Whole numbers are one, two, three, and so on.
Theyíre the answer to questions like ìHow manyÖ?î, ìHow muchÖ?î ìWhich one of
theseÖ?î
Actually, the answer to ìWhich oneÖ?î would be ìthe thirdî or ìthe seventhî, rather
than ìthreeî or ìsevenî.
Thatís an important technical difference and it needs to be taught separately: theyíre
ordinal numbers rather than cardinals.
But for the moment, in this course, weíre really just interested in cardinal numbers.
But while weíre talking philosophy, notice also that number itself is an abstract concept.
The thing we write on paper is not a ìnumberî, but a ìnumeralî.
Itís a symbol that represents a particular number.
The symbol itself is quite arbitrary, and Iíll show you some alternatives in a minute.
We use the Arabic base-ten system.
For example, for the number thirteen, we use the Arabic digits one and three, written in
that order.
But we can combine the same digits in a different way to get a very different number.
For example, one hundred and thirteen is written by combining two ones and a three.
The meaning of the digit one changes depending on where we write it: the first one means
one hundred, the second one means ten ó and the three means just three, in this case (but
it could also mean thirty, three thousand or even three hundredths, depending on where
itís written within the numeral).
This is exactly the same as in normal written languages: the digits are our letters, the
numeral is the word and the number is the meaning of the numeral.
But there are a whole lot of other possible ways to represent whole numbers.
The simplest is just to hold up your hand and start counting on your fingers.
(The word ìdigitî is actually a Latin word meaning ìfinger or toeî!)
You can represent numbers by a collection of objects ó thatís usually the first context
you learn to count with.
Gradually real objects are replaced with counters or something else arbitrary to represent them,
then tallies and number lines, then written symbols, building abstraction layer by layer
until eventually you get to algebra, where a single symbol can denote a whole range of
possible numbers.
But the written symbols donít have to be our Arabic digits.
Here are some tiles from my Mah Jong set.
The blue symbols at the top of each tile are the Chinese representation of the green Arabic
digits in the top-left corner.
Essentially, these are a different graphical representation of those whole numbers.
In fact, if you look at the first row, youíll see that the numbers one to three are drawn
using ìdigitsî that are just horizontal lines.
The symbolic representation is arbitrary; these are just different ways of representing
the same number.
As a teacher, also look for ways to represent numbers using different positions.
Get your students up and moving around and actively engaged in building physical and
volitional representations of numbers.
Make them gather in groups of particular sizes, stand in a position that represents the relative
size of a number, step forwards and backwards along a physical number line to represent
adding and subtracting.
There are also many good ICT tools around for representing and manipulating whole numbers.
These are especially good if you have a decent interactive whiteboard in your classroom.
Ok, so hereís a few hard questions about whole numbers.
Whatís the first whole number?
You might be surprised to find out that people disagree on this.
If weíre counting how many of something there are, then obviously the first number is one.
But if you want to say that you donít have any yetÖ well, then you need zero.
Most mathematicians would say that the first and smallest whole number is one.
Zero isnít a whole number, because you donít actually have a whole of anything!
And indeed, the ancient Romans didnít even have a way to represent zero in their number
system.
Okay, so whatís the biggest whole number?
How many whole numbers are there?
Ask a child and youíll get any of a wide range of possible answers depending on their
age and level of educationóusually a nice round number or one either sideóbut the simple
reply to that is just to add one to whatever they say.
And of course that gives us the real answer to this question: there isnít a biggest whole
number.
Now, you might try to say ìinfinityî.
But be careful: infinity isnít really a whole number.
How can you have an infinite number of real objects?
Youíll always have a particular finite number of them ó it might be large, but itíll still
be a finite number.
Infinity itself is a very tricky concept and it doesnít follow the rules of normal numbers.
I spent a whole semester studying infinity at university, and trust me, itís a very
strange number!
So the best answer to ìwhatís the biggest whole numberî is really just that there isnít
one: whatever big number you find, I can give you one thatís bigger than it.
Okay, so how do you compare whole numbers?
How can you put them in order from smallest to largest?
The simplest way is just to make a number line.
Starting at one (or zero, if youíre a person who believes zero is a whole number), lay
out the numbers in the relevant position.
By convention we put the smaller numbers on the left, and the larger numbers on the right.
(Or if youíre going vertically, put the smaller numbers on the floor, and the larger numbers
up towards the ceiling.)
I mean, the direction is essentially arbitrary; it doesnít really matter.
But we want to reinforce some simple conventions to make it easier for our students to work
reliably and systematically.
And we read from left to right, so why not count in that direction too?
And later on, when they get to coordinates on graphs, and gradients leading to algebra,
the same conventions theyíve used for many years can support and guide them as they explore
new ideas.