The Most Beautiful Equation in the World?

This is the most beautiful equation in mathematics:

eπi + 1 = 0


The Most Beautiful Equation in mathematics

Does it look beautiful to you?  Mathematical beauty is not necessarily aesthetic beauty; a beautiful piece of maths does not need to look beautiful (although it may be possible to represent the mathematics in a visual way which is beautiful; the Mandelbrot set is a good example of this – see http://mathworld.wolfram.com/MandelbrotSet.html.  To begin to understand mathematical beauty you must understand the mathematics; so let’s analyse the equation eπi + 1 = 0.

This equation contains five terms – e, π, i, 1 and 0 – and the first thing to appreciate is that these five terms are all just numbers.  1 and 0 are well-known to everyone; they are the fundamental numbers of arithmetic from which all other integers can be derived (all other positive integers can be made up from 1 by addition – for example 7 = 1 + 1 + 1+ 1 + 1 + 1 + 1; whilst 0 – 1 = -1, and from -1 we can generate all the negative integers too).  Thus, in many ways, 1 and 0 are the most important integers; but what about the other numbers in this equation; e, π and i?

e is a number which is approximately equal to 2.718 (in actual fact e = 2.71828182845904523…; the dots signify that this expression continues forever, the decimal terms never stop).  It is an example of an irrational number – which means that it cannot be written as a fraction.  Now there are lots of irrational numbers (in fact most numbers are irrational) however e is an especially important one; but before I explain why e is so important I have to confess that there are several ways in which to explain why it is special – and mathematicians will often disagree about what is the best way to first introduce e and which of its properties make it truly special.  But since this is my blog I shall ignore other viewpoints and simply explain why e is important in the way which I think is best!  (To learn more about e, please see http://mathworld.wolfram.com/e.html)

To understand e we must first consider the function y = ex (to see what this curve looks like, please go to http://mathworld.wolfram.com/ExponentialFunction.html).  Now this function has a special property; the gradient of the curve at every point is equal to the height of the curve at that point; and this is the only curve which has this property and this is why e is so special!

“…π is fundamental to our understanding of much of the mathematics that underpins modern science and technology!”

Next we come to π – a number which everyone has probably heard of as it’s a number we are all taught at school; π is the number which appears when we calculate the area or circumference of a circle (the area of a circle is given by the formula πr2 where r is the radius of the circle, and the circumference of a circle – the circumference of a circle is the distance around the circle – is given by the formula 2 πr).  But that’s only the first occurrence of π in mathematics; indeed π keeps appearing in all manner of different places in maths and is fundamental to our understanding of much of the mathematics that underpins modern science and technology!

And then there’s i, the most mysterious of these numbers.  i = √-1; it is the imaginary number.

Now in just one blog entry I cannot fully explain why these numbers – e, π, i, 1 and 0 – are so important and instead I just want to give you a flavour of what it is that makes these numbers so special; you’ll just have to trust me on this for now, but I will write subsequent entries that explain the special properties of these numbers in more detail.  For now though, if you can accept that these five numbers – e, π, i, 1 and 0 – are the most special numbers in maths, then the fact that there is just one equation that links these five numbers, is truly wonderful.  But there’s more to this equation that just that!

Let’s take a look at the equation again: eπi + 1 = 0.  When I said this equation was made of five terms that was only partially true; although there are five numbers in this equation, there is more to it than just that – there are also the operations of multiplication, addition and raising to a power that are used to combine these expressions.  To put it another way, this equation also involves addition – we have to add the term eπi to the 1; and we have multiplication – we have π multiplied by i; and we have a power – e is raised to the power πi.  So not only does this equation contain the five most important numbers in mathematics, it combines them using the three most fundamental (and simplest) mathematical operations.

“…it’s a bit like asking Rembrandt, Picasso, Dali, Turner and Munch to each paint part of a picture…”

And this is remarkable; it’s a bit like asking Rembrandt, Picasso, Dali, Turner and Munch to each paint part of a picture, in any style they might choose, and finding that they all painted something in a different style but yet finding that the different parts of the picture all fitted together to produce something aesthetically beautiful; or like asking Hendrix, Elvis, Eminem, Mozart and Usher to each write and perform a piece of music and then play all five pieces of music simultaneously and finding that all the different parts harmonise and work as one piece.  And that’s what’s so remarkable about this equation; the five numbers involved – e, π, i, 1 and 0 – just shouldn’t fit together in such a simple way, yet they do.  And that’s what makes this equation so beautiful.

Blog by Dr John McDarbyA Level Maths teacher at Bellerbys College London

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