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The beauty of Euler’s Number and Euler’s identity“Who has not been amazed to learn that the function y=ex, like a phoenix rising again from its own ashes, is its own derivative?” – Francois Le LionnaisLeonhard Euler, a Swiss mathematician, engineer and physicist of the mid eighteenth century, is arguably one of the most prolific modern mathematicians. His mathematical discoveries and findings that he provided to the subject are some of the most widely used to this day, most notably: the introduction of the concept of a function, popularizing the notation of fx; the use of many mathematical letters such as π for the ratio of a circle’s diameter to its circumference, as well as the imaginary number i and finally, but most importantly, the invention of Euler’s Number e. As stated by Robin Wilson, "Most of modern mathematics and physics derives from work of Leonhard Euler" outlining the importance of his work to the subject. e is defined as being approximately equal to 2.71828…, or as limn→∞1+1nn, however, this does not give it nearly the credit it deserves, with its derivation and uses being much deeper than just an irrational, transcendental number on a page.One of Euler’s numbers’ most prolific uses and where it is seen very often is in the equation eiπ=-1. This is known as Euler’s identity and is commonly known as the most beautiful equation in mathematics, as a seemingly complex equation tying together two irrational numbers as well as an imaginary number to give the result so simple as negative one, its beauty is hard to overlook. “Like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than just skin deep, Euler's equation reaches down into the very depths of existence.” - Keith DevlinThe derivation and proof of Euler’s identity accentuates the sheer wonder of this equation, once again tying together many different areas of mathematics. One area of mathematics that can be used to explore this is group theory and how Euler’s identity relates to it. Group theory involves studying the nature of symmetry, where a shape requires some property to remain unchanged or invariant. This, therefore, is seemingly unrelated to the identity, however, the identity again shows the breadth of subjects its covers. Moreover, another way to explore the identity is through calculus and taking the derivative of exponential functions. Finally, the last way in which Euler’s identity can be proved that I will be exploring is through Maclaurin series and trigonometric interpretations.Group Theory approachAs stated earlier, group theory is the study of the nature of symmetry. However, this is not just simply about the fact that a square has four lines of symmetry or a rectangle has 2, but rather exploring what actions that can be put on a square, rectangle or circle that leave it looking indistinguishable from its original. For example, consider the actions that can be put on a square that will leave it looking the same as the original shape, you can rotate it 90° clockwise, 180° or flip it along its horizontal line of symmetry. These are just a few examples of the different possible actions that will leave the shape unchanged. Each one of these actions is called a symmetry of the square and all of the different symmetries make up a group. For the square, it consists of 8 symmetries – a dihedral group of order 8. Secondly, consider the rotations on a circle and the possibilities that can arise when working with a shape like a circle rather than a square. Every possible rotation lies between 0 and 2π radians, and hence gives an infinite number of possible rotations on the circle. This can be hard to visualise and hence can be helpful to pick an arbitrary point on the circle and see what happens to it when an action is performed on the circle. For example, the new projected point may be half a circumference away from the original point and hence the circle has been rotated by π radians. This outlines how group theory is not just what a group of symmetries is, but how the symmetries interact with each other. On the circle, a 270° clockwise rotation, followed by a 120° clockwise rotation will result in the same image as a single 30° clockwise rotation. Moreover, for the square, if you first rotate the square 90° anticlockwise and follow it with a flip along the vertical axis, you get the same result as if you were to have just flipped along the diagonal axis. These interactions of symmetries and the results that you get out of them is what really defines group theory.The same idea can be applied to the number line, as transformations of the number line can correspond to the values on it. For example, shifting the number line 5 units to the left will result in the number 5 now taking up the space where originally the origin, or point 0, would have been. As with the square or circle, this is a symmetry as the number line has not changed shape and the image of it is still identical to the original. This group of sliding actions, each of which is associated with a unique real number is called the additive group of real numbers, as just as the interactions of symmetries with the square and circle resulted in different symmetries, the same applies with the number line. Considering a slide of 3 units to the left, and then a slide of 7 units to the right, the resultant symmetry is the same as a 4-unit slide to the left, outlining the possibility to add together translations of the number line.This idea can then be extended into the complex plane, where instead of shifting the line of numbers keeping 0 as the reference point, you can shift the whole coordinate plane whilst keeping the coordinate (0,0) as a reference point. A horizontal slide will refer to the real numbers as before with the real number line, whereas a vertical slide of the plane will refer to the imaginary numbers. Hence, for example, the point 2+3i comes from a shift 2 units to the right and then followed by a shift 3 units vertically up. The same interaction of symmetries can be continued into the imaginary plane and further shown with the interaction of two of these complex numbers, such as 2+3i and 1-3i, giving an overall shift 1 unit to the right. This group is called the additive group of complex numbers, referring to the successive shifts in both axes. As well as shifting, a number line or complex plan can also stretch onto new values, giving the multiplicative group of real numbers or complex numbers. For example, by using the point 1 as a reference point and keeping 0 fixed, a stretch of factor 2 will bring it to the point 2, or a stretch of factor 3 will bring it to 3, with each stretch referring to only one real number on the number line, and the reverse can be said that all real numbers correspond to a stretch of the number line. The same interaction rules also apply to the multiplicative group, as a stretch of factor 2 followed by a stretch factor 3 corresponds to the same point as a stretch factor 6.Taking this to the complex plane, when stretch the point (1,0) to a new point, it is only possible if a rotation is also applied to the plane, and hence the transformation has both a scale factor and a rotation angle to consider. Consider the point 1i on the imaginary axis, to get the point (1,0) onto that point requires a rotation of π/2, and hence the multiplicative action that refers to 1i is a π/2 rotation, and with the same interactive rules, by multiplying by i again, it makes the rotation into a 180° rotation and hence brings (1,0) to (-1,0), reflected by the fact that i2=-1. Each complex number therefore can be constructed from a stretch along the real axis followed by a rotation on the unit circle. This is the multiplicative group of complex numbers. Now we can consider exponentials and apply them to this group theory, for example take 2x+y=2x2y which comes from a common indices rule. Now we can consider how this will affect the groups, as the inputs being added correspond to outputs being multiplies, which can be thought of on two number lines, one for the adding components and one for the multiplying components. In order to see how the inputs, affect the outputs, it is useful to see how the shift on the input axis will cause a stretch on the output axis. For example, a shift of 2 units in the input followed by a shift of -3 units will cause a stretch of factor 22 followed by a stretch factor 2-3 on the output axis. This can be considered as preserving its group structure, as the structure of the group is defined by the arithmetic behind it. Functions that, between groups, preserve their structure and arithmetic in this way are called homomorphisms, and are highly useful in group theory.As previously shown, we can extrapolate the ideas from the number line onto our complex plane and therefore the ideas with exponentials can also apply. In the additive group, the 2 components are shifts in the horizontal, real direction as well as shifts in the vertical, complex direction, whereas in the multiplicative group there are rotations along the unit circle and stretches along the real axis. Therefore, one of the shifts in the additive group will cause a stretch in the multiplicative group and one of the shifts will cause the rotations. As a real input into 2x always gives a real output, we can deduce that the shift in the real axis causes a stretch in the real axis, and therefore the shift in the vertical direction will cause the rotations along the unit circle. From this, we can then see what distance around the unit circle the shift in the imaginary axis on the additive group corresponds to. For 2x, the distance comes out as 0.693 radians along the unit circle, whereas for 5x, it comes out as 1.609 radians. The most interesting part is that for ex, a 1i shift corresponds to a 1 radian distance along the unit circle, meaning that a πi vertical shift will cause a full 180°/π rotation and will map the point to -1 and hence, eiπ=-1. Calculus approach By considering the derivative of fx=ex, you can therefore analyse the gradient of the graph at any point. However, in order to make this more useful, let’s consider ft=et, with t being the time elapsed. As ddtet=et, the velocity at any point is equal to the position at that time, or more generally ddtekt=k*ekt, hence the velocity is equal to k times the position. Now, consider how this can be represented if the exponent was i, giving eit. The position is always eit, and hence the derivative/velocity will be ddteit=i*eit. As seen previously, this outlines how the velocity is going to be a 90° rotation of the position vector, as a multiplication of i results in this transformation on the complex plane. Starting at t=0, eit=1 which gives the starting point of 1+0i on the complex plane. If you plot the vector field that this corresponds to, there is only one trajectory starting from 1,0 where the velocity is always matching the vector that it is passing through. This, if traced, will give the unit circle, which goes through the point (-1,0). In order to get to a point of (-1,0), a rotation of π is required and therefore at t=π, eiπ=-1. By considering the derivatives of exponential functions, you are able to derive the key fact that the multiplication of i will cause a rotation of 90° and apply that to the relation between the position and velocity vectors in order to map the movement of the function ft=eit in order to prove Euler’s identity.Trigonometric approachAny function, f(x) can be written and represented as its Maclaurin Series, defined as n=0∞fn0xnn! where fn is the nth derivative of the function. For example, fx=sinx, can be written as fx=x-x33!+x55!-x77!… or more generally sinx=n=0∞-1nx2n+12n+1!. Furthermore, cosx= n=1∞-1nx2n2n!. The easy part about finding the Maclaurin series for fx=ex is that every order derivative (fnx) is equal to ex, and hence fn0 will always equal 1 and hence simplifying the Maclaurin series to ex=1+x1!+x22!+x33!+x44!… or n=0∞xnn!. There are clear similarities between the Maclaurin series of sin, cos and ex, hence why ex can also be written in terms of cos and sin. If we replace x with ix, we can write the series as 1+ix+ix22!+ix33!…, and therefore, by considering the fact that i2=-1, i3=-i, i4=1, the series can therefore become 1+ix-x22!-ix33!+x44!+ix55!…. By taking out a factor of i, we can separate half the terms and write it as 1-x22!+x44!-x66!+x88!…+ix-x33!+x55!-x77!+x99!Therefore, by substituting in the Maclaurin series for sin and cos: eix=cosx+isin xFinally, by substituting in π for x, given that sinπ=0 and cosπ=-1: eiπ=-1Another way that this can be seen is with the unit circle. As previously seen, the function fx=eix represents the track of the unit circle on the complex plane. As with the nature of the unit circle, the horizontal component of the vector is cosx and then the vertical component of the vector is isin x and hence the output is cosx+isin x given the rules of adding complex numbers. ................
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