Consider.

If you're content with the counting numbers then we can construct the negative numbers. These have the specific property that when added to the positive number of the same size we get zero,

But most people are happy with the integers, so move on. We're reasonably happy with addition, but what is multiplication? If you think of it as repeated addition you're screwed when you want to multiply by 2 1/2. It's better to think of it as a scaling. Multiplying by 2 means that you scale things up to be twice as big. Thus 1 goes to 2, 2 goes to 4, and 5 goes to 10. Also, -1 goes to -2, -6 goes to -12, and so on.

So what do we mean when we scale by -1? We look at the sequence of scaling by 4, then by 3, then by 2, and so on, each time asking where the number 1 gets sent.

    Scale by 4 and 1 -> 4
    Scale by 3 and 1 -> 3
    Scale by 2 and 1 -> 2
    Scale by 1 and 1 -> 1
    Scale by 0 and 1 -> 0

Scaling by -1 sends something to the same distance on the other side of zero.

So where does -1 get sent under a scaling of -1? It gets sent the same distance the other side of zero. -1 gets sent to 1.

Therefore it makes sense to say that -1 scaled by -1 is 1.

(-1) * (-1) = 1

Wecan use this to ask about the square root of -1. What geometric operation can we perform on the number line, such that doing it twice is the same as multiplying by -1?

An answer is to rotate anti-clockwise by 90 degrees. Another answer is to rotate clockwise by 90 degrees.

Pursue this, and you start to construct the Agrand diagram, and the complex numbers.

Someone once said that a lot of confusion could have been avoided if, instead of the terms positive, negative, and imaginary, they had instead used the terms forward, backward, and lateral.

Sure, imaginary numbers are based on the axioms for real numbers. But using other axioms ("number line" is a visualization aid appropriate for grade school, not for a serious mathematical discussion) would most definitely NOT allow us to solve exactly the same physics and engineering problems, or have the same outcomes for the same problems.

a·0 = 0 for all numbers a

Thus it is true that

–1·0 = 0

Replace 0 on left hand side with (–1 + 1) to get

–1·(–1+1) = 0

Now use the distributive property

–1·(–1) + (–1)·1 = 0

Keeping in mind that 1·a = a for all numbers gives us

–1(–1) + (–1) = 0

Thus –1(–1) is a number such that when added to –1 gives 0. Hence –1(–1) must be the opposite of –1 which is 1.

  > You are assuming that additive inverse is unique

    a + x = 0 = x + a

    a + y = 0 = y + a

    x + a + y = (x + a) + y = 0 + y = y
    x + a + y = x + (a + y) = x + 0 = x

  > you are assuming that -1(-1) is some value at all;

  > you could write an extremely similar proof that
  > leaves you with 0/0 = 1 and it would be faulty to
  > conclude that 0/0 actually is 1.