Euler's Identity, the beauty of Mathematics

Beauty - a combination of qualities, such as shape, colour, or form, that pleases the aesthetic senses, especially the sight - is definitely not the first word that springs to most minds about maths... But for the lucky few whose studies, work or interests have allowed them to explore maths further, beauty is not just a happy coincidence or a personal view of their work: it is motivation and a road map. Constructing new systems in search of it may seem extremely abstract and a waste of time and resource when the rewards of developing the more applied areas are clearer and more immediate. 

However, consider zero. The very idea wasn’t present in European mathematics until the 1200s. Romans lived in the same world we do today, however ‘zero’ simply did not exist. The reality is that zero is far more unusual than we might appreciate. How do we define nothing? Can I have zero apples? How can I define no apples without first having one apple? Does this mean zero is just relative? Quite simply yes. In today's world our brains are so used to the symbol for zero and the structure it brings to our number system that we don’t even question it. The motivation for its invention in maths was to help us model our lives in a more simplistic way. We wanted more beauty.  

Similarly, we are all quite comfortable with the existence of negative numbers, as we use them daily to describe the things around us. Being indebted to someone for example. My bank balance may read -£100, a negative number, but I don’t have -£100. Here the ‘–‘ symbol simply represents the direction that money is owed. I owe the bank £100 but if it were defined in the opposite direction my balance could just as easily read positive £100. 

Slightly more abstract are ‘complex numbers’. When teaching in schools we refer to them as ‘imaginary’ as we struggle to represent the square root of a negative number in the world we live in. However, this has not held us back from finding infinite (another mathematical definition) consequences, theorems and applications of this so-called imaginary system in real life. It begs the question whether complex numbers are really any more imaginary than negative ones. As again, what does -3 apples look like? 

In terms of beauty, Euler’s identity is held in the highest regard by countless mathematicians across a range of areas with so many describing its elegance and poetry. There is little disagreement that it is the most beautiful equation. But if beauty is such an overwhelming theme of all mathematics, then what makes Euler’s identity SO special? 

Among other things, the definition of mathematical beauty differs from the conventional one by its inclusion of two words, purity and simplicity. Short and sweet, Euler’s identity contains just three basic operations and five constants exactly once. 

Addition, multiplication and constants 0 and 1 are foundations of the everyday mathematics we all use and are the first mathematical topics we teach our children. Intertwined with this are less understood but similarly pure and fundamental concepts: i, the symbol used to represent the square root of -1, that quite literally forms the basis for the complex number system; e, the base of the natural logarithm; and the well known , whose use in geometry can be traced back to the ancient Greeks. appears so often in seemingly unrelated areas of maths, however its derivation is simple, representing the ratio between a circle’s circumference and its diameter.

These numbers and operations are all beautiful in their own right and are all hugely important in their respective fields. This identity not only links them but provides a bridge that allows countless strategies and theorems to be translated across. It is the most satisfying example of the interconnectedness of mathematics and, in perfect mathematical beauty, does so using simply one line.