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I'm a high school student, who started learning about complex numbers literally today. Sorry if I say anything naive.

My question is this: sqrt(-1) x sqrt(-1) = sqrt(1) = 1 [simple algebraic manipulation]

however i = sqrt(-1), and is literally defined as i^2 = -1

yet the algebraic result says that i^2 = 1 ??

have i done a mistake somewhere? if not, can someome explain this?

rohiths18
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  • Well, as you venture into the complex number territory, the said simple algebraic manipulation ceases to be simple, or indeed possible. – Ivan Neretin May 22 '20 at 06:07
  • See e.g. https://math.stackexchange.com/questions/438/why-sqrt-1-times-1-neq-sqrt-12 or https://math.stackexchange.com/questions/49169/why-sqrt-1-times-1-neq-sqrt-12 – Minus One-Twelfth May 22 '20 at 06:08

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You are making a mistake first made by Euler in 1770 (see https://www.jstor.org/stable/27642191). In fact, the rule $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$ does not apply when both $a$ and $b$ are negative. Therefore, you cannot say $\sqrt{-1} \times \sqrt{-1} = \sqrt{1}$.

Prasiortle
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  • Thats interesting. Historically, do you think, mathematicians only decides that said algebraic rule doesn't apply to cases where both are negative, was only discovered/implemented AFTER Euler made this mistake? In other words, did they modify the rule so it stays consistent with complex numbers, or is there a completely different, unrelated to complex numbers, reason why the rule has said condition? If so, what is that? Thanks – rohiths18 May 22 '20 at 06:12
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    Well, this gets into some rather deep complex analysis. The issue is that it's not entirely clear how to unambiguously define the square root for negative numbers. With real numbers, we have the convention that you take the positive square root, so $\sqrt{9} = 3$ and not $-3$. But with e.g. $\sqrt{-1}$, you have no easy way to distinguish between $i$ and $-i$. It turns out that, for deep reasons, if you want to have an unambiguous way of defining $\sqrt{}$ for all numbers, then you have to give up some desirable properties, in particular the property that $\sqrt{a}\sqrt{b}=\sqrt{ab}$. – Prasiortle May 22 '20 at 06:18
  • See also https://math.stackexchange.com/questions/144364/is-the-square-root-of-a-negative-number-defined and https://artofproblemsolving.com/community/c5h1207196p5973817 – Prasiortle May 22 '20 at 06:28
  • Complex numbers (apart from zero) have two square roots each. A square root of $a$ times a square root of $b$ is.a square root of $ab$, but it may not be your favourite square root of $ab$. @rohiths18 – Angina Seng May 22 '20 at 06:28