And the Geometry of Numbers

This is part two of a series about Leonard Euler (1707–1783). You can read the first part here.

Before we begin our quest towards understanding the beautiful fact called Euler’s Identity, let’s warm up with an amazing piece of history.

Numbers in Love

About 500 b.c. the Greeks regarded some numbers as more important than others. In particular, they knew of two numbers with a remarkable property. The two numbers are 220 and 284.

Before explaining why these numbers are so interesting, we need to know what a proper divisor is. Well, it is very simple. A proper divisor of say, n, is…

What is the factorial of 1/2?

Fredrik alleged / Wikimedia Commons

What is your favorite function?

If your answer is not the gamma function, then I’ll ask you again after you have read this article.

Your answer might have changed…


In the late 1720s, Leonhard Euler was thinking about how to extend the factorial to non-integer values.

This was the start of a rich theory used all over the scientific world. A theory of one of the most important functions in mathematics.

Leonhard Euler is, without doubt, one of the greatest mathematicians in history. To give you an idea of Euler’s powers here follows some examples that show his brilliance.


A hunt for a new operation

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In school, we are taught a couple of basic operations: addition, subtraction, multiplication, division, and maybe exponentiation. Only later we find out that really, subtraction is addition by a negative number and division is multiplication by a fraction.

That leaves us with addition, multiplication, and exponentiation.

But of course, the story doesn’t stop there because there are relations between the operations.

For example, addition and multiplication have an interesting relationship via the distributive law:

  • a ⋅ (b + c) = a ⋅ b + a ⋅ c.

So multiplication distributes over addition. But actually, multiplication and exponentiation have a similar…

How flowers found mathematics millions of years before humans

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The Fibonacci Numbers

The Fibonacci sequence starts 1, 1, 2, 3, 5, 8, 13, 21, 34, … and the characteristic feature is that the sum of two consecutive numbers gives the next number in the sequence.

I clearly remember my uncle, who was also a mathematician, writing the start of this sequence on paper when I was a little kid, asking me for the next number.

I don’t think I was able to answer correctly but when he explained the sequence to me, I became quite fascinated by this, and I have since enjoyed mathematical knowledge sharing, puzzles, and discussions with my uncle…

And the nature of probability

Image by Suzy Hazelwood

I hope you’re sitting down for this one!

When I was in high school, I heard about this phenomenon called the birthday paradox. It is loosely stating that in a room of only 23 people, the probability that two or more people have their birthday on the same day is more than 1/2, i.e. there is a chance of at least 50% that two or more people’s birthdays coincide.

This was one of the cases where I really needed to see a proof before I could believe it. Only 23 people and 365 ish days to choose from?

I will…

The nature of subsymmetries

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Have you ever wondered what restrictions symmetries inside bigger symmetries have to obey?

In this article, we will dive into the beginning of subgroup theory by proving a result first discovered in the 18'th century by the brilliant mathematician Joseph Louis Lagrange.

In fact, he is so popular that the French call him French and the Italians call him Italian, but I will leave that discussion to the respective countries and simply admire his beautiful (and non-demographic) mathematics…

When I first heard about the concept of a subgroup, I imagined a fractal-like image of a pattern inside a bigger pattern…

The nature of symmetry and the symmetry of nature

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Group theory is the language of many of the mathematical disciplines. An indispensable tool in understanding the underlying nature of nature.

A theory that holds the secrets of the fundamental particles and forces of the Universe itself!

We use it to understand shapes in higher dimensions, the proof of insolvability of higher degree polynomials, and the structure of number systems, but the use of this beautiful theory doesn’t stop there.

Group theory is the study of symmetry and as you will soon see, symmetry is everywhere in mathematics and in nature.

My own encounter with groups started when I studied…

Generalizing Leibniz' formula for π, the alternating harmonic series, and the Basel problem

Image by author

A couple of years ago I was thinking about how to generalize many of the infinite series that we know so well. It was clear to me that it was possible to generalize them since some of my studies into the field of Fourier series had led me to some of these generalizations in the past. But it was clear that I needed another way - a more controlled way to generate such formulas.

Not long ago, I found a method and I couldn’t sleep that night because I was calculating like crazy in my head. …

An ancient riddle

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In mathematics, especially when it comes to number theory, there are many basic things that we don’t know.

Simple questions that we can’t answer.

This can be realized by considering a few of the famous unanswered questions that we mathematicians would like to know the answer to.

A few of them are

  • Are there infinitely many primes p such that p+2 is also prime?
  • Are there infinitely many Mersenne primes?
  • Can every even number greater than 2 be written as a sum of two primes?
  • Are there infinitely many amicable numbers?
  • Is the Collatz conjecture true?

These questions seem deceptively…

Use Neo4j to calculate a safety score — how safe is your code?

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As a mathematician and working data scientist, I am fascinated by programming languages, machine learning, data, and of course, mathematics.

These technologies, arts and tools, are of course of huge importance to society and they are transforming our lives as you read this article, but another emerging technology is growing. And it is growing fast!

It is a technology based on a mathematical field that I studied at university and which was first discovered (or invented..? let’s safe that talk for another time, shall we?) …

Kasper Müller

Mathematician and Data Scientist interested in the mysteries of the Universe, fascinated by the human mind, music and things that I don’t understand.

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