The magical powers of ‘e’

e is a number. But it’s not just any old number. Sure, nine out of ten people you might ask on the street would see nothing special about this crazy phenomenon lying between 2.7 and 2.8. But, like that secret ingredient hiding behind a Michelin star chef’s perfect soufflé, the number 2.718… lies behind a plethora of mathematical delights.

e is an actual number, but, like pi, it is irrational – you can find the first few thousand digits at

The property that might be considered as a defining feature of e is that the gradient of the ex curve is the same as its value at ANY point along the curve – so say we took the number 729, and raised e to it, I could tell you without doubt that the gradient of the curve at that point would be 729 too!

…and so the function ex differentiates to itself – pretty self-explanatory given the last point, but if you’re given a messy differentiation or integration that involves a lot of e powers, you know you can tidy it up at least a little bit.

This ex curve is exponential, but it strangely calms down to a gentle sine curve when raised to imaginary numbers.

e can be actually be defined as the limit as n approaches infinity of
And yet ex can also be found using the infinite sum:


Euler is pronounced as Oiler – which is just weird, and something I still have to get used to, especially when I’m trying to ask my teacher a question and I completely mispronounce it without noticing, to much confusion – and is named after Leonhard Euler, the 18th century mathematician who also found the Euler constant and introduced the concept of a function and the way that we write trigonometric functions.

And finally, the area of the region bounded by y=1/x, the x-axis, x=1 and x=e is 1. Neat.


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