1=2

[przyk]

I just found it comical that rpenner and others felt compelled to point out that 1 does not in fact equal 2.

I think it's useful and interesting to point out exactly why fake proofs like the one in the OP don't work. Someone who isn't already familiar with basic logic could learn a lot about it that way.


But in the spirit of trolling:

so now you've broken the square function's inverse??

Like you're probably aware, the square function defined over the reals isn't bijective and thus doesn't have an inverse.

$$-1^2 = 1$$

Actually in mathematics and most programming languages I know, the power operator binds more tightly than the negative sign, so $$-1^2 = -1$$. Problem solved!

 

[RJBeery]

Actually in mathematics and most programming languages I know, the power operator binds more tightly than the negative sign, so $$-1^2 = -1$$. Problem solved!

Well played

 

[przyk]

Yet another argument for RPN.

 

[Cavalier]

Another good one (for kids):

Assume a = b

Then it follows that 2=1 because...

Multiplying both sides by a, we get: a^2 = ab

Subtracting b^2 from both sides: a^2 - b^2 = ab - b^2

Factoring, that is: (a + b)(a - b) = b (a - b)

Simplifying, we get: (a + b) = b

Since a = b, by assumption: (a + b) = (b + b), and so we get: (b + b) = b

That is: 2b = b

Dividing both sides by "b": 2 = 1

 

[prometheus]

Another good one (for kids):

Assume a = b

Then it follows that 2=1 because...

Multiplying both sides by a, we get: a^2 = ab

Subtracting b^2 from both sides: a^2 - b^2 = ab - b^2

Factoring, that is: (a + b)(a - b) = b (a - b)

Simplifying, we get: (a + b) = b

Since a = b, by assumption: (a + b) = (b + b), and so we get: (b + b) = b

That is: 2b = b

Dividing both sides by "b": 2 = 1

This is a very nice example of what przyk was talking about, and it's a lot more subtle than the OP.

I'll give you a clue as to where the logical misstep lies so as not to spoil it for anyone who wants to think it through. In this step:

Simplifying, we get: (a + b) = b

What operation is being done?

 

[pmb]

but if root 1 = root 1 ... what's stopping me from saying -1 = 1 ?

Because there's no logical reason to say it. If sqrtr(1)= sqrt(1) then all we know is that 1 = 1.

You must be confusing this with (-1)[sup]2[/sup] = (+1)[sup]2[/sup] and then taking the sqrt of both sides. Once you do that you'd get |-1| = |+1| ===> 1 = 1

 

[Emil]

Another good one (for kids):

Assume a = b

Then it follows that 2=1 because...

Multiplying both sides by a, we get: a^2 = ab

Subtracting b^2 from both sides: a^2 - b^2 = ab - b^2

Factoring, that is: (a + b)(a - b) = b (a - b)

Simplifying, we get: (a + b) = b

Since a = b, by assumption: (a + b) = (b + b), and so we get: (b + b) = b

That is: 2b = b

Dividing both sides by "b": 2 = 1

Wow .... beautiful. :bravo:

Prometheus properly noticed.