Numbers are pretty neat. We use them to count the things we’re buying at the store. We use them to track the days and weeks as time passes. We use them more abstractly to discover new theorems and laws about the way the world and even the universe works. At the core of all of this is a number system that we all agree on. These numbers are represented by a numeral system, a set of symbols that represent numbers, organised in such a way that we can represent any number easily. Looking back though, all the numbers we use today weren’t always there. We didn’t always have a number 0 to describe, well, nothing. We didn’t always have fractions to accurately represent proportions. We didn’t always have the complex numbers to fill in the gaps that the real numbers couldn’t. The number system we use now took centuries, even millennia, to develop (or discover if that’s your interpretation of mathematics). Let’s take a look at how we got here.

## The First Counting Systems

Early humans likely counted objects with a tally system. Tally marks would be carved into stone or wood or cave walls, with each mark standing for the number 1 and each 5th mark scoring the other 4 (like a seat-belt) to make it easier to keep track of. For small numbers, this system is perfectly fine, and indeed in the Paleolithic age, we probably weren’t keeping count of 5 million humans to feed or 8 billion leaves in a forest. However, as civilisation grew, we needed a different system in order to write larger numbers more conveniently.

The systems that came about from early civilisations were extensions to the tallying system. As an example, the ancient Egyptian numeral system had more characters and symbols than just line strokes. For example, the number 265 was written down as:

But this didn’t exactly solve the problem, writing long numbers was still an arduous process.

One solution was a positional system. A system such as this would have a few characters, but their position would give them different values. An abacus is a physical example of this. Several different civilisations independently created positional systems (the Mesopotamians more than 5000 years ago are an example). The earliest known base 10 system goes back to 3100 BC in Egypt. A base 10 system is one we use today, each new place has values of increasing powers of 10. As an example, . This allows us to write really big numbers fairly easily. 265 is just 3 characters compared to the 13 the ancient Egyptians would have to use. This isn’t to say we never used any other system; the Babylonians had a base 60 number system (which is how we count seconds and minutes).

## The Negative Numbers

The idea of negative numbers is a bit abstract. It’s easy to visualise the positive numbers as things in the real world. A dog has 4 legs, a spider 8. Negative numbers don’t have that kind of interpretation. Nevertheless, they were recognised in China around 100 BC – 50 AD. In *The Nine Chapters on the Mathematical Art (Jiuzhang Suanshu)*, negative numbers were used to solve systems of simultaneous equations. This was done using a number rod system, with red rods representing positive numbers and black rods negative numbers.

In 7th Century India, negative numbers were used to represent debts, whereas positive numbers represented assets. Indian mathematician Brahmagupta (598-668 AD) would then go to create the rules for operations with negative numbers, in terms of debts (for negatives) and fortunes (for positives) in his work *Brahmasphutasiddhanta*.

In the West, it took a while before the concept of negative numbers became accepted. The first person to use them in a European piece of work would be Nicolas Chuquet (1445-1488), a French mathematician. He used these to write out negative powers or exponents (i.e , but he called these numbers “absurd”. As scholars began to uncover and translate texts from the Islamic and Byzantine sources in the 15th Century, the idea slowly became more accepted (to varying degrees amongst different mathematicians). Some mathematicians, such as Cardano (1501-1576) used and accepted negative numbers in most cases except for negative coefficients in quadratic equations. This is because he interpreted these quadratic equations geometrically, with squares of different lengths, so negative coefficients meant that there would be sides of negative length, what he deemed an “absurdity” (we’ll see an example of this later on). Nonetheless, as the centuries went on, confidence, acceptance, and use in negative numbers grew as we adopted them into our number system.

## The Number for Nothing

In modern notation, we just read the number from left to right and in our heads keep track of the positions and the powers of 10, and sometimes we’ll use commas or other symbols to split the number up so it’s more readable. The first evidence we have of 0 comes from Mesopotamia, with a slanted double wedge used. However, this 0 was used more as a symbol in this way of writing numbers, to show the absence of a certain power, the same way the 0 in 403 means there’s no multiple of 10 in the number. The first recorded use seems to be from the Babylonians, who used it in the same way, but with their base 60 number system. However, the idea of 0 as its own number hadn’t been uncovered yet. There are records to show that even the ancient Greeks were unsure about whether 0 should be a number or not. “How can ‘nothing’ be something?”

The idea of 0 being its own number with its own properties and rules came about in the 7th century in India. Brahmagupta was also the first known documented user of 0. He would use small dots under numbers as a placeholder value, but also as something with its own “null” value, called “sunya”. He also went on to define operations surrounding 0, such as how the subtraction of a number by itself gives you 0. However, he could not explain division by 0; this impossibility was explained far later by George Berkeley in 1734. This idea spread across China and the Middle East, where Mohammed ibn-Musa al-Khowarizmi (a Persian polymath) in 773, who some describe as the father of algebra, being the first to treat it as an independent discipline and introducing the ideas of balancing equations and reducing. He played a major role in creating Indian arithmetic and showed how 0 can be used in algebraic equations. By the 9th century, 0 had been incorporated into the Arabic numeral system as the round oval we use today. It then continued to migrate across the West, with prominent mathematicians like Fibonacci bringing it into the mainstream.

A debt minus zero is a debt,

Brahmagupta,

A fortune minus zero is a fortune,

Zero minus zero is a zero…Brahmasphutasiddhanta

## Rationals and Irrationals

Here’s a question: there’s a right-angle triangle with a base and height of unit 1. What’s the length of its hypotenuse? Now if you remember the famous Pythagorean formula, the answer is . Indeed, a Greek philosopher of Pythagoras’ school of thought (which just means he followed the same ideas and aims as him), Hippasus of Metapontum (530-450 BC), sought to use his teacher’s theorem to solve this problem. Now to us, this seems almost trivial, as we know what these numbers are, but Hippasus is considered to be the first person to discover these “irrational” numbers.

With all the numbers we’ve discovered so far, there are two ways we can categorise them. Rational numbers are numbers that can be written as a ratio or fraction of two integers. So , , and are all rational. However, some numbers, like , cannot be written as a ratio of two integers (proof of this can be found here). These are irrational numbers.

Now, this posed a problem for Hippasus, as many of members of this school of thought believed that **only **positive rational numbers existed, so Hippasus’ discovery went against all of their teachings. What happened to Hippasus after this discovery can only be speculated. Some stories say that after this discovery, he died of natural causes on an ocean voyage, while others say other members of this school of thought threw him overboard a ship to silence him and his heretical ideas, or that he was silenced by the gods. It’s also entirely possible that none of these stories is true and that they have been embellished to make this pivotal moment in history seem more exciting and epic. The reason that his peers could’ve gone so far is that numbers played a big part in their philosophy and religion. As we’ve discussed, we’ve had numbers from our practical need to count and measure things since the early stages of humanity. However, the Pythagoreans believed that spiritualism, physics, ethics- everything- could be described and understood through rational numbers. So, this discovery wasn’t just going against what they thought about maths; it was their entire philosophy of life.

Regardless, this isn’t true. This is probably the first time they had been discovered, but how Hippasus proved that the is irrational is hard to tell. Nevertheless, another philosopher, Euclid, proved it (linked above) around mid 4th century BC. Nowadays there are a few more ways to prove it (that are much more complicated).

All the numbers we’ve now discovered thus far, rationals and irrationals, are grouped under the “real” numbers category. A lot of these discoveries have been made as a consequence of something, usually the rules of mathematics as we know them. We can prove with all the basic rules that the square root of 2 is not rational, so as a consequence it must be irrational, a whole new type of number. 0 and negative numbers are the same; we lacked but needed something that had a “null” value, so we created 0. The next group of numbers is a bit more involved in its discovery, but it’s the same story, a consequence of the unbreakable axioms of mathematics.

## Complex Numbers

From the way the square root of a number is defined, square rooting a negative number doesn’t really make sense with all the numbers we’ve discovered thus far. In simple words, a square root of a number is some other number that squares (is multiplied by itself) to make it. The square root of 4 is 2 because 2 squared makes 4. The square root of 2 is its own number, which results in 2 when squared. What’s the square root of ? Thinking about it, the square of any number is always positive, so it doesn’t look like there’s a solution to this. Now, one might just conclude that this is just some rule, that there’s just no solution to this, no square root of negative numbers. However, this problem has been popping up in mathematics for millennia. Heron, an Alexandrian Greek Mathematician (10-70 AD) got involved in a calculation on a pyramid design (which involved a frustum, written about in his book Stereometrica AD 75), but evaluating this led to , square rooting a negative number. Another famous Greek Mathematician, Diophantus, met this issue in another problem that he wrote about in his book Arithmetica (AD 275):

A right-angled triangle has an area of 7 square units and a perimeter of 12 units. Find its sides.

You can give this a try if you want. If you do, the result is a quadratic equation where, again, you have to square root a negative number. At the time, mathematicians either didn’t know how to handle this or just ignored the issue. Over time, a few quotes from several Hindu mathematicians were shared, affirming that the square root of a negative number simply can’t exist.

As in the nature of things, a negative (quantity) is not a square (quantity), it has, therefore, no square root.

Mahavira Acharya

Yet, problems that involved it just kept cropping up over time. Was this just a coincidence or a hindrance in the field we just had to accept?

Jumping forward in time, we get to Girolamo Cardano (1501-1576). He plays an important part in this story. As you might expect, he came up with a problem and found that the solution involved square rooting a negative number, which was thought to be an impossibility.

Split the number 10 into 2 numbers such that their product is 40. Written in symbols and modern notation, this asks to solve the pair of equations and .

What Cardano did differently to everyone else was that he didn’t immediately stop once he ran into this roadblock. He *pretended* that the square root of a negative number was just a normal number, and used the rules of algebra to show that it is a solution. But because he *pretended* that it was a real number when it’s not, he deemed it an impossible solution.

Cardano would run into this problem again with something a bit more general: solving cubic polynomial equations. At the time, mathematicians were trying to find the solutions to cubics of the form . Cardano had a method of solving these that worked. His method can be generalised in the fittingly called *Cardano’s formula*:

x=\sqrt[3]{\frac{b}{2} + \sqrt{{\left( \frac b 2 \right)^2-\left(\frac a 3 \right)^3}}} + \sqrt[3]{\frac{b}{2} - \sqrt{{\left( \frac b 2 \right)^2-\left(\frac a 3 \right)^3}}}

This is quite a complicated formula, and we need not worry too much about where it comes from (you can see a proof of its derivation here if you’re interested). What’s more important is that this formula has been proven to work. However, there’s a particular cubic equation which causes the problem we’ve spent this section talking about to occur again: . If you have the time (or a calculator nowadays), you could solve this equation and you’d get 3 real solutions. Specifically, these are and . However, Cardon got an answer with . So either the formula and Cardano’s method are all wrong, or something is up with this idea of square rooting negative numbers. But we know that this formula works for many other cubic equations, so why does it fail when it comes to this one?

Rafael Bombelli (1526-73), a hydraulic engineer, was the one who figured out what was happening. Like Cardano in his other problem, he assumed that the square root of a negative number was some number going against the consensus at the time. As well as another more complicated assumption, he found that the solution given by Cardano’s formula does indeed return , one of the real solutions to this equation. To get here though, Bombelli assumed that the square root of a negative number was its own number. So, much like how we came up with the negative and irrational numbers, Bombelli had just discovered the complex numbers.

This point could be described as the beginning of complex numbers. Now, everyone **had** to accept that square rooting a negative number meant something, as it had just solved this mystery and is the solution to many of the problems that have plagued mathematicians for centuries. Bombelli later went on to formalise this discovery and develop some of the algebraic rules.

Several more mathematicians over time made significant contributions to this branch of mathematics, developing the idea, terminology and notation of how it is introduced and taught in schools nowadays. Carl Friedrich Gauss introduced the term *complex number* in 1832, while René Descartes (1596-1650) before him split these numbers into “real” and “imaginary” parts (imaginary parts are just the part of the complex number with the negative square root). Leonard Euler (1707-1783) was the first to use the notation for the square root of -1. To make complex numbers, we just factor out an whenever it turns up and basically treat it like its own number or identity. He also went on to find and prove more relations linking complex numbers to other areas of mathematics, such as the trigonometric function, which all play a big part in the famous Euler equation:

e^{i\pi}+1=0

This equation links almost everything we’ve talked about in this article into one nice equation. It relates rational numbers, irrational numbers like e and , the number 0 and the imaginary number , showing that all of these numbers are linked to each other.

When I first read about Eulers Identity it was because of how it was compared to a Shakespearean Sonnet 😍😍. It’s just the most amazing thing ever in maths. Probably the most interesting part of A level Further Maths as well 😂 nice work 👍👍

Great work bro! Super interesting article, an absolute pleasure to read!

Great work bro! Super interesting article, an absolute pleasure to read!!