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August 7th, 2018, 05:23 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 05:29 AM.
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August 7th, 2018, 06:58 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 01:34 AM.
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