A CHARTERED UNIVERSITY & INTERGOVERNMENTAL INSTITUTE WITHIN EUCLID

Euler’s identity is often hailed as the most beautiful formula in mathematics. People wear it on T-shirts and get it tattooed on their bodies. Why?

Leonhard Euler, 1707-1783. Portrait by Johann Georg Brucker.

where  is the base of the natural logarithm,  is the ratio between a circle’s circumference and diameter, and . These three constants are extremely important in maths — and since the identity also involves  and , we have a formula that connects five of the most important numbers in mathematics using four of the most important mathematical operations and relations – addition, multiplication, exponentiation and equality. That’s why mathematicians love Euler’s identity so much.

But where does it come from and what does it mean? As we mentioned above, . This might seem shocking because negative numbers are not supposed to have square roots. However, if we simply decree that  does have a square root and call it , then we can build a whole new class of numbers, called the complex numbers. Complex numbers have the form  where  and  are ordinary real numbers (for the complex number  we have  and ). See here for a quick introduction to complex numbers and how to calculate with them. Note that a real number can also be viewed as a complex number. The number , for example, is a complex number with  and .

Just like a real number is represented by a point on a number line, a complex number  is represented by a point on the plane. To the complex number  we associate the point with coordinates .

In this description we used Cartesian coordinates: they describe the location of a point by telling you how far to walk along the horizontal direction and how far to walk along the vertical direction. Sometimes, however, it’s more convenient to describe the location of a point in terms of the arrow starting at the crossing point of the two axes as shown below.

To define that arrow you need its length  and the angle  it makes with the positive -axis (measured anti-clockwise). These are the polar coordinates of our point. Basic trigonometry (see the diagram below) tells us that if a point has Cartesian coordinates  and polar coordinates , then

and

Therefore the complex number  represented by our point, , can also be written as

Here comes the crucial point. It just so happens that for real numbers  and

You can prove this using power series, see here to find out more. It’s a beautiful fact that the exponential function and the two trigonometric functions sine and cosine are linked in this way. And it means that any complex number  can be written as  where  is the length of the line connecting the point on the plane that is associated to  to the crossing point of the axes, and  is the angle that line makes with the positive -axis (measured anti-clockwise).

This now makes Euler’s identity crystal clear. The complex number  represents the point on the plane at distance  from the crossing point of the axes with an associated angle of . That’s the point with Cartesian coordinates  which represents the complex number .

Putting all this together, we see that

which means that

And that’s Euler’s identity.

### How is AI affecting the work of university professors in terms of assessing and grading academic papers?

Introduction Artificial Intelligence (AI) has revolutionized various aspects of our lives, and the realm of academia is no exception. From simplifying administrative tasks to enhancing

### Greening the Campus: Sustainable Practices and Research Collaborations

Introduction: Why Sustainable Practices Matter on Campus In today’s rapidly evolving world, the need for sustainable practices has become more crucial than ever. As we

### Using Legos as a means to train future engineers

Welcome to our comprehensive guide on using Legos as a means to train future engineers. In this article, we will explore the numerous benefits of

### Universities in the Netherlands: Centuries of Global Leadership

Introduction Universities in the Netherlands: Centuries of Global Leadership The Netherlands, with its rich history and vibrant culture, has long been a hub of academic

### What is a unitary state? The case of the Netherlands.

Introduction In this article, we will delve into the concept of a unitary state, using the case of the Netherlands as an example. A unitary