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Andante

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I've been having a little "problem" with the concept of equivalent mathematical statements. As far as I can tell, normally in algebra any given number multiplied by 1 (1n) and the given number (n) are equal, and therefore anything function you perform on (1n) can also be done to (n) with the exact same results. However, I have stumbled across a case where this does not seem to be true, and my algebra teacher won't even deal with the question because she says "we haven't begun to deal with imaginary numbers yet" (apparently implying that because the rest of the class doesn't even know they exist, I shouldn't even be thinking about them until she introduces them to the class :grumpy:)

Okay, so the issue she won't address is this:

1. If you multiply a number by 1, it has the exact same value as the number itself.

--So, 25 and 25(1) are the exact same thing.

2. Therefore, if you perform any given operation on these two equivalent statements, the answer should be the same.

--So, the square root of 25 and the square root of 25(1) are both 5.

3. Therefore, in theory you should be able to multiply a number by something that is EQUIVALENT to 1, and the number should still equal (1n) and (n).

--So, 25(i^4) and 25(1) and (25) all have the exact same value. [Where i is an imaginary number and i^4= (i^2)(i^2)=(-1)(-1)=1]

4. Therefore #2 should still hold true. But it doesn't seem to in this case:

--The square roots of 25 and 25(1) are both

My problem: As far as I understand, (or have been led to understand), the basic rules of mathematics are supposed to apply universally...at least within algebra. Identity statements should always remain identity statements, no matter what you do to the two equivalent statements. That is what we have always been taught in school, ever since kindergarten. So what I'm wondering is this: do the rules really NOT always apply, and we've been lied to all along, or do they actually still apply in this case and I'm just not seeing how?

And if the rules DON'T apply, then why is mathematics dealt with as "truth" when the truths it relies upon are not universal?

Okay, so the issue she won't address is this:

1. If you multiply a number by 1, it has the exact same value as the number itself.

--So, 25 and 25(1) are the exact same thing.

2. Therefore, if you perform any given operation on these two equivalent statements, the answer should be the same.

--So, the square root of 25 and the square root of 25(1) are both 5.

3. Therefore, in theory you should be able to multiply a number by something that is EQUIVALENT to 1, and the number should still equal (1n) and (n).

--So, 25(i^4) and 25(1) and (25) all have the exact same value. [Where i is an imaginary number and i^4= (i^2)(i^2)=(-1)(-1)=1]

4. Therefore #2 should still hold true. But it doesn't seem to in this case:

--The square roots of 25 and 25(1) are both

**5**, but the square root of 25(i^4) is 5(i^2) or 5(-1) or**-5**.My problem: As far as I understand, (or have been led to understand), the basic rules of mathematics are supposed to apply universally...at least within algebra. Identity statements should always remain identity statements, no matter what you do to the two equivalent statements. That is what we have always been taught in school, ever since kindergarten. So what I'm wondering is this: do the rules really NOT always apply, and we've been lied to all along, or do they actually still apply in this case and I'm just not seeing how?

And if the rules DON'T apply, then why is mathematics dealt with as "truth" when the truths it relies upon are not universal?

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