Euler s Identity: why and how does e πi = 1? Abstract In this dissertation, I want to show how e " = 1, reviewing every element that makes this possible and explore various examples of explaining this formula, and its applications. Introduction This question has often been described as the most beautiful and important equation ever to be discovered, it ties 4 different constants into one elegant phrase: which has revolutionised mathematical thinking. Even though Euler s Identity has been proved before, I want to understand it from a very basic and historical point of view. I will reference each constant* 1 (e, π, -1 and i), and explain all concepts used (infinite series, negative numbers, complex numbers, irrational constants and limits) to explain: why does e " = 1? General history surrounding the discovery of Euler s Identity Leonard Euler was an incredible Swiss mathematician during the 18 th century. He revolutionised calculus, geometry and all of mathematics with his brilliant discoveries: as well as Euler s Identity there is Euler s constant, Euler s number, the use of sigma, functions and part of algebraic notation. He has given many contributions to modern day mathematics, and is one of the most brilliant mathematicians whose legacy lives on today. However, it is not only Euler that played a large part in discovering his Identity, DeMoivre s formulae* 2 also played a small part in inspiring Euler. Euler s work is also based upon other incredible historical figures such as Archimedes (3 rd century BC), given that Euler s Identity required the discovery of π; which the Babylonians (another culture of incredible genius from approximately 1900 BC) had partially discovered to one decimal place. Without simple notation of numbers which emerged 150,000 years ago in Congo, none of this work would have been possible at all. Without the development of numbers themselves and base 10 (first recorded in India circa 500 BC) of course, the basic principles of mathematics would never have been possible. With all the evidence above, we can conclude that there were many contributions to Euler s Identity that influenced his work. Complex and Irrational Numbers We now need to investigate Complex and Irrational Numbers as they play a large part in Euler s Identity. Firstly, Irrational Numbers are numbers that continue infinitely in a random sequence: such as π and e. There is no pattern to how they reoccur, this is because they require an Infinite Series (which I will discuss later) to be calculated. They are usually referred to as constants because they have many uses within Geometry and Economics.
There are in fact an infinite number of Irrational Constants: the square root of a number that isn t already the power of a whole number will be an Irrational Number this is also true if you root it by any other power. For example, 2 and 3 will have an irrational square root, whereas a number like 4 has a square root of 2. They are used all around us, in fact on all A4 paper, the length is exactly 2 times the width. Complex numbers are far more difficult to comprehend. To begin with you must understand that from a basic view of mathematics it was thought that it was not possible to find the square root of a negative number* 3. However, this was found not to be the case and a whole new realm of numbers was conceived: the complex realm. The square root of - 1 is i. This is the unit for complex numbers. The best explanation is to imagine you have a graph of x 2, and at one point y=0. This is the case with all quadratic equations. But what about the graph of x 2 +1? At no point in the graph does y=0. Except if you say x=i (i.e. the square root of -1); then it will equal 0. Irrational and complex numbers will become very useful for working out Euler s Identity as we will go on to find out. π We need briefly discuss Pi as it occurs in Euler s Identity. This is possibly one of the most important and interesting numbers to be discovered, it is roughly equal to 3.141592653. But what is its purpose? Pi is one of the most fundamental constants in geometry, as it is used to work out the area (πr 2 ), perimeter (2πr) and volume ( πr3 ) of spheres and circles. Pi is necessary in all these processes, but it also has applications in trigonometry* 4. The practical applications of Pi influence engineering, architecture and observational science. e Euler s number is the basis of Euler s Identity. e is fundamental in calculus and is used in economics. It is also discovered through an infinite series (which I will reference when I explain why e " = 1). Euler s number roughly equals 2.718281 The most important thing about e, is that if you graph e x, at all points, the gradient* 5 and area, from - until x, are equal. I will explain more about e at the end.
Limiting in calculus What is the value of e x as x approaches -? That is what limiting is used for. In the graph of e x, it shows the curve infinitely approaching 0, but does it ever reach 0? The premise behind limiting is we assume 0.99999... =1, the same goes with sequences: a sequence could infinitely approach a value, but it would be illogical to work out the value for an eternity, so instead you work out the limit of a sequence as it approaches a number. For instance, why does 2 0 =1. If you plot 2 x on a graph, you will notice it just passes over 1, so we can assume it does equal 1: this is the reasoning behind limiting. Infinite series As previously mentioned, infinite series play a large role in calculating irrational numbers, and also in answering why does e " = 1? An infinite series is a calculation that continues infinitely, i.e. + +... (the exclamation meaning all integers from 0-n multiplied together, e.g. 5=1*2*3*4*5) when continued forever and limited (this is another use for limits, you limit an infinite series to get a constant answer ) you can get certain constants. For instance, the sequence I just showed was in fact the same as e x, this will become useful in the final segment. When writing out an infinite series, you usually use Σ (sigma). This is written as: 4( 1 ), which is the infinite series for π. So, with the sequence for working out e x, you would write it out as: This method simplifies most infinite series, but what does it actually mean? Where it says n=0 below the sigma, and above, those are the parameters of the sequence between which we need to sum, using as the sequence itself. So, with this, all we are doing is following a sequence, then adding the resulting values of the sequence together. x n Euler s identity Now that the foundations for the question have been laid, what is needed is to work out the value of e " The first step in working this out is to replace as x (in e " ) which gives us e ". But how does this help? As previously mentioned, Euler s explanation of this may have been based on DeMoirve s formulae, as both are very similar, but to work out Euler s explanation, you first should go through the process of simplifying the infinite series within the sequence. As we already know the infinite series for e x, we now have a replacement for x (in the series): ix, so we have a new infinite series:
ix n Which then gives: i x n To show this as an infinite series, and further simplified this would be: i x 1 x x i 2 3 + x 4 To explain this, if you work out the sequence i x, it would be i, -1, i, 1 So, the sequence will alternate between adding and subtracting, giving us the above infinite series. And so now we can finally give a value for x: π. Just as predicted, if you carry this formulae on infinitely, and give the value of pi you get, -1. This solves the equation. However, to represent this in another way, we can use trigonometry, by grouping the infinite series like this: i x 1 x 3 + x 5 + x 2 + x 4 x 6 The two groups become the same as i sin π + cos π. Meaning we now have a new formulae* 6 : e " = i sin x + cos x Which also gives us 0i+-1, which also values at -1. Application and further research Euler s Identity was revolutionary for mathematics and changed the general opinion on how modern day problems are approached. However, in itself, it only had one use. This was the relationship between sine and cosine shown in the formulae above. For further research I will look into the resonance which lets this formula works: how three of the most important constants came together to form Euler s Identity. Conclusion Euler s identity is a beautiful formula, that revolutionised mathematical thinking, it is made up of many elegant parts that come together to make an even more elegant formula. To understand it is an enormous step to incredible concepts like the Taylor series; and helps understand even simpler parts of mathematics.
* 1 a constant is any number that does not change, 1 is a constant however the speed of an object is not, as it could change over time, therefore it is not just a singular unit. * 2 (cos x + i sin x) = cos xn + i sin xn Euler s formulae (which was used to explain his Identity) could possibly be derived from this. * 3 This is because (for instance) -1*-1=1, the only way to make a negative number through rooting is using an odd power: 1 is -1. * 4 In fact it is mainly used in trigonometry to explain Euler s Identity, I will reference this in the last segment. * 5 Gradient is the steepness of a line, it is used a lot in calculus. * 6 You cannot jump straight to this conclusion, you must use the Taylor series, but it is far too complex to explain here. Bibliography Unknown. 2007. How Euler did it. [ONLINE] Available at: http://eulerarchive.maa.org/hedi/hedi-2007-08.pdf [accessed 04.03.17] Spencer, P. 1999. Why is e^(pi i) = -1? [ONLINE] Available at: http://www.math.toronto.edu/mathnet/questioncorner/epii.html [accessed 04.03.17] Unknown. N/A. Trigonometry-higher. [ONLINE] Available at: http://www.bbc.co.uk/schools/gcsebitesize/maths/geometry/trigonometryrev1.shtml [accessed 04.03.17]