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If you have written f(x), used e in a logarithm or turned a complex number into polar form, you have used Leonhard Euler's work. Much of it is so familiar that it feels like it was always there. This post explains where some of these ideas came from, works through examples you will meet in JEE and board maths, and ends with what a student can take from Euler's way of working.
Who was Leonhard Euler?
The MacTutor biography from the University of St Andrews gives his dates as 15 April 1707, Basel, to 18 September 1783, St Petersburg. He worked at the St Petersburg Academy, then in Berlin, and returned to St Petersburg in 1766. Soon after his return he became almost entirely blind. MacTutor notes that, helped by a remarkable memory and by his sons, he "produced almost half his total works" after returning to St Petersburg at the age of 59.
Which notation do we owe to Euler?
MacTutor lists the notation as follows:
f(x) for a function (1734)
e for the base of natural logarithms (1727)
i for the square root of -1 (1777)
the summation sign, capital sigma (1755)
pi for the circle constant.
Good notation is not decoration. Writing f(x) lets you talk about a rule without saying what it is, and the summation sign compresses a long sum into one symbol you can manipulate.
What is Euler's formula, and why does JEE care?
MacTutor notes that in his 1748 book Introductio in analysin infinitorum, Euler gave the formula e^(ix) = cos x + i sin x. This is the bridge between exponentials and trigonometry, and it is why the polar form of a complex number works.
Worked example. Find (1 + i)^8.
Modulus: |1 + i| = square root of (1 + 1) = square root of 2.
Argument: the point (1, 1) makes an angle of pi/4 with the real axis.
So 1 + i = (square root of 2) x e^(i pi/4).
Raise to the 8th power: (square root of 2)^8 x e^(i x 8 x pi/4) = 16 x e^(2 pi i).
Since e^(2 pi i) = cos 2 pi + i sin 2 pi = 1, the answer is 16.
Check by brute force: (1 + i)^2 = 2i, so (1 + i)^4 = (2i)^2 = -4, and (1 + i)^8 = 16. The two methods agree.
Put x = pi in the formula and you get e^(i pi) = -1. Our guide to complex numbers for JEE has more problems of this type.
How did Euler solve the Königsberg bridge problem?
The city of Königsberg had seven bridges over the River Pregel, linking an island and the land around it, and people asked whether you could walk across each bridge exactly once. The Euler Archive lists his paper on it as written in 1735 and published in 1741, and describes it as "often cited as the earliest paper in both topology and graph theory". In it Euler reduces the problem to counting how many bridges meet each land area.
Worked example. On the usual map, the island touches 5 bridges and each of the other three land areas touches 3. Every time you walk through an area, you use one bridge in and one bridge out, so an area you pass through needs an even number of bridges. Only the start and end of the walk may have an odd number. Here all four are odd, so the walk is impossible. In modern terms, MacTutor's history of topology puts the general result this way: a graph has a path using each edge exactly once if exactly two vertices have odd degree (or, for a closed walk, none do).
The same history credits Euler with the polyhedron formula v - e + f = 2, which he gave in a 1750 letter to Goldbach. Check it on a cube: 8 vertices, 12 edges, 6 faces, and 8 - 12 + 6 = 2.
What else did he do?
MacTutor notes that Euler showed in 1735 that 1 + 1/4 + 1/9 + 1/16 + ... equals pi^2/6, the "Basel problem" that had defeated Jacob, Johann and Daniel Bernoulli and that Leibniz had also studied without success. The first four terms already give 1.4236, approaching pi^2/6, which is about 1.6449. He also worked on mechanics, optics, astronomy and music.
What can a student learn from how Euler worked?
Invent notation that saves effort. Euler's symbols survive because they make the next step easier. Your own notes improve when you use one clear symbol consistently.
Strip a problem to its structure. Königsberg became easy once distances and shapes were ignored and only connections counted.
Keep a working memory of results. His output after losing his sight depended on knowing his mathematics well enough to work without paper.
When a step in a complex numbers problem goes wrong, the Fermi Problem Library has a step-by-step solver that explains the reasoning rather than printing an answer.
Practice routine: compute (1 - i)^6 and (square root of 3 + i)^6 by polar form, check v - e + f for a tetrahedron and an octahedron, and decide whether a house plan with five rooms and given doors allows a walk through every door once.
Next step: draw the four land areas of Königsberg as dots and the seven bridges as lines, and count the degree of each dot yourself.
