Is the product of 2 square roots of negative numbers a negative or positive value???

Question

I couldn't quite tell if the product of two square roots of negative numbers in ℂ (complex numbers) should be a positive real number or a negative one.. and this is because of the following two possibilities of solving that.

For example:

√−4×√−9=√(2i)²×√(3i)²
   =|2i|×|3i|
   =|6i²|
   =|−6|
   =6

Or

√−4×√−9=√(2i)²×√(3i)²
   =2i×3i
   =6i²
   =−6

So which one is correct and why?

The second is correct. In the first one you’re using the incorrect definition of the absolute value of a complex number — Adam Martens, Apr 10 '20 at 21:12
Oh I see.. can you tell me more about the correct definition of the absolute value? — Sabrine Bina, Apr 10 '20 at 21:25
Um... those are both real numbers. A negative number is a real number. — fleablood, Apr 10 '20 at 22:04
I am sorry for my mistake guys.. I edited it.. I meant to ask whether it is a positive or negative, and not real or complex!! — Sabrine Bina, Apr 11 '20 at 01:13
My answer on a similar question gives you a good idea: https://math.stackexchange.com/a/4849127/267917 — Dominique, Jan 23 '24 at 14:51

score 3 · Answer 1 · answered Apr 10 '20 at 21:14

3

Your dilemma comes from using the square root symbol as if it singled out a particular complex number, but that's not the case.

When $a$ is real and positive the equation $$ x^2 = a $$ has two real solutions. Just one of those is positive, and we call that one $\sqrt{a}$. The other one is $-\sqrt{a}$.

When $a$ is negative it has two complex square roots but there is no reasonable consistent way to call one of them $\sqrt{a}$, so that expression isn't used.

So for example both $2i$ and $-2i$ square to $-4$ but neither of them is "the" square root. Similarly, $1-i$ and $-1+i$ each square to $-2i$ but neither of them is "the" square root.

answered Apr 10 '20 at 21:14

Ethan Bolker

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Oh yes.. you are right!! How come that I missed that point .. so we say that sqrt(-4)=|2|i or what? – Sabrine Bina Apr 10 '20 at 21:26
1

You missed the point: "that expression", namely the expression sqrt(-4), "isn't used". – Lee Mosher Apr 10 '20 at 22:26
I am working with complex numbers here.. that's why it is used – Sabrine Bina Apr 11 '20 at 01:15
what is "the" square root of a negative number? – orangeskid Apr 11 '20 at 01:36
@orangeskid Sometimes $i$ rather than $-i$ is considered "the" square root of $-1$ but in general there is no preferred root of a complex number that isn't real and positive. – Ethan Bolker Apr 11 '20 at 01:55
I see, thank you. But if we consider this root function from $\mathbb{R}$ to $\mathbb{C}$ we cannot make it multiplicative. – orangeskid Apr 11 '20 at 04:01

score 1 · Answer 2 · answered Apr 11 '20 at 02:32

Adding to Ethan's answer: Beginners sometimes write things like "$\sqrt{-4}=\pm2i$", but this isn't actually an equation; it's shorthand for the sentence "If $z^2=-4$, then $z=2i$ or $z=-2i$".

You shouldn't do algebra with an expression like $\sqrt{-4}$ that doesn't denote exactly one number, because you'll quickly lose track of the "or".

For example, say we know that $a^2=-1$ and $b^2=-1$.

Is it possible that $a\times b=1$? Yes, because we could have $a=i$ and $b=-i$.

Is it possible that $a\times a=1$? No, because this is $a^2$, which we've assumed is $-1$.

But if you write $a=\sqrt{-1}$ and $b=\sqrt{-1}$ and substitute these "values" into the expressions above, then you get "$\sqrt{-1}\times\sqrt{-1}$" for both, and you've lost information. The best way to avoid this kind of confusion is to explicitly keep track of the cases instead of trying to use "$\sqrt{}$" (or "$\pm$") expressions to denote multiple values at once.

I understand now. Thank you. So the first expression is not correct. But that's what I was given in an exercise: √-4×√-9+√-2×√-8 and the answer is supposed to be -10 and I couldn't understand how they got this.. can you pleaaaase explain to me how? (If it's possible to solve it of course) — Sabrine Bina, Apr 11 '20 at 08:42
@SabrineBina We can't explain how this is the answer since the person who wrote the question did not really understand the subtlety of complex square roots and made some assumptions that work sometimes but not always. — Ethan Bolker, Apr 11 '20 at 11:19

Is the product of 2 square roots of negative numbers a negative or positive value???

2 Answers2