What i am doing wrong here . and if i am correct then can any one explain to me how $1=-1$

Question

Let suppose

$(-1)(-1)=(-1)²$

if i take square root on both sides then

$\sqrt{(-1)(-1)}=\sqrt{(-1)²}$

after multiplication and cancelation

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

which is wrong although i suppose a correct equation

$1=-1$

where i am wrong and if i am correct then I just want to know why it is so

You have run into a trick issue in mathematical analysis, which is that exponentiation with rational numbers can result in things that might seem unexpected to us - in this case, it's the fact that $(x^{2})^{\frac{1}{2}} \neq x$ for $x < 0$. To avoid ambiguity, in most real analysis texts (a branch of math that often answers questions like these), one DEFINES $\sqrt[n]{x}$ to be the POSITIVE number $y$ such that $y^{n} = x$. — Alex Wertheim, Aug 09 '13 at 06:40

score 5 · Accepted Answer · answered Aug 09 '13 at 06:34

5

The square root of $(-1)^2$ is $1$, not $-1$. The identity $\sqrt{x^2} = x$ requires $x \geq 0$, and the general fact is $\sqrt{x^2} = |x|$.

answered Aug 09 '13 at 06:34

score 2 · Answer 2 · answered Aug 09 '13 at 06:36

2

You have to be careful.. $\sqrt{x^2} = |x|$ for all real numbers $x$. So $\sqrt{(-1)^2} = 1,$ not $-1$ as stated.

answered Aug 09 '13 at 06:36

Euler....IS_ALIVE

2 Answers2