Understanding Number Systems

Whichever IB Mathematics level you are following, you need to understand the concept of number systems.

The simplest system is the set of Natural Numbers, sometimes called the Counting Numbers. Using set notation, these numbers are: {1, 2, 3, 4 ……} and are called the natural numbers because they are the only ones which occur naturally. Negative numbers are an invention (you can’t have less than nothing), although we can easily use negative numbers to represent real things. For example, a temperature of -10°C is real enough, but only relates to an artificially created zero (the freezing point of water); it’s impossible to have temperatures below absolute zero. Similarly, it’s possible to represent an overdraft at the bank using negative numbers, but there are no such things as negative banknotes!

Zero is itself a man-made number, but it has become common to include it in the set of natural numbers, so our set above should actually read {0, 1, 2, 3, 4…}.

Before continuing, let’s have a look at the symbols for the different number systems:

NumberSystems

After Natural Numbers comes the set of Integers = {…-3, -2, -1, 0, 1, 2, 3 …}, in other words, all the whole numbers. You’ll notice that all Natural Numbers are also Integers, a fact we can express in set notation as:

NumberSystems2

Rational Numbers are fractions (rational here meaning “to do with ratios” rather than “logical”). More specifically, the numerator and denominator of a rational number must both be integers, and the fraction must be in its simplest form. So, \frac{{1.8}}{{3}} is not a rational number, nor is \frac{{2}}{{4}} since it is not in simplest form – in fact, it’s just another way of writing \frac{{1}}{{2}}, rather than a number in its own right. When written as decimals, rational numbers will either terminate (eg 1.875) or recur (eg 0.\dot 14\dot 2): in fact, all recurring decimals can be written as fractions. Worth noting is that all integers are in fact rational, since any integer can be written as a fraction with denominator 1, and hence fits the rational definition.

So any numbers which can’t be written as fractions in their simplest form – as decimals, they are non-terminating and non-recurring – aren’t rational, and these form the set of Irrational Numbers. These include the square roots of all integers which aren’t perfect squares (eg \sqrt{10} or \sqrt{55}), and well-known numbers such as \pi and e. However, don’t get the impression that there aren’t many irrational numbers: between any two rationals there will be an infinite number of irrationals; and between any two irrationals there will be an infinite number of rationals – mind-boggling!

Put together all the rationals and all the irrationals and you will get the set of Real Numbers: basically, all the numbers we deal with on a daily basis in Mathematics.

Unless you are an HL Student, in which case you will also have to cope with Complex Numbers, those based on \sqrt{-1}. Incidentally, real numbers are a subset of complex numbers since any real number a can be written as a +0i.

To tie this all together, I have drawn a Venn Diagram showing how all the number systems are related.

NumberSystemsVenn

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