What is the Flawed Proof That 1 Equals -1?

[_Mayday_]

Hey Everyone,

A while back I found this cool little proof that showed 1 = -1. Now I am fully aware there was a little cheat in there somewhere, but I have lost the little proof. Has anyone come across it, or have anything similar? I just think it's cool, even though in one of the steps there is a mistake. I know it starts with like rooting one, and then putting 1 = (-1)(-1) etc.

Cheers.

_Mayday_

 

[dirk_mec1]

I got another version:

$$\frac{-1}{1}=-1$$ and $$\frac{1}{-1}=-1$$

so:

$$\frac{-1}{1}=\frac{1}{-1}$$

if $$\sqrt{-1}=i$$

then

$$\sqrt{\frac{-1}{1}}=\sqrt{\frac{1}{-1}}$$

so:

$$\frac{\sqrt{-1}}{\sqrt{1}}=\frac{\sqrt{1}}{\sqrt{-1}}$$

$$\frac{1}{2}(\frac{i}{1})=\frac{1}{2}(\frac{1}{i})$$

becomes

$$\frac{i}{2}=\frac{1}{2i}$$

$$\frac{i}{2} + \frac{3}{2i} = \frac{1}{2i} + \frac{3}{2i}$$

$$i(\frac{i}{2} + \frac{3}{2i}) = i(\frac{1}{2i} + \frac{3}{2i})$$

$$\frac{-1}{2}+\frac{3}{2}=\frac{1}{2}+\frac{3}{2}$$

$$\frac{2}{2} = \frac{4}{2}$$

$$1 = 2$$

 

[robert Ihnot]

YOu got a problem right here: $$\frac{\sqrt{-1}}{\sqrt{1}}=\frac{\sqrt{1}}{\sqrt{-1}}$$ Since this gives i=1/i.


$$\sqrt{\frac{-1}{1}}=\sqrt{\frac{1}{-1}}$$

Just because a=b doesn't mean that $$\sqrt a = \sqrt b.$$

 

[LukeD]

$$\sqrt{\frac{-1}{1}}=\sqrt{\frac{1}{-1}}$$

Just because a=b doesn't mean that $$\sqrt a = \sqrt b.$$

yes it does (as long as we've agreed on some convention so that $$\sqrt{x}$$ is a function, which we have)

And that line is correct. The problem is that in the complex numbers $$\sqrt{\frac{a}{b}} = \sqrt{\frac{c}{d}}$$ does not imply that $$\frac{\sqrt{a}}{\sqrt{b}}=\frac{\sqrt{c}}{\sqrt{d}}$$. This is true for the positive real numbers, but not for complex numbers in general.

But the poster was just asking for "proofs" that 1 = -1. Of course they are all flawed. But to the OP: There are a lot of "proofs" of this, so any more description, if you could remember any part of it, would be useful.

 

[Diffy]

Hey Mayday,

You pretty much have a proof in the one that Dirk_mec1 posted.

Once you get to this step:

$$\frac{\sqrt{-1}}{\sqrt{1}}=\frac{\sqrt{1}}{\sqrt{-1}}$$

you now have

i/1 = 1/i

multiply both sides by i

you have (i^2)/1 = i/i --> -1/1 = 1/1 --> -1 = 1.

 

[_Mayday_]

Thank you! Wait till my class see this stuff!

 

[Alex48674]

Another one that freaks people out

a=b
aa=ab
aa-bb=ab-bb
(a+b)(a-b)=b(a-b)
divide by a-b
a+b=b
since a=b then
2b=b
2=1
Naturally this is completely fake, the error in this logic is that when you divide by a-b you are dividing by zero. If you want to can keep repeating this and get like 1=4 and stuff. Kinda freaks people out but make sure you explain it to them in the end =P.