mathematical paradox

[Aramil of Elixia]

This is a mathematical paradox for the square root of -1.

i²=-1
-1*1²=-1*-1
-1*1²=1
(i*i)*i²=1
i²*i²=1
i⁴=1

Then square root both sides and you get:

i²=1

square root again:

i = 1

so i = 1, but go back to 1²=1. The first line is i²=-1. So -1=1

[shacktar]

You lost one of the roots after square rooting both sides. It should be:

i² = ±1

You can then discard the positive root.

[Aramil of Elixia]

shoot i forgot about that. Thank you

[LB]

With imaginary numbers and square roots, sometimes you get minus or plus instead of plus or minus, which is weird but it happens. ;)

[helios]

And the difference is...?

[LB]

If you only consider the positive versions, you end up with a negativer answer due to it being minus then plus instead of plus then minus.

[Stupebrett]

What about division by zero?

With the following assumptions:

0 * 1 = 0
0 * 2 = 0

The following must be true:

0 * 1 = 0 * 2

Dividing by zero gives:

0/0 * 1 = 0/0 * 2

Simplified, yields:

1 = 2

Not really a mathematical paradox, but it shows that dividing by zero is not a legitimate operation.

[Gaminic]

By the way, I think you mistyped on the second and third lines:

-1*1²=-1*-1
-1*1²=1

I'm guessing you meant i² and not 1².

Like shacktar said, you can't say "if x² == y², then x == y", only "if x² == y², then |x| == |y|", just like you can't divide by zero or multiply by infinity to "prove" anything.

[helios]

I mean the difference between minus-plus versus plus-minus. Wikipedia says it doesn't have a special mathematical meaning; it's just a notational convenience to save writing expressions.
For example, a±b∓c=k means that both a+b-c=k and a-b+c=k are valid, and both a+b+c=k and a-b-c=k are invalid.

[LB]

Stupebrett, 0/0 is undefined.

[Aramil of Elixia]

@gaminic:thank you yes it is supposed to be i not one