Can anybody ponder about this algebraic paradox?

Question

consider this: [1] x^2-x^2=x^2-x^2 is always true for every real number. [2] Factoring both sides in different ways, it becomes (x-x)(x+x)=x(x-x). [3] Dividing both sides with (x-x) {which by the way is equal to 0 for every real number}, the equation becomes 1(x+x)=x(1) {or, (0/0)(x+x)=x(0/0)}. [4] By convention, 0/0 is indeterminate, which in turn makes the whole equation undefined. but, by the 3rd step, it turns out that 0/0 and 1 are equal because (x-x)/(x-x) can be simplified to 0/0 {since the numerator and denominator are both equal to 0}. [5] Working with (x-x)/(x-x), since the numerator and denominator are equal, it is equal to 1. but if simplified first, this becomes 0/0. by convention {and reality!}, 0/0 is NOT equal to 1. [6] continuing from step 3, that equation becomes x+x=x, which is 2x=x. Since it was stated in step 1 that x can be any real no., if x=1, then the equation becomes 2=1, which we all know {hopefully!} is NEVER true. What is happening here?!

Answer 1

Since x-x (for all real numbers) is inevitably 0 then you can't go on beyond that point. Step three is null and void.

You're supposed to divide by x-x in that step, but knowing that it can ONLY be zero means you are trying to divide by 0, which you cant do!

Answer 2

I came, I saw, I pondered ...

It is not a true paradox. Invalid logic makes it only seem that way.

You simply violated some of the cardinal rules of math by performing some invalid logic - after which everything that follows is invalid, no matter how seemingly correct.

"0/0 is indeterminate"

But then you proceed to equate it to 1, contradicting yourself and formal mathematics rules. Therein lies the flaw from that point onward and all that follows.

If the proper rules of math were followed, your conclusion wouldn't derive.

In math, the stuff of reality = the stuff of fantasy and there's no clear dividing line between them so some seemingly logical conclusions can still be totally wrong if you are not careful. The real art is telling mathematical fantasy from reality.

This falacy is very common in matters of ratios involving zero and infinity.

The Math Fairy is out to get you !

Answer 3

In step 2 it became 0=0 and that was the end of that.

Answer 4

You've already given yourself the answer. Any time you divide by zero you've screwed the pooch and everything after that is pure BS.

Take something simple. (2x²-2x)/(x-1) This can be 'simplified' to as follows

(2x²-2x)/(x-1) => (2x*(x-1))/(x-1) => 2x

So you might naively think there are no problems with this function. But there is.

When x = 1 the function fails to exist because it is 0/0 and it is *not* equal to 2 (as you might think from the simplification)

This is a good example of what Einstein meant when he said, "Things should be made as simple as possible, but no simpler."


Doug

Answer 5

you can never cancel 0/0 and say it is equal to 1 and so
x-xdivided by x-x is not equal to 1

Answer 6

Who's to say that one is actually one. Why can't the number 1 stand for five. Or hell, why can't the number 1 stand for the word dog. Now that would really screw up your equation wouldn't it?

Answer 7

No, you simply cannot mathematically divide any number by zero. Step three is not allowed, algebraically, and everything else that follows is garbage.

Answer 8

x^2-x^2 = x^2-x^2
(x+x)(x-x) = (x+x)(x-x)
1=1

since you screwed up on step [2], all the rest of your argument is gibberish.

Answer 9

yes, i ponder what you just typed could be translated to:

bla bla bla, bla bla bla, bla.

:-)