah, was also translated into Latin in the 12th century, and it was this translation which introduced this new science to the West "completely unknown till then". He astronomical tables were also translated into European languages and, later, into Chinese. His geography captioned Kitab Surat-al-Ard, together with its maps, was also translated. In addition, he wrote a book on the Jewish calendar Istikhraj Tarikh al-Yahud, and two books on the astrolabe. He also wrote Kitab al-Tarikh and his book on sun-dials was captioned Kitab al-Rukhmat, but both of them have been lost.
The influence of Khawarizmi on the growth of science, in general, and mathematics, astronomy and geography in particular, is well established in history. Several of his books were readily translated into a number of other languages, and, in fact, constituted the university textbooks till the 16th century. His approach was systematic and logical, and not only did he bring together the then prevailing knowledge on various branches of science, particularly mathematics, but also enriched it through his original contribution. No doubt he has been held in high repute throughout the centuries since then.
History Of Mathematics
Every culture on earth has developed some mathematics. In some cases, this mathematics has spread from one culture to another. Now there is one predominant international mathematics, and this mathematics has quite a history. It has roots in ancient Egypt and Babylonia, then grew rapidly in ancient Greece. Mathematics written in ancient Greek was translated into Arabic. About the same time some mathematics of India was translated into Arabic. Later some of this mathematics was translated into Latin and became the mathematics of Western Europe. Over a period of several hundred years, it became the mathematics of the world.
There are other places in the world that developed significant mathematics, such as China, southern India, and Japan, and they are interesting to study, but the mathematics of the other regions have not had much influence on current international mathematics. There is, of course, much mathematics being done these and other regions, but it is not the traditional math of the regions, but international mathematics.
By far, the most significant development in mathematics was giving it firm logical foundations. This took place in ancient Greece in the centuries preceding Euclid. Logical foundations give mathematics more than just certainty-they are a tool to investigate the unknown.
By the 20th century the edge of that unknown had receded to where only a few could see. One was David Hilbert, a leading mathematician of the turn of the century. In 1900 he addressed the International Congress of Mathematicians in Paris, and described 23 important mathematical problems.
Mathematics continues to grow at a phenomenal rate. There is no end in sight, and the application of mathematics to science becomes greater all the time.
There are other places in the world that developed significant mathematics, such as China, southern India, and Japan, and they are interesting to study, but the mathematics of the other regions have not had much influence on current international mathematics. There is, of course, much mathematics being done these and other regions, but it is not the traditional math of the regions, but international mathematics.
By far, the most significant development in mathematics was giving it firm logical foundations. This took place in ancient Greece in the centuries preceding Euclid. Logical foundations give mathematics more than just certainty-they are a tool to investigate the unknown.
By the 20th century the edge of that unknown had receded to where only a few could see. One was David Hilbert, a leading mathematician of the turn of the century. In 1900 he addressed the International Congress of Mathematicians in Paris, and described 23 important mathematical problems.
Mathematics continues to grow at a phenomenal rate. There is no end in sight, and the application of mathematics to science becomes greater all the time.
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