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It isn't often that a mathematical equation makes the national press, far less popular radio, or most astonishingly of all, is the subject of a debate in the UK parliament. However, in 2003 the good old quadratic equation, which we all learned about in school, was all of those things.
Where we begin
Where would it all end? Was the quadratic equation really dead? Did anyone care? Are mathematicians really evil monsters who only want to inflict quadratic equations on a younger generation as a means of corrupting their immortal souls?
Maybe so, but it's not really the quadratic equation's fault. In fact, the quadratic equation has played a pivotal part in not only the whole of human civilisation as we know it, but in the possible detection of other alien civilisations and even such vital modern activities as watching satellite television. What else, apart from the nature of divine revelation, could be considered to have had such an impact on life as we know it? Indeed, in a very real sense, quadratic equations can save your life.
Babylonian cuneiform tablets recording the 9 times tables
Let's suppose that you are a Babylonian farmer. Somewhere on your farm you have a square field on which you grow some crop. What amount of your crop can you grow on the field? Double the length of each side of the field and you find that you can grow four times as much of the crop as before. The reason for this is that the amount of the crop that you can grow is proportional to the area of the field, which is in turn proportional to the square of the length of the side. In mathematical terms, if is the length of the side of the field, is the amount of crop you can grow on a square field of sidelength 1, and is the amount of crop that you can grow, then
Finding square roots by using a calculator is easy for us, but was more of a problem for the Babylonians. In fact they developed a method of successive approximation to the answer which is identical to the algorithm (called the Newton-Raphson method) used by modern computers to solve much harder problems than quadratic equations.
For appropriate values of and the amount of crop that the farmer can grow in this field is given by
This looks a lot more like the quadratic equation that we are used to, and even under the evil eye of the tax man, it’s a lot harder to solve. Yet the Babylonians came up with the answer again. First we divide by to give
Now we complete the square by using the fact that
Combining this with the original equation we have
This is now an equation that we can solve by taking square roots. The result is the famous " formula":
which can be rewritten as
(The formula usually has "-4ac" because the quadratic equation is more usually written in the form "".)
The fact that taking a square root can give a positive or a negative answer leads to the remarkable result that a quadratic equation has two solutions. So much for mathematical puzzles only having one solution!
Now, this is where the teaching of quadratic equations often stops. We have reached that object beloved of all journalists when they interview mathematicians - a formula. Endless questions can be made up which involve putting values of and into the formula to give (two) answers. But this isn’t what mathematics is about at all. Finding a formula is only the first step on a long road. We have to ask, what does the formula mean; what does it tell us about the universe; does having a formula really matter? Let’s now see where this formula will take us.
A surprise for the Greeks, a bit of mathematical origami and a sense of proportionWe now fastforward 1000 years to the Ancient Greeks and see what they made of quadratic equations. The Greeks were superb mathematicians and discovered much of the mathematics we still use today. One of the equations they were interested in solving was the (simple) quadratic equation
They knew that this equation had a solution. In fact it is the length of the hypotenuse of a right angled triangle which had sides of length one.
It follows from Pythagoras’ theorem that if a right-angled triangle has shorter sides and and hypotenuse then
Putting and then . Thus
So, what is in this case? Or, to ask the question that the Greeks asked, what sort of number is it? The reason that this mattered lay in the Greek’s sense of proportion. They believed that all numbers were in proportion with each other. To be precise, this meant that all numbers were fractions of the form where and are whole numbers. Numbers like 1/2, 3/4 and 355/113 are all examples of fractions. It was natural to expect that was also a fraction. The huge surprise was that it isn’t. In fact
where the dots mean that the decimal expansion of continues to infinity without any discernible pattern. (We will meet this situation again later when we learn about chaos.)
was the first irrational number (that is, a number which is not a fraction, or rational), to be recognised as such. Other examples include ,, and in fact "most" numbers. It took until the 19th century before we had a good way of thinking about these numbers. The discovery that was not a rational number caused both great excitement (100 oxen were sacrificed as a result) and great shock, with the discoverer having to commit suicide. (Let this be an awful warning to the mathematically keen!) At this point the Greeks gave up algebra and turned to geometry.
Far from being an obscure number, we meet regularly: whenever we use a piece of A4 paper. In Europe, paper sizes are measured in A sizes, with A0 being the largest with an area of . The A sizes have a special relationship between them. If we now do a bit of origami, taking a sheet of A1 paper and then folding it in half (along its longest side), we get A2 paper. Folding it in half again gives A3, and again gives A4 etc. However, the paper is designed so that the proportions of each of the A sizes is the same - that is, each piece of paper has the same shape.
The proportions of the first piece of paper are and those of the second are or . We want these two proportions to be equal. This means that
Paper used in the United States, called foolscap, has a different proportion. To see why, we return to the Greeks and another quadratic equation. Having caused such grief, the quadratic equation redeems itself in the search for the perfect proportions: a search that continues today in the design of film sets, and can be seen in many aspects of nature.
Let’s start with a rectangle, and then remove a square from it with the same side length as the shortest side of the rectangle. If the longest side of the rectangle has length 1 and the shortest side has length , then the square has sides of length . Removing it from the rectangle gives a smaller rectangle with longest side and smallest side . So far, so abstract. However, the Greeks believed that the rectangle which had the most aesthetic proportions (the so called Golden Rectangle) was that for which the large and the small rectangles constructed above have the same proportions. For this to be possible we must have
This is yet another quadratic equation: a very important one that comes up in all sort of applications. It has the (positive) solution
The number is called the golden ratio and is often denoted by the Greek letter .
Sunflower seeds, arranged using Fibonacci numbers
The Parthenon, embodying the Golden Ratio
In this sequence each term is the sum of the previous two terms. Fibonacci discovered it in the 15th century in an attempt to predict the future population of rabbits. If you take the ratio of each term to the one after it, you get the sequence of numbers
and these numbers get closer and closer to (you guessed it) the Golden Ratio .
By finding both of the roots of the above quadratic equation we can actually find a formula for the nth term in the Fibonacci sequence. If is the th such number with and then is given by the formula
Conics link quadratic equations to the stars
Half of the cone can be visualised as the spread of light coming from a torch. Now, if you shine a torch onto a flat surface such as a wall then you will see various shapes as you move the torch around. These shapes are called conic sections and are the curves that you obtain if you take a slice through a cone at various different angles. Precisely these curves were studied by the Greeks, and they recognised that there were basically four types of conic section. If you take a horizontal section through the cone then you get a circle. A section at a small angle to the horizontal gives you an ellipse. If you take a vertical section then you get a hyperbola and if you take a section parallel to one side of the cone then you get a parabola. These curves are illustrated below.
A cross-section of a cone can be a circle ...
... an ellipse ...
... a parabola ...
... or a hyperbola.
Conic sections come into our story because each of them is described by a quadratic equation. In particular, if represents a point on each curve, then a quadratic equation links and . We have:
The circle: ;
The ellipse: ;
The hyperbola: ;
These curves were known and studied since the Greeks, but apart from the circle they did not seem to have any practical application. However, as we shall see in the next issue of Plus, a link between quadratic equations and conics, coupled with a mighty lucky fluke, led to an understanding of the way that the universe worked, and in the 16th century the time came for conics to change the world.
About the authors
Chris Budd is Professor of Applied Mathematics in the Department of Mathematical Sciences at the University of Bath, and the Chair of Mathematics at the Royal Institution in London.
Chris Sangwin is a member of staff in the School of Mathematics and Statistics at the University of Birmingham. He is a Research Fellow in the Learning and Teaching Support Network centre for Mathematics, Statistics, and Operational Research.
They have recently written the popular mathematics book Mathematics Galore!, published by Oxford University Press.
This article was inspired in part by a remarkable debate in the British House of Commons on the subject of quadratic equations. The record of this debate can be found in Hansard, United Kingdom House of Commons, 26 June 2003, Columns 1259-1269, 2003, which is available online at the House of Commons Hansard Debate website.