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What is a number? |
What is an integer?I conceived of this topic when I asked myself the question: What is an integer? This was when I was looking at Fermat's Last Theorem and thought I needed to define an integer in order to make progress. (See M1, the Whole-number right-angled triangles article)Definitions of 'integer' (For example, in Wikipedia) don't help. ( ... -3, -2, -1, 0, 1, 2, 3, etc.) Definitions of 'number' (For example, in Wikipedia) don't help. I mean, what is it about integers that makes them different from all the other types of number on the number line? What makes them special? I suppose it boils down to: What is the number 1 ? This must be one of the basic questions in mathematics.Is the number one a starting point - an axiom - indefinable? You could say 1 = n0 but you have to know a lot of maths before you get to that stage. Imagine Imagine a horizontal row of pigeon holes or Slots stretching from left to right as far as the eye can see in either direction.
We can look at some of the properties of the Number Line without specifying any particular number. For instance, a slot with a particular value is equivalent to an Ordinal Number by virtue of its position in the row. The difference in value between the No Value Slot and any other slot is equivalent to a Cardinal Number, in the sense that arithmetic can be performed using such Differences in Value. The dimensions of Ordinal and Cardinal Numbers I have made a distinction between the Value of a number, it's position in the Number Line - which is zero-dimensional, and a number we use for arithmetic - which is, in this context, one-dimensional. Zero is a special case - a number that is zero-dimensional. So, Ordinal Numbers which are equivalent to the Values of the slots are zero-dimensional. Cardinal Numbers which are equivalent to a finite length of the Number Line are one-dimensional. Now we can do some arithmetic. Redefine the Slot with no Value as the ‘Zero Slot’, let it contain the Symbol 0. To add one number to another:
The Counting numbers So far I have made no mention of other numbers. How do we decide which slot represents the number one? It seems to me, that the only way, is to arbitrarily select a Slot a finite distance to the right of the ‘Zero Slot’, let it contain the Symbol 1, and call it the ‘One Slot’. So, to answer one of my original questions, “How do you define the ‘number one’ mathematically?” The answer is, “you don’t”. It has to be a starting point.” Add the number 1 to the number 1, as described earlier and define the answer as the number 2. Let its slot contain the symbol 2. Continue this process till you have as many numbers as you wish. Call these the Counting Numbers. How we learn to count As toddlers, we are taught to count by our parents. It goes something like this: Parent points to an object, and says, “Onnnne”. Parent points to another object, and says, “Twoooo”. Parent points to another object, and says, “Threeee”. Parent points to another object, and says, “Foooour”, etc. This sequence is like an ordered list. After a while we associate the first object in a sequence with “Onnnne”, the second object with “Twoooo”, etc. This is equivalent to giving a Slot in the Number Line a Symbol (The sound of each number). The sequence always comes in the same order. Two always comes after one, three always comes after two etc. This is equivalent to giving a Slot in the Number Line a Value. Later we realise that after we have counted the fourth object then the number of objects we have counted is called four, etc. This is equivalent to adding 1 to 1, to 1, to 1, and getting the answer 4, or, associating the number 4 with the distance along the Number Line between the ‘Zero Slot’ and the ‘Four Slot’. We learn about concept of zero from the game ’ Peep–oh’. Now I’m here, now I’m not here, now I’m here. Not here – here. No object – object. Nothing – thing. Zero – one. Numbering the slots Start with the zero slot. Question: How do we define which slot is the Zero Slot? Answer: Choose any slot, at random. Question: How do we define the slot representing the number 'one'? Answer: Choose any slot, a finite distance to the right of the Zero Slot. Question: How do we define the slot representing the number 'two'? Answer: Perform the addition procedure. Add one to one. The answer will be the slot representing the number 'two'. The slots representing the other counting numbers can be defined by continuing this process. We have now defined the Counting Numbers on the Number Line. Mike Holden - Mar 2010 |
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