The Master of Us All (Part 2)

By Nowsherwan Khan
Team Research

This is a continuation of The Master of Us All.

Euler’s Identity: (1748)

The famous American physicist, Richard feynman refers to one formula in particular which he called ‘’Our Jewel’’ . He was referring to Euler’s identity. First expressed in 1748 by Leonhard Euler.

The identity is e^ix = cosx +isinx   where i is the imaginary number, e is the base of natural logarithm, cos and sin are trigonometric operators. a special case of this identity is when u plug in x=pi.

You get e^ i.pi = -1. (cos(pi)= -1 and sin(pi)=0) 

Which gives us e^i.pi +1 = 0. Now if you had a party and you would invite 5 numbers, you would surely invite 0 since its additive identity, 1 since its multiplicative identity, e if you love calculus, pi if you love trigonometry, and i if you are doing complex numbers. The dream team of 5 numbers (i,e,pi,0,1) all in one beautiful formula. This one formula is relating 5 different branches of maths in one simple equation, how Euler got to it? Only through his genius mind and efforts.

In 1988, the journal mathematical intelligencer did a survey of the world mathematical community for the most beautiful equation or formula. Of all the mathematics ever done, this formula was was on number 1 and it was ranked as the most beautiful equation and its from Euler. Someone got so much inspired by this that he wrote a poem on it 

E to the power iota pi plus one equal 0
Made the mathematician Euler a hero
From the real to complex
With out brain in great flex
He led us with zest but no fearo

It was from this identity that Euler deduced such strange consequences as i^i = 1/sqrt(e^pi) , about which Harvard mathematician Benjamin pierce is reported to have said, “Gentlemen, we have not the slightest idea of what this equation means, but we may be certain that it means something very important”.

Euler’s Polyhedral formula (1752)

V + F = E + 2

Or V – E + F = 2. 

V is vertices, F is faces and E is edges, the above formula states for any polyhedral, the formula V-E+F=2, always hold. Or the sum of Vertices Plus Faces = Edges Plus 2.

Now we know how many edges, faces, etc can polyhedras have ? many right? 6,8,10,12,18,20 ( like dodecahedron, icosahedron, octahedron etc). Now how Euler found this for every Polyhedra is just amazing!.

In the math intelligence survey, this formula was listed as second Most beautiful equation!. Top 2 formulas in all maths are given by Euler.

About the polyhedral formula, Euler stated, “ I find it surprising that these general results in solid geometry have not previously been noticed by anyone, so far as I am aware.”

The second most beautiful equation was actually a general result to Euler.

The Basel Problem (1734)

Back in 1689, Jakob Bernoulli who was the brother of Johann challenged the mathematical community to find the exact sum of 1+1/4+1/9+1/16…..+1/k^2.

He issued the challenged from Basel, so it was the Basel problem. Jakob, johan, Leibnitz couldn’t figure it out. They knew it was around 2 or less than 2 but no one could figure exact answer, well Euler did the exact answer and the answer was pi^2/6.

Now we know pi comes only in circles, what is pi doing  here in infinite series, no one knew. Euler gave around 4 different proofs in his life career and all were giving same answer. Its so strange that infinite series is having pi in it, which means it has something to do with circles here right?. Yes! How Euler proved it? He used the infinite sin function series. (sinx/x)

In the math intelligencer pole, the most beautiful equation, this one was ranked 5th!. So in top 5 most beautiful equations, Euler has 3 contributions, its enough to see his skills and genius in field of maths in all time.

Euler Line (1767)

Euler alone published 4 volumes in geometry, many people had published a lot in geometry, but Euler was busy publishing more.

In any triangle, if u take the intersection of altitudes, medians, and perpendicular bisector, you will get a line joining those points which is called Euler line.

Euler didn’t just gave the relation or the line, he also proved that the ratio of those 2 line segments are always 1:2.

3 points and 2 line segments, the ratio was always 1 to 2 and now we can find a lot about the orthocenter and medians, etc. with this beautiful proof.

Amicable Numbers

In math, amicable numbers mean that if there are 2 numbers eg M and N, then they are amicable if each is the sum of  the proper divisors of the other.

e.g. 220, 284, 

If you take the proper divisors of 220 and 284, they should be less than 220 and 284 themselves, and you add those divisors, you get the other number, and so they are amicable.

Now, there is a little problem with these numbers: the 2 numbers or this pair of 220 and 284 were known to Greeks long time ago and no other pair was known, since its very hard to find them.

In 9th C, Islamic mathematician Thabit ibn Gurra gave a rule which which found 2 more amicable numbers but this rule didn’t come to Europe after the renaissance.

In 1636, Fermat found another pair which was 17,296 and 18,416. Fermat had a rivalry, the mathematician Descartes, since fermat found a pair so Descartes also wanted to find a pair.

In 1638, Descartes found a pair which were 9,363,584 and 9,437,056. But that was it , these 3 were the only pairs found and what fermat and Descartes found were also pair found by Thabit.

So Euler thought to give it a try and in 1750, Euler gave 58 more such pairs.

58 more pairs!! This is how Euler would work, he would take a challenge and just blow it up. From 3 pairs to 61 pairs, That’s how Euler worked.

Applied Mathematics 

Even though Euler did a lot in Pure math, He also contributed to applied math. About half of those 75 volumes are on mechanics, optics, acoustics etc

The Concept of a Function: 

It was Euler who gave the concept of function in analysis. Before that people had applied calculus to a curve by geometrical methods but Euler emphasized on functions and gave the concepts of trigonometric, exponential, logarithmic functions etc which are still used today and we all know their roles, without functions, we cant hope to achieve great results.

These are few of the hundreds of his contributions! Euler is considered as the greatest mathematician of all time. There was no match to his speed, memory and genius mind,  what Newton, Leibnitz, johann couldn’t do, Euler would do it with ease. His contributions are breathtaking.

If Newton was the greatest scientist of all time, Euler was the greatest mathematician of all time!

The famous mathematician Laplace said about Euler:

“Read Euler, Read Euler, he is the master of us all”

References:

A tribute to Euler By William Dunham
Koehler Professor of Mathematics Muhlenberg College
Visiting Professor of Mathematics Harvard University.