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This statement is not an axiom of arithmetic; it is proven from the axioms of arithmetic, and the proof only works for when a and b are non-negative reals. When a and b are not non-negative reals, the proof does not work and the identity is in fact incorrect.
Therefore, I reiterate my reiteration: there was no creation of a special case. The truth for a and b non-negative and the falsity otherwise are both deduced from the axioms of arithmetic.
There is nothing wrong with this and it is not unusual.
To help show this, here is another identity:
a^2 = a*|a| for a non-negative.
|a| means the absolute value of a, which means its size irrespective of whether it is positive or negative. You can write it as sqrt(a^2) if you want.
This is exactly analogous. The identity is not an axiom of arithmetic: it is proven from the axioms of arithmetic, and the proof only works when a is non-negative. If a is negative, the proof fails, and the correct identity is in fact
a^2 = -a*|a|
What your argument says is that we have invented a special case because the identity doesn't work for negative numbers; and therefore, negative numbers are somehow fake. As you don't believe that negative numbers have the same problem as complex numbers, hopefully you now see why your argument doesn't make sense.
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Originally Posted by
Universal Mind
That exception is even more bizarre than the other one. Getting the square root of a fraction by putting the square root of the numerator over the square root of the denominator seems like it should always be the logical method because squaring a fraction involves getting the square of the numerator over the square of the denominator. So, going in the reverse direction, it would make sense to get the ratio of the square root of the numerator to the square root of the denominator. But what we have run into is a situation of, "Oh no, we are dealing with imaginary numbers, so we can't do that here." You answered my question, but now I am even more convinced that the "proof" is a reductio ad absurdum of imaginary numbers.
I am repeating myself yet again by saying this: the identity does not apply to matrices either. If it 'disproves the existence of complex numbers' then it disproves the existence of matrices too. You repeatedly insist on referring to it as if it is some kind of ancillary axiom necessitated by consistency when it has been clearly explained to be no such thing; like all mathematics, propositions are proven from the axioms. This particular proposition can be proven for non-negative integers, but it doesn't happen to go through for other entities.