Maths in a minute: Euler’s identity

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.

where is the base of the natural logarithm, is the ratio between a circle’s circumference and diameter, 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 ). 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 x=-1 and y=0

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 (x,y).

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.