False Proof with Imaginary Numbers

The incorrect step is the "cross-multiplication" between lines 3 and 4.

A negative times a negative = a positive....?

Yes. (Algebra 1)

Step 3 (ignoring i=i). 1/i does not equal i/1; in the same way that 1/8 does not equal 8/1.

But then again, I'm not a mathematician, so, what do I know.

Good Answer.

I took the path of not dealing with 'i' by adding a few more steps before the last line:

√(1*1) = √(-1*-1)

√(1) = √(1)

1 = 1

Of the six lines, line 3 is true, line 4 is false. Your error is there.

Lines 3 and 4 (actually 4 and 5) are identical to each other, except maybe for bold-facing. I too neglected i = i as line 1, but it seems we agree on the location of the misstep. Let's just blame the OP for not numbering the lines in the first place.

To elaborate, to prove sqrt (a/b) = sqrt (a)/sqrr (b) requires squaring both sides and then taking the square roots. Since there are 2 square roots, + and -, squaring is not reversible.

The only things I find stuff like this to be proof of is that how we define the rules of mathematics, real or imagined, have flaws and or poorly defined mathematical concepts that need better definition.

I think the correct definitions exist, but they aren't always taught. It can be quite tricky to make math/geometry be fully rigorous. For instance, Euclid is still classic, but he missed some things.

So basically people who obsess over this nonsense are exercising ignorance for the sake of seeing where it takes them?

I guess that sounds like the majority of math professors and math geeks I have been around.

I get it. So the fact I want to divide cats by rocks to see how many imainary trees I will get so I can find a cure for cancer totally justifiable use of my time and resourcs and I must publish this work so that others can benefit and build on it as well.

Sure. That makes perfect sense in reality now and all this time I was just figuring that these sort exercises were just practices in half wittery done by pompous halfwits that were out of touch with reality.

A fairly common theme in false proofs is dividing by zero without realizing it. (This one is a different issue.)

I haven't seen the correct answer listed in the comments above (if I missed it, let me know). Here is a link explaining the flaw in the proof:

Flaw in Proof

I had no idea that this rule existed for square roots which is why I found this false proof interesting. I certainly didn't figure it out on my own but had to read the explanation. You can find the explanation all over by googling "1=-1 imaginary number false proof". I chose the one above because I thought they did a good job explaining it. If you find a better explanation, please post it in the comments.

IMHO Dr. Math is overcomplicating it in his explanation by referring to roots of complex numbers. Any number, real or complex, has + or - square roots. You have to choose the one which makes sense (sometimes both make sense). I think Amy and the "official" answer are both correct.

His "proof" of 1 = -1 is different from the one you posted, and using the right √ clears the contradiction. Yours contains an error, as I posted in #11 (to give credit to Kilowatt, he drew attention to it in #3).

Sometimes using the wrong √ leads to an inconsistency eg in the well-known formula e^2*pi*i = 1, you can't take the √ and say e^pi*i = 1, because it doesn't. It comes from the other direction, e^pi*i = cos(pi) + i*sin(pi) = -1.

Could not help but think of common core math and what its gurus would make of this.

lol, it's confusing enough. Common core would make it impossible to understand.

As codemaster pointed out, line 3 is an error. √1/-1 ≠ √-1/1

You are not extending the radical sign over the whole quotient.

Is there a way to do that in our limited text editor on CR4? I left out the over line symbol, sorry.

I'm usually lazy and don't use this, but it can be very useful if you want to post an equation or derivation to this forum.

https://www.codecogs.com/latex/eqneditor.php

It allows you to generate the equations and then save them as an image which you can then upload to CR4 (see below):

Thanks for the tip, formerly Roger.

Has CR4 improved on the image insertion front? I've tried in the past and found it usually more trouble than it's worth, and I believe you've had problems. But I can now insert images from Mathcad, some examples, relevant to this thread, below. That's assuming it works, I hit Preview then back to edit and it's still there, which wasn't always the case.

Oh cool, that's good to know. I'll give it a try.

I'm a big Mathcad fan/user.

Shoot, I just tried it, but don't seem to be able to paste in from Mathcad. I'll keep playing with it.