Types of Number

Pick a number between 1 and 20

If you were given the instruction above, the person giving it would probably be surprised, to say the least, if you were to answer "π", or "√2", but from a mathematical point of view these would both be perfectly good responses. Rather as the word "animal" is frequently used as a synonym for "mammal", as opposed to "bird" or "fish", the word "number" is used as a synonym for the much less common word "integer". Mathematicans use many adjectives to differentiate between different types or class of number. Here are a few of them:

  • Natural
  • Whole
  • Perfect
  • Prime
  • Square
  • Triangular
  • Positive
  • Negative
Don't worry if you don't know what most of these words mean when applied to numbers

Unlike most of the terms used by mathematicians, many of the words above have rather emotive connotations. Were such things possible, perhaps we would rather buy a used car from a "natural" rather than an "irrational" number. We would prefer a numeric bank manager to be a "real" rather than an "imaginary" (or possibly not, if we were heavily in debt). Those of a spiritual tendency might listen eagerly to the musings of a transcendental, and despise a more practical square. No doubt most of us would try to avoid a complex number altogether, or suggest that it, and any miserable negative we came across, consult a therapist.

Of course, all these adjectives have a very specific mathematical meaning when applied to numbers, and we should not let the non-mathematical usage of the words influence our thinking. However, it is certainly arguable that these connotations do affect how we feel about numbers - perhaps we are likely to think that some of the more intimidating types of number are best left to the experts. Some of these adjectives reflect the fact that new types of number have initially been treated with great suspicion (not to say confusion) by mathematicians themselves, and have only gained wide acceptance once they are found to be useful, and some way of understanding them more intuitively has been found.

In this context I am particularly sorry for the imaginary and complex numbers, whose most famous representive is √-1, better known as i. In most ways complex numbers are no more difficult to deal with than the so-called reals, who must have had a particularly good P.R. agent when they acquired their name. There are straight-forward ways of calculating with complex numbers. Learning how to use them is very like learning to calculate with rational numbers (i.e. fractions) when you already know how to calculate with whole numbers. Similarly, the process by which mathematicians construct the complex number system from the rational or real number system is analogous to the way in which the rational numbers are created from the integers (the integers are the whole numbers, both positive and negative). If anything, it is the real numbers which are the most difficult to grasp intellectually, and which have the most paradoxical properties.

There are many books, and many web sites, where you can learn about the hierarchy of number systems, and how the integers are constructed from the natural numbers, the rationals from the integers, the reals from the rationals, and the complex numbers from the reals. Wikipedia pages on mathematics seem particularly good, and I will be including links to them in many places.

The next few pages summarise this process, pointing out a few significant facts along the way. Note that when mathematicians want to specify whether some number is natural, real, complex or whatever, they usually do so by designating the set it belongs to using a single capital letter, usually bold and in a distinctive type face. I have put the standard letter used for each type of number at the start of each section. We begin with the Natural, or Whole, numbers.