# Exclaim explain

Q) A pupil asked me the other day why 0! = 1. Can you enlighten me?

For non-mathematicians let me explain what is meant by a factorial number.

This is best done by looking at the pattern formed by the numbers: 1! = 1 = 1 2! = 1 x 2 = 2 3! = 1 x 2 x 3 = 6 4! = 1 x 2 x 3 x 4 = 24 We can reverse this to get: 4! V 4 = 3!

3! V 3 = 2!

2! V 2 = 1!

1! V 1 = 0! ( 1! V 1 = 1) By looking at the pattern we arrive at the conclusion that 0! = 1 This is written in general terms as a definition: 0! = 1 and n! = n(n - 1)! for n

0.

On the calculator this is usually written x! To find 3!, press 3 on the calculator and follow this with the x! button; you might need to press = afterwards to get the answer.

The ability to work out 0! is useful when working out combinations, as this involves calculating factorials. For example, the number of combinations of r objects out of n objects is given by the following formula The formula has to work for all cases so let us consider a very simple case: that of the number of possible combinations of 1 object from a set of 1 objects. Substituting into the formula gives us: We know that there is only one way for this to happen so this tells us that 0! has to have a value of 1 to make the formula work.

I was working with one of my A-level students who had been tackling revision papers in preparation for his exam. One of the questions asked him to show that a particular quadratic had "real and distinct roots". He asked me what "real" meant and what where the roots called if they were not real.

The square roots of 1 ( C 1) are +1 and - 1; the square roots of - 1 (C - 1) are +i and - i. The square root of a positive number is real.

The square root of a negative number gives rise to another set of numbers that mathematicians have called imaginary numbers.

The reason suggested for the term "imaginary" is that mathematicians had doubts as to their validity as numbers, so the name stuck.

The two sets of numbers, "real" and "imaginary", are mutually exclusive (except for 0 which could be considered as both, though it is generally accepted as a real number).

When numbers are made up of both real and imaginary parts, for instance 6 + 3i (6 is the real part and 3i the imaginary part) a new set of numbers is created, and these are called complex numbers.

A while ago I was listening to Radio 4's Open Book programme in which they were discussing a book related to this very topic: Imagining Numbers (particularly the square root of minus fifteen) by Barry Mazur (Allen Lane pound;9.99).

The book was written for non-mathematicians "who may wish to experience an act of mathematical imagining". It's a good read for the non-mathematician teaching A-level maths as it provides an alternative perspective in accessible language.

Finally, on the topic of numbers, a poetic thought on integers: Integers : Count each amount that is whole.

+4 There are no bits attached.

+3 Count in opposite directions +2 So the numbers are hatched.

+1 Positive and negative numbers 0 Seemingly dispatched - 1 With zero in the middle.

- 2 Opposites are thus matched - 3 As they head towards infinity.

- 4 Together the numbers batched : By definition called "Integers".

Wendy Fortescue-Hubbard is a teacher and game inventor. She has been awarded a three-year fellowship by the National Endowment for Science, Technology and the Arts (NESTA) to spread maths to the masses.www.nesta.org.uk Email your questions to Mathagony Aunt at teacher@tes.co.ukOr write to TES Teacher, Admiral House, 66-68 East Smithfield, London E1W 1BX

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