By Phillip Newbold, La Trobe University

One of my earliest memories involves mathematics. I was in primary school. It was just after lunch time on a Friday and that meant it was time for the weekly puzzle. My teacher explained the problem, which involved a frog leaping up and then slipping part way back down an incline. We were tasked with figuring out how many leaps it would take for the frog to get to the top of the incline. On that day, I was the only one in my class to figure it out. The enjoyment I felt figuring out this, now simple, problem was the same I felt when first solving algebra problems in early high school and again when figuring out tough calculus and trigonometry problems in late high school.

However, up until this point, I had seen these areas of mathematics as being linked only through the shared name of the subject in which I learnt about them. Not that they were all parts of a whole. Similarly, I had never thought that mathematics was something you could do as a passion. I saw it merely as a tool for other, more practical (or so I thought) topics such as physics and chemistry. This all changed when I first encountered Euler’s identity:

It will not be immediately apparent, to those reading this who have not studied mathematics, why this equation changed the way I thought about mathematics. To explain this, I need to give some brief background.

Although it may not appear so, this expression is made up of five numbers. These numbers are; e, i, π, and more obviously 0 and 1. e, known as Euler’s number, is the base of the natural logarithm. It is easy to ignore when you are busy trying to understand the way logarithms work and I gave it little thought when I first encountered it. i, which is the unit imaginary number, is, amongst the concepts explained here, possibly the most difficult to grasp for newcomers. It is best understood as the solution to the equation  That is, it is equal to -1 when squared. It is a bizarre concept when first encountered but is a fundamental element of mathematics. π is the best know of these three numbers, it being the ratio of a circle’s circumference to its diameter.

I had seen these numbers as disconnected, belonging to distinct areas of mathematics. As fish swimming in separate, disconnected ponds. However, Euler’s identity showed me that, when combined, they gave a neat and elegant expression. That all these mathematical concepts are related. Fish swimming in one great ocean. It’s this vision of mathematics, as beautiful, elegant and interconnected, that lead me to study it in University and choose a career with numbers at its core.

I will never forget how Euler’s identity changed the way I saw mathematics, just as I will never forget that puzzle I solved one Friday afternoon in primary school.


Phillip Newbold was a recipient of a 2018/19 AMSI Vacation Research Scholarship.

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