Removing the mysteries of e

Introduction

In all the explanations that I have seen about e^ix, there is always an unexplained gap between the meaning of the exponential e^x and the meaning of the Complex exponential e^ix. There is never any sensible reason given as to why an exponential curve should end up as a circle on the Complex plane. In most school education, we are just told to accept it all without question. It is hard to visualise why e^ix does what it does.

In this explanation, I will try to remove the obscurity of the meaning of e^ix and its related formulas by demonstrating how it works.

To get the most from the following explanation, it will help if you have at least a trivial understanding of Complex numbers and exponentials, and it might help if you have some experience of e^ix. A good test for whether you will get what I am trying to say or not is whether you understand how Sine and Cosine indicate the y-axis and x-axis values of points on the circumference of a unit-radius circle at particular angles from the centre. It is possible to calculate the Sine and Cosine of a number by drawing and measuring a circle. If you understand this, then this explanation should be straightforward. If you do not understand this, then you might benefit from my book about waves.

Notes

• Given that "x" in e^ix is actually an angle, in this explanation, I will write the exponential as e^iθ to be consistent with using θ for angles. For example, instead of e^ix = cos x + i sin x, I will write e^iθ = cos θ + i sin θ.
• The explanation in the PDF is meant to be easy to understand. Therefore, there are things that will be obvious to people who are good at maths, and also things that might be obvious to everyone. It might seem overly repetitive at times.
• I frequently include unnecessary ones and zeroes in Complex numbers in order to make the explanation clearer. This is to reinforce the idea that we are working in the Complex plane, and to benefit people who aren't as mathematical. I'm aware that this makes some people flinch.
• I tend to capitalise certain words to make ideas clearer and less ambiguous. For example, I refer to "Complex" numbers rather than "complex" numbers.

Summary

The main part of the PDF is a simple exploration of the following important ideas. (If these seem complicated, they are explained far more simply and in far more depth in the PDF.)

• It is possible to identify any point on a unit-radius circle by saying by how much the point at 1 + 0i would need to be rotated to get there.


• It is possible to rotate a point on the Complex plane by 90 degrees (0.5π radians) by multiplying the point by "i".


• It is possible to rotate a point on the Complex plane by a particular angle in an angle system that divides a circle into four pieces, by multiplying the point by a power of "i" where the exponent is that angle. In other words, if we multiply a point on the Complex plane by i^θ where θ is an angle in an angle system based on a circle being divided into four pieces, that point will be rotated by that angle. We will refer to angles in a system where a circle is divided into four pieces as "quarter-circle angle units".


• We can identify a point on a unit-radius circle by saying by how much the point at 1 + 0i would need to be multiplied by i^θ to get there. That point will be 1 unit away from the origin and at an angle of θ quarter-circle angle units.


• Therefore, we can identify any point on a unit-radius circle solely in terms of i^θ.


• Therefore, we can say that any point indicated by i^θ can also be indicated by "cos θ + i sin θ", where Cosine and Sine are working in quarter-circle angle units. As the point indicated by i^θ is one unit away from the origin and at an angle of θ quarter-circle angle units, then it must be the case that it is equal to "cos θ + i sin θ", which identifies the same point when Cosine and Sine are working in quarter-circle angle units.


• We can identify any point on the Complex plane by saying by how much the point at "1 + 0i" must be rotated and scaled to get there in terms of multiples of i^θ. This means that ai^θ identifies any point on the Complex plane, where "a" is the distance from the origin, and θ is an angle in quarter-circle angle units.


• We can rephrase i^θ so that θ can be a value in degrees or radians or any other way of dividing up a circle. For example, i^(θ/90) works with θ as an angle in degrees. If we enter an angle in degrees into i^(θ/90), the result will be a point on a unit radius circle at an angle of θ degrees. Therefore, the exponential i^(θ/90) must be equal to "cos θ + i sin θ" when θ is an angle in degrees, and Cosine and Sine are working in degrees. The exponential i^(2θ/π) works with θ in radians. If we enter an angle in radians, it will identify a point on a unit radius circle at angle of θ radians. Given that, i^(2θ/π) must be equal to "cos θ + i sin θ" when θ is an angle in radians, and Cosine and Sine are working in radians.


• We can rephrase the different versions of the exponentials to be a Real base raised to an Imaginary power, and keep the meanings exactly the same. For example, works in degrees and works in radians. We might not be able to solve these with the knowledge we have so far in this explanation, but we can prove that they are correct. It is still the case that these are equal to "cos θ + i sin θ", where θ, Cosine and Sine are working in that particular way of dividing up a circle.


• From knowing that e^iθ also identifies a point on a unit radius circle at an angle of θ radians, we know that must be equal to "e". From that, we have the step from Real powers of "i" to Imaginary powers of "e".


• Therefore, we can think of e^iθ as really being a way of identifying the position of a point on a unit-radius circle in terms of how much the point at "1 + 0i" would need to be rotated to get there. The amount of rotation is given by a multiplication by e^iθ.


• We can use what we have learnt to calculate other bases to Imaginary powers. For example, 1.01761^iθ identifies a point on a unit-radius circle at an angle of θ degrees. Therefore, 1.01761^iθ must be equal to "cos θ + i sin θ", where Cosine and Sine are working in degrees.


• We can use what we have learnt to solve seemingly difficult calculations such as the i^th root of "i".

All of the above ideas mean that the properties of e^iθ are not mysterious or complicated^*. They are also not unique. The equivalence of e^iθ to "cos θ + i sin θ" is something that should be expected, and not a surprising revelation. (* OK, they might seem complicated here, but my explanation is simpler in the PDF, and it is much simpler in my book about waves, but you will need to do more reading there.

The PDF makes the above much easier to understand, and starts with the basics of what angles are. It goes into more depth. Even if the ideas in the PDF are not particularly profound, they will help some people better understand some aspects of maths.

The number 4.3045... is a sort of landmark in all of this because it connects different aspects of "i" and "e". For the sole reason that search engines often don't find this page when someone searches for 4.304530... here is the number to various levels of decimal points: