Math paradox?

So, I was doing some math. And I wondered, "What if you try to solve the second-degree equation 'x^2=0' algebraically?"

So I sat down to work. I tried doing it with the quadratic equation that we all love (or maybe hate). I worked it out, and somehow I got to "x = +/- 2i/2", which yields x = i or x = -i. But obviously that leaves us with x squared=-1. That makes no sense.

Crazier is, if we first substitute b and c in our quadratic equation as 0, and substitute 1 for a later on, we do get x=0. So what is going on here?

Is it that 2i/2 doesn't yield i? But in that case, xi/x is 0 by definition. Or am i wrong?

Post your opinion below. I'd love to hear what other people think.

In response to @Jisu101 , who is probably a good dude, since he (or she) likes math.

Sorrow, pal. Xi / x is not zero. It equals i -- the square root of -1

Now let's look at this step by step:

You say X^2 = 0.

Factor the left side, to get X * X = 0. (I'll use * to mean "multiply" -- I don't really need the symbol, of course).

If you multiply two numbers to get a product of zero, at least one of them MUST be zero.

So either the X on the left is zero, or the X on the right is zero.

But both Xs represent the same thing.

Sooooooo

X = 0.

Done. It is a quadratic with two identical solutions, and those solutions are zero.

EDIT: I screwed up one of my sentences above, typing too fast and not proof-reading. But it's fixed now.

@Noflaps said in #2:
> Sorrow, pal. Xi / x is not zero. It equals i -- the square root of -1
>
> Now let's look at this step by step:
>
> You say X^2 = 0.
>
> Factor the left side, to get X * X = 0. (I'll use * to mean "multiply" -- I don't really need the symbol, of course).
>
> If you multiply two numbers times zero, at least one of them MUST be zero.
>
> So either the X on the left is zero, or the X on the right is zero.
>
> But both Xs represent the same thing.
>
> Sooooooo
>
> X = 0.
>
> Done. It is a quadratic with two identical solutions, and those solutions are zero.

The interesting part is when you work out the entire x=(b+/-sqrt(b^2-4ac))/2a. There I came up with x=i. But that doesn't make sense? That's why I decided to make this thread.

Math. Not even once!

@Jisu101 -- You mean you used the quadratic formula, and think you got "i" for a solution. Check it again. I suspect you've made a small mistake. The solutions really ARE zero and zero. They MUST be.

And the quadratic formula, properly used, makes no mistakes. I'll save this, and then come back and do another post to show the solution that way.

yeah I actually just realised
In the 4ac I had written "4" but it actually needs to be 0 since it's a multiplication. Sorry ... I make that mistake often when multiplying 0*something, I just add 0 instead of making the whole answer 0.

@Jisu101 I present: The quadratic formula, from memory (please check me):

x = ((-b +/- (b^2 - 4 ac)) / 2a

If X^2 = 0, then a = 1, b = 0 and c = 0

so x = (-0 +/- (0^2 - 4*1*0)) / 2*1) = (0 +/- 0-0) / 2 = (0+0)/2 OR (0-0)/2

= 0/2 or 0/2 = 0 or 0

= 0

Yep. The quadratic formula works fine -- to get x = 0

Please don't believe me: check my work. Carefully. But...this isn't my first mathematical rodeo. Just sayin'

You should be proud of caring about math, as you seem to do.

EDIT: you realized your mistake before I got done with this and posted it! Good job! :)

If you really want to know about math paradoxes, just look into dividing by zero.

I don't really see dividing by zero as a "paradox," but I'm willing to read why it strikes you so, and I don't feel strongly that you can't possibly make an argument that it is.

Perhaps because if we ASSUME it is valid it leads to a mathematical contradiction, which feels paradoxical -- but I wouldn't call that a "paradox" because it doesn't seem to be proper on its face to begin with. I think of paradoxes as apparently true things that lead to falsity, or apparently false things that lead to truth.

I think of dividing by zero simply as an undefined activity. It leads to no answer.

But, for fun, let's pretend it does: and call that answer "A" (for answer).

Well, if we assume that, say, 8 / 0 = A

Then we have, by simple algebra, 8 = 0 times A

But 0 times ANYTHING is not 8. It is zero. "zero" examples of 8 cannot add to provide 8 -- and multiplication, after all, is merely repeated addition, made quicker for the fruitfully impatient.

So if we assume that we can divide by 0, we have to throw algebra in the trash, or pretend that multiplying by zero yields something other than zero.

So we must admit that division by zero cannot be done. Not exactly a paradox, but rather a wise community decision.

Wise community decisions are increasingly rare, so it becomes harder to recognize them.