In post 18.15, we saw that the number *e* raised to the power *x* (see post 18.2) is defined by the series (post 17.11)

* e ^{x}* = 1 +

*x*+

*x*

^{2}/2! +

*x*

^{3}/3! +

*x*

^{4}/4! +

*x*

^{5}/5! + …

Here, for example, 5! = 5 × 4 × 3 × 2 × 1.

In post 18.6, we saw that the sine and cosine of the angle *ϑ*, measured in radians (post 17.11) could also be expressed as the series

sin*ϑ = ϑ* – *ϑ*^{3}/3! + *ϑ*^{5}/5! – *ϑ*^{7}/7! + *ϑ*^{9}/9! -…

cos*ϑ* = 1 – *ϑ*^{2}/2! + *ϑ*^{4}/4! – *ϑ*^{6}/6! + *ϑ*^{8}/8! – *ϑ*^{10}/10! +…

Now let’s think about the number *e ^{i}*

*, where*

^{ϑ}*i*represents the square root of -1, as discussed in post 18.16; this means that, by definition,

*i*

^{2}= -1. It follows, from equation 1, that

* e ^{i}*

*= 1 +*

^{ϑ}*i*

*ϑ*+ (

*i*

*ϑ*)

^{2}/2! + (

*i*

*ϑ*)

^{3}/3! + (

*i*

*ϑ*)

^{4}/4! + (

*i*

*ϑ*)

^{5}/5! + (

*i*

*ϑ*)

^{6}/6! + (

*i*

*ϑ*)

^{7}/7! + (

*i*

*ϑ*)

^{8}/8! …

= 1 + *i**ϑ* – *ϑ*^{2}/2! – *i**ϑ*^{3}/3! + *ϑ*^{4}/4! + *i**ϑ*^{5}/5! – *ϑ*^{6}/6! – *i**ϑ*^{7}/7! + *ϑ*^{8}/8! …

= (1 – *ϑ*^{2}/2! + *ϑ*^{4}/4! – *ϑ*^{6}/6! + *ϑ*^{8}/8! …) + (*i**ϑ* – *i**ϑ*^{3}/3! + *i**ϑ*^{5}/5! – *i**ϑ*^{7}/7! + …)

= (1 – *ϑ*^{2}/2! + *ϑ*^{4}/4! – *ϑ*^{6}/6! + *ϑ*^{8}/8! …) + *i*(*ϑ* – *ϑ*^{3}/3! + *ϑ*^{5}/5! – *ϑ*^{7}/7! + …)

= cos*ϑ* + *i*sin*ϑ.*

This result is called “Euler’s relation”, named after the Swiss mathematician Leonhard Euler (his last name is pronounced like the English word “oiler” https://forvo.com/word/leonhard_euler/#de) who lived from 1707 to 1783.

According to post 18.16, we can represent cos*ϑ* + *i*sin*ϑ* and, therefore, *e ^{i}*

*, in two dimensions by the diagram below.*

^{ϑ}

Its modulus (post 18.16) is then given by cos^{2}*ϑ* + sin^{2}*ϑ* = 1 (see appendix 4 of post 16.50) and its argument (post 18.16) is *ϑ*. As *ϑ* increases, the point P moves round in a circle of radius 1. So the complex number (post 18.16) *ae ^{i}*

*represents the position of a point P that moves around a circle of radius*

^{ϑ}*a*as

*ϑ*increases.

The picture below is copied from post 18.11; it represents a point, initially at P moving around a circle, of radius *a*, with an angular speed (post 17.12) *ω*.

After a time *t* has elapsed, it reaches P’. From the previous paragraph, we can see that, at any given time *t*, we can represent the position of P’ by *a**e** ^{iωt}*. Sometimes this is written

*a*exp(

*iωt*) to avoid writing so much information in the superscript

*iωt*. This method of representing the position of an object going round in a circle is an alternative to the trigonometric representation given in post 18.11.

We also saw, in post 18.11, that the projections of OP’ can represent a simple harmonic oscillator or a wave. The projection on OX gives the cosine form of the time-dependence of the position of the oscillator or amplitude of the wave; the projection on OY gives the sine form. So, the expression

* x* = *a**e*^{iωt}

could be used to specify the position of a simple harmonic oscillator. Also, the expression

* ψ* = *a**e*^{iωt}

could be used to represent a wave. These two equations contain both the sine and cosine forms of post 18.11, in a single equation for a simple harmonic oscillator and a single equation for a wave.

In post 18.11, we saw that the sine and cosine forms were solutions of a differential equation. If you think about it, you will see that if *a*cos(*ωt*) and *a*sin(*ωt*) are solutions of the differential equation then *n**a*cos(*ωt*) + *m**a*sin(*ωt*), where *m* and *n* are any numbers, must be a solution too. Putting *n* = 1 and *m* = *i*, we can see why *a**e** ^{iωt}* is a solution too.

The picture below is also copied from post 18.11.

In this picture, there is a phase difference, *ϑ*, between Q and Q’. So our general expression for the position of an object going round in a circle, a simple harmonic oscillator or a wave has the form *a*exp*i*(*ωt* + *ϑ*) where *ϑ* is a phase term.

So a very abstract idea, like the square root of minus – 1, enables us to reduce the number of equations we need to describe everyday observations.

__Related posts__

18.11 Motion in a circle…

18.10 Waves

18.8 Natural frequency and resonance

18.7 The simple pendulum

18.6 The pendulum: a simple harmonic oscillator

17.13 Centripetal force

17.12 Going round in circles

16.14 Aliasing