Proving -1 = 1 wrong

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1. Mar 11, 2015

Rectifier

Hey there!
These is this falsidical paradox that I cant seem to prove wrong.

$-1 = (-1)^1 = (-1)^\frac{1}{1}= (-1)^\frac{2}{2} = (-1)^{\frac{2}{1} \cdot \frac{1}{2}} = (-1)^{2 \cdot \frac{1}{2}} = ((-1)^2)^{\frac{1}{2}} = (-1)^{2 \cdot \frac{1}{2}} = ((-1)^2)^{\frac{1}{2}} = (1)^{\frac{1}{2}} = 1$

Any ideas?

2. Mar 11, 2015

Gil

You have (1)1/2, and it can be +1 or -1. The mistake you do is that you sould consider ((-1)2)1/2=|-1| = 1

3. Mar 11, 2015

micromass

No it can't. It's always 1.

4. Mar 11, 2015

symbolipoint

Someone will say it eventually. $$i^2=-1$$
$$(i^2)^(1/2)=+-(-1)^(1/2)$$, the text is there but it does not look right.
$$√(i^2)=+-√(-1)=i$$

5. Mar 11, 2015

Gil

The square root of 1 is ±√1 = ±1, isn't it?

6. Mar 11, 2015

micromass

1 has two square roots: -1 and 1. But $1^{1/2} = 1$.

7. Mar 11, 2015

axmls

A number raised to the power of 1/2 is the exact same thing as a square root. So $(-1)^{1/2} = \pm 1$. Information is lost upon squaring. That's where I'd say the issue comes from.

Namely, the OP starts with a number $x$, and takes $$x = x^{(2)(1/2)} = (x^2)^{1/2}$$

but information is lost when we square, because $$x = a \implies x^2=a^2$$

but $$x^2 = a^2 \nRightarrow x = a$$

Though someone else may see farther than me regarding this problem.

8. Mar 11, 2015

Staff: Mentor

micromass is correct. $\sqrt{1} = +1$.

9. Mar 11, 2015

Stephen Tashi

It isn't in general true that $x^{ab} = (x^a)^b$.

Last edited: Mar 11, 2015
10. Mar 11, 2015

Staff: Mentor

No. The sqaure root of -1 is the imaginary unit i.
That's irrelvant to this question. We're taking the square root, not squaring something.

11. Mar 11, 2015

Staff: Mentor

There is a misconception about square roots that shows up here quite often. An expression such as $\sqrt{4} = 2$, not $\pm 2$. While it's true that 4 has two square roots, one positive and one negative, the symbol $\sqrt{4}$ represents the positive square root.

More generally, for any positive real number a, the expression $\sqrt{a}$ represents the positive number b such that b2 = a.