Negatives (i)

There is a long history behind the introduction of negative numbers, which I am not going to deal with here. Instead, I’ll use them to demonstrate one of the methods by which mathematicians extend the concepts of number.

So far, we’ve added zero to the numbers, giving this:

0, 1, 2, 3, 4, 5, 6, 7, 8, …

These are the natural numbers. Sometimes 0 is excluded from their company, but I’m going to include it for convenience.

It’s easy to see that when you add two of these numbers, the result is always another natural number:

1 + 1 = 2
3 + 4 = 7
89 + 1 = 90

However, if we try subtracting, we run into trouble:

1 – 1 = 0 (OK)
3 – 4 = ?
89 – 1 = 88

If the number subtracted is greater than the number we are subtracting from, we can’t answer the question with any of the numbers in the original set.

One way of describing this situation is that addition is closed over the natural
numbers, but subtraction is not. “Closed” means that the operation keeps you within the domain under question: this has an appealing symmetry and elegance, and can also lead to much greater generalisations than what we’re describing today.

So there are two options here: we can say that subtracting a greater amount from a lesser amount has no answer, so Don’t Do That.

The second and more productive way is to say: OK, so far, 3 – 4 has no answer. Let’s pretend that it does, and then see how that pretend number behaves.

We’ll invent an arcane notation with which to depict our pretend number: [3-4]. Let’s also pretend that it can add and subtract in the same way that our honest, god-fearing natural numbers do, although we have to make up a pretendy rule for this. In a sum, we can expand our pretend number [3-4] into 3 – 4, and see if we get anything sensible as a result.

[3-4] + 1 = 3 – 4 + 1 = 4 – 4 = 0

[3-4] + 2 = 3 – 4 + 2 = 5 – 4 = 1

[3-4] + 5 = 3 – 4 + 5 = 8 – 4 = 4

[3-4] + 10 = 3 – 4 + 10 = 13 – 4 = 9

[3-4] + 0 = 3 – 4 + 0 = [3-4]

It seems that if we add our pretend number to any number other than zero, we can get a natural number as an answer. The other pattern that stands out is the relation between the natural number we’re adding to [3-4] and the answer:

1 -> 0
2 -> 1
5 -> 4
10 -> 9

We could construct lots of other pretend numbers which will give exactly the same results in the above as [3-4]: [7-8], [5-6], [999999-1000000]. For reasons of economy, let’s take the smallest of these, which is [0-1], and allow it to come out from behind the curtain as:

–1

And once we’ve allowed a single negative number to take off its brackets, the rest follow. Every expression like 3 – 4 can be expressed by [3-4] and reduced to its lowest form:

[0-2] = –2
[0-3] = –3
[0-4] = –4
[0-5] = –5

Both addition and subtraction are now closed, since any expression X + Y or X – Y will have a value Z which is also from our set.

We could also say that subtraction of a number N is the same as addition of −N. This means that addition and subtraction are actually the same operation.

Negative apples

The idea of negative numbers met with resistance: if natural numbers refer to collections of Things, then what can a negative number stand for? Zero can mean “no apples”, but what could be represented by −1 or −2 apples? You can get around this by regarding −1 as an operation, so that it means “take one apple away”. This did not convince everyone, and there were still a few holdouts against negatives up until the 19th century.

The other way of looking at negative numbers, and the one which was really responsible for their adoption, was that they give a natural way to represent credits and debits.

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