Different types of numbers and the birth of i

Numbers are something we use all the time, but we rarely pay attention to the different types of numbers or think about their history. The most common numbers that appear in day-to-day life are what mathematicians formally call the natural numbers. Those are the whole non-negative numbers — i.e., 0,1,2,3,4 etc. There are some debates in the mathematical community whether 0 should be included in the set of the natural numbers or not. Generally, it depends on the convention and also whether you are a number theorist or you work in another area of maths. The natural numbers are denoted by the letter ℕ. We call this “blackboard N”. There is a story explaining why it is used and how it came about. Back in the day when everything was either written on a typewriter and /or printed by a printing press scientists needed to distinguish between the standard letter N, which could be used to denote other mathematical or physical quantities, and the set of numbers used. So to create a new symbol they started typing the letter N twice — the second on top of the first, thus creating what we now refer to as blackboard letters.

There are numbers we use which live outside the natural numbers. The next bigger set we commonly encounter are the integers. The word integer comes from Latin and means “whole”. Those are the positive whole numbers, zero, as well as the negative whole numbers. So we’ve got -1, -2, -3, 0, 1, 2, 3 etc. The symbol for the set of integers is ℤ, coming from the German word Zahlen, which means numbers. We can see that the natural numbers are contained in the integers. The need for creating the integers became apparent when people realised they cannot do some basic arithmetic with the current numbers at hand, for example subtract 10 from 5. They did not have anything to denote the result by. Imagine what life without negative numbers would be — you will not have a way of describing you have a negative balance on your bank account for example. Maybe overdrafts wouldn’t even exist, who knows!

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The next step in expanding the domain of numbers is adding fractions (or decimals). Again, this is a logical step in the development of the number system since we clearly need a way to explain parts of a whole numerically. The set obtained from adding fractions to the set of integers is the rational numbers. They are denoted by ℚ, the notation originating from the Italian word quoziente, meaning “quotient”. (Very roughly speaking, a quotient is another way of calling a fraction — something that has a numerator and a denominator, i.e. something of the form x/y, where x and y are numbers). Also for those of you who are crazy about languages, rational comes from the word ratio, which again talks about proportion and is closely related to fractions.

Having the three sets of numbers we described above — natural, integers and rational, seems like all the numbers we need to cover our day-to-day arithmetic needs. However, unfortunately they are not enough if one wants to do science. Since ancient Babylon times it has been known that one needs a special number to calculate the length (or circumference) of a circle. The formula we use today is twice the number 𝛑 (this is the Greek letter pi, pronounced “pie”) times the radius of the circle. The first formula for finding the circumference can be traced back to Ancient Babylon. Back then, they first calculated the area by computing the radius squared and adding this to itself three times. This gave them the idea that there should exist a constant (i.e., a fixed number) with numerical value around 3, that is used to calculate both the area and the circumference. Pretty good guess considering this was done nearly 4000 years ago! The actual value of 𝛑 we know today is 3.14…

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(If you are interested to check out more in depth details about the history of the number 𝛑, I recommend this video — https://www.youtube.com/watch?v=gMlf1ELvRzc).

Nowadays, 𝛑 has become one of the symbols of mathematics. There are competitions where people try to memorise and recite as many digits of 𝛑 as they can, pushing the boundaries of the human brain further than we would have thought possible. There is even an International Pi Day on March 14th, celebrating the number (and an excuse for people to eat pie).

However, 𝛑 does not belong to any of the sets of numbers we defined above — it is neither a whole number, nor can it be expressed as a fraction. The first proof of this fact was done in the 1760s by Jonathan Heinrich Lambert, who was a German mathematician, physicist, philosopher and astronomer. This pointed out to the existence of another larger sets of numbers. We call those numbers irrational numbers, with the name correctly suggesting that those are numbers that are not rational, i.e., they cannot be expressed in a fraction (or finite decimal) form. However, they cannot exist on their own so mathematicians had to define an even larger set of numbers consisting of the rational and irrational numbers. We call it the real numbers. The set of real numbers is denoted by ℝ. In addition, the irrational numbers include all sets of roots of rational numbers, as well as other famous numbers like e.

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Again it seemed that mathematicians have everything they need by defining the set of real numbers. However, there was another final piece missing. There were no numbers such that when squared (i.e., multiplied by themselves) give a negative number. If we square a positive number, we get a positive number. If we square a negative number, we again get a positive number, since a minus times a minus is a plus. Thus, mathematicians introduced a new number, denoted i, defined by the property that it squares to -1. This gave rise to new type of numbers called “imaginary numbers”, denoted 𝕀. The need for such numbers was first conceived in mid 16th century, when scientists needed to understand solutions of polynomial equations. However, it wasn’t until 1637 when René Descartes coined the term imaginary. Thus, a new fundamental set of numbers, called the complex numbers ℂ, was introduced. A complex number has two parts — a real part and an imaginary part. Mathematicians write complex numbers as a + bi, where a and b are real numbers. This representation tells us that the real part has value a, and the imaginary part has value b. For example, 2+3i is a complex number. Due to the fact that the numbers have 2 parts, we can think of them geometrically as points in the plane.

Complex numbers play a crucial part in the development of modern mathematics. However, they caused a lot of dispute in the scientific circles when first introduced. There were people protesting against them and not seeing the need for their introduction. The complex numbers are indeed abstract, however, without them most of modern day algebra, number theory and physics would not exist. And we would have probably never had developed the quaternions.

Maths PhD | Writer | Educator