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August 7th, 2018, 06:23 AM   #41
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I do not know whether this idea contributes any light on the subject at all.

In the arithmetic of real numbers we define negative as follows:

$x \text { is negative } \iff x < 0.$

But the < is not defined with respect to complex numbers so does it even make sense to say that negative numbers have square roots in the complex numbers?

The Platonism implied in much discussion of mathematics seems to me to get in the way of understanding. To say that the real number - 1 has square roots, but that they are not real numbers at least seems like a category mistake and thus is confusing.

What seems to me less confusing is to say that a certain subset of the complex numbers behaves exactly like the set of real numbers. The complex number - 1 + 0i is a member of that subset, but that the solution of the equation x^2 + 1 - 0i i = 0 is not a member of that subset does not imply any contradiction.

The equation x + 1 = 0 has no solution in the system of natural numbers, but no one gets excited that it does have a solution in the system of integers.

The equation 6x = 13 has no solution in the system of integers, but no one gets excited that it does have a solution in the system of rational numbers.

The equation x^2 = 2 has no solution in the system of rational numbers, but no one (except Pythagoreans) gets excited that it does have a solution in the system of real numbers.

The equation x^2 + 1 = 0 has no solution in the system of real numbers, but it bothers people that it does have a solution in a different system. Odd.
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Last edited by JeffM1; August 7th, 2018 at 06:29 AM.
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August 7th, 2018, 07:58 PM   #42
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Originally Posted by Micrm@ss View Post
But all of that depends on whether you want the extension of $\mathbb{R}$ to be a field to begin with. That is a choice. A very good choice that I agree with, and a choice that gives many many good results. But it is still a choice to make.
This was, essentially, my point. If you decide that you don't want a field, you are basically making substantial changes to the system.

If you decide that you do want a field, you add a root of $x^2+1=0$ to the reals, give it a name, $i$, and set about finding its emergent properties.

Any other approach involves deciding what properties you want for your new number system.
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Last edited by skipjack; August 8th, 2018 at 02:34 AM.
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August 18th, 2018, 06:19 PM   #43
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Originally Posted by v8archie View Post
This was, essentially, my point. If you decide that you don't want a field, you are basically making substantial changes to the system.

If you decide that you do want a field, you add a root of $x^2+1=0$ to the reals, give it a name, $i$, and set about finding its emergent properties.

Any other approach involves deciding what properties you want for your new number system.
Here is the point I think you are missing.

How one looks at numbers and what they mean is a subjective thing.

One view of looking at the issue here is to see that x^2 = y is an operation and thus needs an inverse operation, but because for every result of x^2 there are two possible values for x, then so too are there two results for every y when put through the inverse operation.

Normally the two values of x would be x and -x and both would give +y.

But the question arises of how one can get -y. The answer is simply instead of
+x or -x,
you get +x and -x. (logical "or" vs logical "and")

Simply, if x^2 = +x * +x = -x * -x = y, then +x * -x = -x * +x = -y.

If you plot these functions as multiplication, you get four quadrants, two quadrants of same sign, and two quadrants of opposing sign. When rooting a positive number, you get only values of matching sign, but the same results except for opposing signs give the same answer except negative.

This could very well have been chosen as the solution to rooting negative numbers, in which case, it could have worked and simply have resulted in a very different set of emergent properties.

As a logic system, any emergent properties will be logical. Thus using the discovered emergent properties does not mean that they are the only valid way.

A good analogy I heard once, was that humanity is like a puddle. We see the hole we are in and become amazed at how perfectly well it fits us, ignoring the fact that we fit it so well because it shapes us, and that no matter what we do, we will always fit the hole. I'd say the same applies to math, any logical choice would have emergent properties that would seem natural and perfectly chosen from the perspective that made the choice in the first place.

Last edited by MystMage; August 18th, 2018 at 06:22 PM.
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