Definition 0.1.1 Sets of numbers
This is not really a definition, but you should know these symbols
- \(\mathbb{N} = \) the natural numbers,
- \(\mathbb{Z} = \) the integers,
- \(\mathbb{Q} = \) the rationals, and
- \(\mathbb{R} = \) the reals.
Before we do anything else, it is very important that we agree on the definitions and names of some important collections of numbers.
Natural numbers — These are the “whole numbers” 1,2,3,… that we learn first at about the same time as we learn the alphabet. We will denote this collection of numbers by the symbol “\(\mathbb{N}\)”. The symbol \(\mathbb{N}\) is written in a type of bold-face font that we call “black-board bold” (and is definitely not the same symbol as \(N\)). You should become used to writing a few letters in this way since it is typically used to denote collections of important numbers. Unfortunately there is often some confusion as to whether or not zero should be included ^{ 1 }This lack of agreement comes from some debate over how “natural” zero is — “how can nothing be something?” It was certainly not used by the ancient Greeks who really first looked at proof and number. If you are a mathematician then generally \(0\) is not a natural number. If you are a computer scientist then \(0\) generally is.. In this text the natural numbers does not include zero.
Notice that the set of natural numbers is closed under addition and multiplication. This means that if you take any two natural numbers and add them you get another natural number. Similarly if you take any two natural numbers and multiply them you get another natural number. However the set is not closed under subtraction or division; we need negative numbers and fractions to make collections of numbers closed under subtraction and division.
Two important subsets of natural numbers are:
Hence the number \(7\) is prime, but \(6 = 3\times 2\) is composite.
Integers — all positive and negative numbers together with the number zero. We denote the collection of all integers by the symbol “\(\mathbb{Z}\)”. Again, note that this is not the same symbol as “\(Z\)”, and we must write it in the same black-board bold font. The \(\mathbb{Z}\) stands for the German Zahlen meaning numbers ^{ 3 }Some schools (and even some provinces!!) may use “\(I\)” for integers, but this is extremely non-standard and they really should use correct notation.. Note that \(\mathbb{Z}\) is closed under addition, subtraction and multiplication, but not division.
Two important subsets of integers are:
Rational numbers — this is all numbers that can be written as the ratio of two integers. That is, any rational number \(r\) can be written as \(p/q\) where \(p,q\) are integers. We denote this collection by \(\mathbb{Q}\) standing for quoziente which is Italian for quotient or ratio. Now we finally have a set of numbers which is closed under addition, subtraction, multiplication and division (of course you still need to be careful not to divide by zero).
Real numbers — generally we think of these numbers as numbers that can be written as decimal expansions and we denote it by \(\mathbb{R}\text{.}\) It is beyond the scope of this text to go into the details of how to give a precise definition of real numbers, and the notion that a real number can be written as a decimal expansion will be sufficient.
It took mathematicians quite a long time to realise that there were numbers that could not be written as ratios of integers ^{ 4 }The existence of such numbers caused mathematicians (particularly the ancient Greeks) all sorts of philosophical problems. They thought that the natural numbers were somehow fundamental and beautiful and “natural”. The rational numbers you can get very easily by taking “ratios” — a process that is still somehow quite sensible. There were quite influential philosophers (in Greece at least) called Pythagoreans (disciples of Pythagoras originally) who saw numbers as almost mystical objects explaining all the phenomena in the universe, including beauty — famously they found fractions in musical notes etc and “numbers constitute the entire heavens”. They believed that everything could be explained by whole numbers and their ratios. But soon after Pythagoras' theorem was discovered, so were numbers that are not rational. The first proof of the existence of irrational numbers is sometimes attributed to Hippasus in around 400BCE (not really known). It seems that his philosopher “friends” were not very happy about this and essentially exiled him. Some accounts suggest that he was drowned by them.. The first numbers that were shown to be not-rational are square-roots of prime numbers, like \(\sqrt{2}\text{.}\) Other well known examples are \(\pi\) and \(e\text{.}\) Usually the fact that some numbers cannot be represented as ratios of integers is harmless because those numbers can be approximated by rational numbers to any desired precision.
The reason that we can approximate real numbers in this way is the surprising fact that between any two real numbers, one can always find a rational number. So if we are interested in a particular real number we can always find a rational number that is extremely close. Mathematicians refer to this property by saying that \(\mathbb{Q}\) is dense in \(\mathbb{R}\text{.}\)
So to summarise
This is not really a definition, but you should know these symbols