Why does (-1)*(-1)=(+1) ? It is arbitrary, and there really is no good reason. We could construct number lines that work differently, so that imaginary numbers never appear. Such alternative number lines would still allow us to solve the exact same physics and engineering problems. Sure, the computations would work differently, but the way we would measure and use the initial conditions in our equations would be different too. In the end, the predicted outcome would still be the same.
Imaginary numbers are an artifact of how our number line is constructed. We could construct alternative number lines where imaginary numbers do no exists. The computations involving such alternative number lines would be different, but the outcome would be the same.
This sounds profound, but is wrong on so many levels. In a sense everything about mathematics is arbitrary, but there's a consistency and structure that makes such a statement unhelpful and misleading.
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
Following this progression we see that it's natural in some sense to say that scaling by -1 means that 1goes to -1. And indeed, 2 goes to -2, and 73 goes to -73.
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.
Beautiful exposition. Even in the late 1700s, many mathematicians rejected the use of mere negative numbers, viewing them as anomalies which indicated that one had phrased a problem wrong to begin with. On the other hand, Euler understood everything very well and even calmly explained how to take logarithms of complex numbers, which bewildered most of his contemporaries.
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.
Thanks for this citation. Forward, backward and lateral.. for some reason, this is what make the most sense to me in all these explanations of complex numbers. I guess you could also go upward/downward. And then, in a fourth or even nth dimension.
That's a mix of complete nonsense and meaningless truism.
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.
You are assuming that additive inverse is unique. Also 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.
> You are assuming that additive inverse is unique
Suppose that some number a has two additive inverses,x and y.
a + x = 0 = x + a
a + y = 0 = y + a
Consider
x + a + y = (x + a) + y = 0 + y = y
x + a + y = x + (a + y) = x + 0 = x
Thus x=y, and so the additive inverse is unique.
> you are assuming that -1(-1) is some value at all;
We can define it to have a value and then derive the properties of the value. We can show it to be consistent by creating an explicit model. Such things are relatively easy to do, but require a level of detail inappropriate for this context.
> 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.
Additive inverses are unique in any ring. The subtle things I skipped over because of the technicalities involved is how do I know the distributive property holds for negative integers. Indeed, what is a negative integer? How does one get them from the natural numbers? Suffice it to say that this can all be defined in a consistent, precise way and everything works out.
Actually, we only think that it can be described in a consistent way. We have proven it consistent using set theory , but Godwell's theorems tell us that we cannot prove a system to be consistent without using something outside of said system (unless the system is inconsistent). This means that there is some level in our chain of proofs that cannot be proven consistent.
Maybe I am very wrong on this issue. The arguments outlined below demonstrate that given a set of axioms or properties, there is only one consistent way to define (-1)*(-1).
Axiomatics are entirely arbitrary if you will. Thats the whole point: I restrict myself to some basic rules only because that allows me to show other potentially useful things, consequences and applications.
Imaginary numbers are an artifact of how our number line is constructed. We could construct alternative number lines where imaginary numbers do no exists. The computations involving such alternative number lines would be different, but the outcome would be the same.