Euler’s Identity!

Possibly the most beautiful formula in Mathematics…

When I first learned of this identity (i.e. an equation which is always true), I was blown away.

‘e’ is a constant, irrational number (i.e. never ending – it has infinitely many numbers after the decimal point and you can never write out the whole number) roughly equal to 2.718. For exactly what this number – known as ‘Euler’s number’ is, see this e (mathematical constant) – Wikipedia.

‘i’ is an imaginary number which is defined to be the square root of -1. Think about this for a second. How do you find the square root of a number – you find the number which when multiplied by itself gives you the original number e.g. the square root of 4 is 2 because 2×2=4. But we know that a positive number times a positive number is positive, and a negative number times a negative number is positive. So how do you find the square root of a negative number? Well, only by defining an imaginary number (quite literally, a number that sort of ‘doesn’t exist’) equal to the square root of -1. So if you’ve created this concept to enable you to deal with the seemingly impossible, how on earth does it appear in an equation involving completely real numbers so neatly?? Weird eh?!

And then take pi. You probably know that pi is the irrational number related to the area and circumference of a circle (2pi x radius = circumference of a circle). Again, pi is a number for which you can never write the whole thing down. There are even competitions for who can name the most number of decimal places of pi.

Then put them all together. A never ending number (e), that you multiply by itself ‘i x pi’ times, where ‘i’ is imaginary, and pi is also never ending. How do you even make sense of what that means? And yet, it equals -1!! I think that is beautiful, mind boggling and cool all at once (but then, I am a bit of a nerd).

For more on this, see The Most Beautiful Equation of Math: Euler’s Identity | Science4All.

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