Algebra, Analysis

Types of Numbers, part I

Well it was about time for a Maths related post, after all those introductions, right? My first one will introduce the very basics, the simbols we cannot live without: numbers. A number is just a number, right? Well, not really! Numbers are much like people: they belong to different groups. Just like a high school will have cool kids, the geeks and so on, so do numbers. In fact, some numbers are squares, some are perfect and one is even golden!

First of all, we should talk about number sets. Understanding the differences between the main types of numbers makes Maths way easier to get. I have to admit sets weren’t my friends when i met them, but my awesome personality made them like me. I came to the conclusion that numbers sets evolved with society and its needs.

Let’s go back  to 1000 BC and say that we’re growing trees. We count 1,2,3,4,5,.. apples, which actually represents the most basic of number system, the ‘natural’ or ‘counting’ numbers. These numbers worked very well for many years until the man got bored of what he had and looked in his neighbour’s yard and wanted his apples. But he wanted more apples than he had to exchange so his neighbour said ‘it’s ok, you can have them on debt’, which actually meant you had -n apples. Bam! Suddenly we got negative numbers. The positive and negative whole numers are called ‘integers’. The integer set is …, -3, -2, -1, 0, 1, 2, 3,.. and can also be expressed as 0, ±1, ±2, ±3,… and so on.

As society evolved, it had various products to consume so people couldn’t eat a whole apple or a whole bread, and they started to cut it into pieces and eat portions of it. Half an onion with 2 and a half potatoes , one and a half tomatoes and 3,5 l of water make an excellent soup!  These are fractions, or rational numbers. This set includes any number that can be expressed in the form ab, where a and b are integers and b cannot equal zero – dividing by zero is a big no-no! Another way to express this is to say that rational numbers can be any ‘terminating’ or ‘repeating’ decimal. For example, 1/4 is equal to 0,25 (‘terminating’ decimal) and 1/3 equals 0,(3) (‘repeating’ decimal). But the debt idea is still valid so the rational numbers can be negative too.

As years passed by, the industry has appeared and with it more complicated operations which didn’t stop at division. All these other numbers that appeared and that couldn’t be expressed as a fraction, are called ‘irrational’ numbers. Two good examples of irrational numbers are π and √2 (square root of 2); these are weird numbers because they continue forever, without repeating or terminating.

Now if we take a final look at these sets we can see so many relations between them. The natural set is included in the integer set. The integer set is included in the rational numbers set. I always imagined these sets as Russian dolls, the irrational set being separated from the others.

russian dolls

Another important thing we didn’t talk about is zero’s place. He’s not a part of the natural set because it’s not natural to count nothing, but he is a whole number. We use zero on a daily basis, but few of us know its significance. Zero is a vital part of our place-value system. Without it 206 and 26 would look very similar indeed. Although this might seem obvious to us now, the theoretical leap required to develop a symbol that represents nothing is very impressive –  and neither the ancient Greeks nor the Romans had a representation of zero. The Indian mathematician Grahmagupta was the author of the first text to treat zero as a number. It’s sometimes said that you cannot ponder the infinite until you have pondered zero. In fact, this pondering of zero and the infinite is a big part of calculus – the nightmares of almost every student.In essence, calculus is used in science, economics and engineering to look at the infinitely large and small. So, not to put too fine a point on it, the appearance of zero was a huge moment in the history of Mathematics.