Beauty - a combination of qualities such as shape, color, or form, that pleases the aesthetic senses, especially the sight - is not necessarily the first word that springs to mind when discussing mathematics. But for the lucky few whose studies and work allow them to explore maths further, beauty is not just a happy coincidence or 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 resources, when the rewards of developing the more applied areas are clearer and more immediate.
However, consider zero. Whilst some cultures have been using a symbol to represent ‘relative nothingness’ or for ease in showing higher values (hundreds and millions etc.) for many thousands of years, the 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 that 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 numbers system that we don’t even question or recognize that in reality it is merely something we have defined. The desire for its invention in mathematics was to help us to model our lives in a simpler way. We wanted more beauty.
In schools when we teach about the complex numbers, 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 questions whether complex numbers are really any more imaginary than negative ones. As again, what does -3 apples look like?
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.
Another obvious example is temperature scales. In degrees Celsius it is not uncommon in winter for the temperature to be ‘negative’. However this is actually just a comparison of outside temperature to the one at which water freezes. When using the Kelvin scale, negative temperatures are impossible. It is not possible to have negative energy. When you start to think in this way suddenly the idea of creating maths simply for the sake of it and because ‘it looks nicer’ does not seem so ridiculous.
In terms of beauty, Euler’s identity is held in the highest regard by countless mathematicians across a range of areas, with 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 3 basic operations and 5 constants exactly once.
Addition, multiplication, and constants 0 and 1 are foundations of the everyday mathematics we all use and are the first things we teach to our children. Intertwined with this are less understood but similar 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.
Blog by - Lucy chats maths