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.
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.
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.