# Introduction

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.

First…

# A Beautiful Integral Formula for the nth Derivative

## With some interesting applications

About a year ago I was trying to come up with a formula for the nth Fibonacci number which wasn’t Binet’s well-known formula. Fiddling around with the generating function, I was thinking about how to extract the coefficients without having to differentiate the beast n times.

This is a story about the result of this pursuit. We can prove it and use it to solve interesting problems and maybe even prove other theorems.

In this article, I will show you this beautiful formula for calculating derivatives without having to differentiate a single time. …

# Limits

## The beginning of calculus

This is part two in a series of articles where my goal is to teach calculus from scratch.

You can find the first one here.

The most fundamental and important concept to understand in calculus is that of a limit.

As with many other objects in mathematics, limits can be understood from many different levels. In this article, I will try to provide you with a couple of different levels of understanding so that you are able to understand the inner workings of calculus as we move on in the subject.

As you will see in the coming articles, limits…

# Functions and Continuity

## Prerequisites for calculus

This is the first article in a series of small articles meant to teach the subject of calculus from scratch in a friendly way starting with the very basics.

Calculus may seem intimidating for some because it signifies a leap in mathematical maturity from basic algebra and arithmetic to function theory, however, it is actually a beautiful, useful, and not so hard subject if one takes the time to understand the very basics of it first.

When I first studied it, I remember feeling excited because up until then I felt like the whole subject of mathematics was quite boring…

# Eratosthenes Calculated the Shape and Size of the Earth More Than 2000 Years Ago Without Leaving the Country

## Here’s how he did it!

In the middle of the 20’th century, the space age began. We put satellites in orbit to better measure and map the Earth, and we could finally get a pretty accurate measure of the Earth’s circumference of about 40,075 km at the equator.

That’s is pretty impressive, but it turns out that this result had already been discovered by a mathematician called Eratosthenes of Cyrene more than 2000 years earlier.

# Eratosthenes

Eratosthenes was born in 276 BC in Cyrene, Libya. His studies began in Cyrene. Later he moved to Athens to study further. …

# The Different Kinds of Mathematical Proofs

## Proof techniques, logic, and metamathematics

On 8 September 1930, David Hilbert, one of the greatest and most charismatic mathematicians of the 20’th century, was retiring with the famous words

Wir müssen wissen.
Wir werden wissen.

These are the famous lines he spoke at the conclusion of his retirement address to the Society of German Scientists and Physicians.

He was referring to all the unproven statements and seas of knowledge left for future mathematicians and scientists to tackle. Hilbert was certain that it would be a matter of time before we would have full knowledge and have proven all true statements out there.

In English, his…

# Introduction

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…

# 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

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…

# Subgroups and Lagrange’s Theorem

## The nature of subsymmetries

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… ## 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.