Which step is wrong?

Question

(1)$1=\sqrt{1}$

(2)$=\sqrt{-1×-1}$

(3)$=\sqrt{-1}×\sqrt{-1}$

(4)$=i×i=-1$

$1=-1$

Which step is wrong? I guess step 3 but I dont know why

Yes, you can't just go from (2) to (3), because you haven't proved that this power identity holds for complex numbers (and in fact this example proves that it doesn't if you define $\sqrt{}$ this way). — Izaak van Dongen, Feb 23 '20 at 09:29

score 2 · Answer 1 · answered Feb 23 '20 at 09:30

2

The third step because $$\sqrt{-1}=\{i,-i\}$$ and we can not write $$\sqrt{(-1)(-1)}=i\cdot i.$$

answered Feb 23 '20 at 09:30

Michael Rozenberg

score 1 · Answer 2 · answered Feb 23 '20 at 09:49

1

In $\mathbb C$, $$\sqrt{ab}=\sqrt a\sqrt b,$$ doesn't hold.

answered Feb 23 '20 at 09:49

Todd

2 Answers2