The Constant Known as e

Introduction
Understanding Euler's Identity
From Exponential to Trigonometric
The Final Step to Euler's Identity
The Geometry Behind $e^{ix}$

Introduction

To grasp the concept of $e$ , imagine this scenario: you've invested $100$ units into a financial scheme promising a $100\%$ return over a year. While such a scheme would certainly be a scam in real life, let's overlook that for the sake of explanation.

A year later, your investment doubles to $200$ units, marking a growth factor of $2$ . But what happens if you opt for a semester-long investment instead, achieving a $50\%$ return, then reinvest the total (now $150$ units) for another semester? The growth factor of your investment becomes:

(1.5)^2 = 2.25

Notice how it increases. Let's break down the investment further into quarters, each with a 25% return. The growth now becomes:

(1.25)^4 = 2.441

What if the investment were even more frequent, say monthly? This would yield:

(1 + \frac{1}{12})^{12} = 2.613

The formula for calculating the growth factor is thus:

(1 + \frac{1}{n})^n

Does this mean the growth factor increases infinitely? Not quite. Eventually, it approaches a constant value, known as $e$ . Hence, $e$ is defined by the term:

\lim\limits_{n \to \infty} (1 + \frac{1}{n})^n

Understanding Euler's Identity

Before we connect $e^x$ to the trigonometric world, we first need to expand $\sin(x)$ and $\cos(x)$ as infinite series.

\begin{split} \sin(x) &= x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots \\ \cos(x) &= 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots \end{split}

See Polynomial Approximation for a detailed explanation of how these series are derived using repeated differentiation.

We also need to expand $e^x$ :

e^x = \sum_{k = 0}^{\infty} \frac{x^k}{k!}

See Binomial Expansion for how this form emerges from combinatorial reasoning. Alternatively, the same result can be derived through the Taylor Series, where each term comes from successive derivatives of $e^x$ .

From Exponential to Trigonometric

Now, what if we substitute $x$ with an imaginary number $ix$ (where $i = \sqrt{-1}$ )? We get:

e^{ix} = \sum_{k=0}^{\infty} \frac{(ix)^k}{k!}

Let's expand the first few terms:

\begin{split} e^{ix} &= 1 + ix + \frac{(ix)^2}{2!} + \frac{(ix)^3}{3!} + \frac{(ix)^4}{4!} + \cdots \\ &= 1 + ix - \frac{x^2}{2!} - i\frac{x^3}{3!} + \frac{x^4}{4!} + i\frac{x^5}{5!} - \cdots \end{split}

Now separate the real and imaginary parts:

\begin{split} e^{ix} &= \left(1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots \right) \\ &\quad + i \left(x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots \right) \end{split}

These two grouped series are exactly the definitions of $\cos(x)$ and $\sin(x)$ :

\begin{split} \cos(x) &= 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots \\ \sin(x) &= x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots \end{split}

So we can combine them into the famous relation:

e^{ix} = \cos(x) + i\sin(x)

The Final Step to Euler's Identity

Now, let's substitute $x = \pi$ into that equation:

e^{i\pi} = \cos(\pi) + i\sin(\pi)

We know that $\cos(\pi) = -1$ and $\sin(\pi) = 0$ , so:

e^{i\pi} = -1

And by adding 1 to both sides:

e^{i\pi} + 1 = 0

This is Euler's Identity, often celebrated as one of the most elegant equations in mathematics. It ties together five of the most fundamental constants in a single expression:

$e$ (the base of natural logarithms)
$i$ (the imaginary unit)
$\pi$ (the circle constant)
$1$ (the multiplicative identity)
and $0$ (the additive identity)

The Geometry Behind $e^{ix}$

If we imagine the complex plane, where the horizontal axis represents real numbers and the vertical axis represents imaginary numbers, then any complex number $z = a + bi$ can be visualized as a point at coordinates $(a, b)$ .

Now, for $e^{ix}$ , we can write it as:

z = \cos(x) + i\sin(x)

This means:

The real part (horizontal coordinate) is $\cos(x)$
The imaginary part (vertical coordinate) is $\sin(x)$

If we plot the point $(\cos(x), \sin(x))$ , it always lies on a unit circle, since:

\cos^2(x) + \sin^2(x) = 1

Thus, as $x$ increases, $e^{ix}$ traces a circle of radius 1 centered at the origin. It rotates counterclockwise with angle $x$ measured in radians.

You can think of $e^{ix}$ as describing a continuous rotation on the complex plane.

At:

$x = 0$ : $e^{i0} = 1$ - the point starts on the real axis.
$x = \frac{\pi}{2}$ : $e^{i\pi/2} = i$ - the point has rotated 90° upward.
$x = \pi$ : $e^{i\pi} = -1$ - now it's on the opposite side of the real axis.
$x = 2\pi$ : $e^{i2\pi} = 1$ - completing a full rotation back to the start.

When the exponent has both real and imaginary parts, say $e^{x + iy}$ , we can separate them as:

e^{x + iy} = e^x \cdot e^{iy} = e^x(\cos(y) + i\sin(y))

The term $e^x$ scales the radius while $e^{iy}$ controls the rotation.