Solving a Paradox Involving Complex Numbers

Question

So I encountered an exercise in my algebra textbook and it is somewhat paradoxical. Here is the exercise:

$$1 = \sqrt{1} =\sqrt {(-1)(-1)} = \sqrt{(-1)} \sqrt{(-1)} = i^2 = -1$$

I think it has to do with the first step of the problem. The number $\sqrt{1}$ shouldn't simplify to just 1. Is it possible that $\sqrt{1}$ can also be $± 1$?

Edit: I wasn't aware that someone else had already asked a similar question to what I just asked.

you cant split the roots when numbers inside are negative – asddf Feb 03 '17 at 00:33 — asddf, Feb 03 '17 at 00:33
Yes that is what Arnold said you both are correct. – John W. Smith Feb 03 '17 at 00:36 — John W. Smith, Feb 03 '17 at 00:36

score 2 · Accepted Answer · answered Feb 03 '17 at 00:34

2

$$\sqrt{ab}=\sqrt{a}\sqrt{b}$$

That is true only if $a\ge 0$ and $b\ge 0$.

answered Feb 03 '17 at 00:34

Arnaldo

21,342

I see. So the middle part of the string of inequalities is false then? – John W. Smith Feb 03 '17 at 00:36
Yes, it is wrong! – Arnaldo Feb 03 '17 at 00:37
1

Okay now it makes sense! I knew it had to do with separating the square roots but I wasn't so sure so I decided to ask it as a question. – John W. Smith Feb 03 '17 at 00:38
1

@Oliver821: feel free to ask everything you want. – Arnaldo Feb 03 '17 at 00:39
1

@Arnoldo in the future yes but for now everything is clear. – John W. Smith Feb 03 '17 at 00:41

Solving a Paradox Involving Complex Numbers

1 Answers1