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[Leafsnail]

Ok, so, simple little maths puzzle my friend showed me.  I got it quite quickly, but partly because I've already seen a proof where something like this could've come up.

What is the error in the following proof?

Spoiler: Incorrect maths lies within.  You have been warned (click to show/hide)

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

[RedWarrior0]

Spoiler (click to show/hide)

Since a = x, then a-x = 0

[Jim Groovester]

Divide by zero. That's always what it is.

[Sowelu]

I'll just leave this here.

And this is one reason why I want to homeschool my kids.

EDIT:  ...I disagree with some points he makes though.  He rails about how stupid it is to prove the obvious.  Well, a lot of obvious things aren't true, and if you don't have the correct tools, you can't prove the non-obvious.

Also, it's really annoying not having a common language to talk about stuff with other people.  Artists have common language for things like "vanishing points", after all.

[Christes]

Divide by zero. That's always what it is.

Not in this one: (This one even tormented Euler)

Spoiler (click to show/hide)

1 = sqrt(1)
=sqrt(-1*-1)
=sqrt(-1)*sqrt(-1)
=i*i
=-1

[Sowelu]

Divide by zero. That's always what it is.

Not in this one: (This one even tormented Euler)

Spoiler (click to show/hide)

1 = sqrt(1)
=sqrt(-1*-1)
=sqrt(-1)*sqrt(-1)
=i*i
=-1

I dunno.  By the definition of square root, its inputs can be positive or negative and get the same result.  sqrt(x^2) = (+/-)x is kind of fundamental.

[RedWarrior0]

There's probably one you can do involving infintesimal numbers and infinite numbers multiplying. (and equalling pi)

[Christes]

Divide by zero. That's always what it is.

Not in this one: (This one even tormented Euler)

Spoiler (click to show/hide)

1 = sqrt(1)
=sqrt(-1*-1)
=sqrt(-1)*sqrt(-1)
=i*i
=-1

I dunno.  By the definition of square root, its inputs can be positive or negative and get the same result.  sqrt(x^2) = (+/-)x is kind of fundamental.

sqrt(x^2) = |x|

We typically take sqrt (and other roots) to mean the positive (real) root, and that really is the issue here.

[Vector]

EDIT:  ...I disagree with some points he makes though.  He rails about how stupid it is to prove the obvious.  Well, a lot of obvious things aren't true, and if you don't have the correct tools, you can't prove the non-obvious.

Also, it's really annoying not having a common language to talk about stuff with other people.  Artists have common language for things like "vanishing points", after all.

He's talking about terminology glut and intuition-killing proofs.  It's very important to be rigorous, but there's a certain kind of rigorous that really isn't needed.  Mathematicians basically never do two-column proofs.  Ever.  They also have one hell of a lot of terminology, but don't have names for "whole numbers" versus "mixed numbers" versus "improper fractions," because those aren't important terms to anyone.

[dragnar]

I'll just leave this here.

Ugh, yes. I can personally vouch for how terrible mathematics is in schools. I FAILED early math classes because I couldn't memorize the tables and formulas I was supposed to. The only thing that finally got me to do well in a math class, and ENJOY it, was one teacher who did exactly what he proposes. It was a private school, with it's own curriculum(closed down now...) and her math class involved her giving us interesting problems and just having the class go at it with no help but the occasional nudge in the right direction.

[ed boy]

consider Sn=1!+2!+3!+...+n!

can you show that for n>3, Sn is not a square number?

[Christes]

Nevermind ... I misunderstood your question.

[Leafsnail]

Why n>3?  3! isn't a square number... right?

[Christes]

But 3!+2!+1! = 9 = 3^2

I don't see an obvious way to do it, but I've been wrong before, especially in Number Theory.  Is this for a class?  It might be helpful to know what sort of machinery you have to work with if it is.

[Leafsnail]

Oh, right, yeah.  Derp.

I did do a question sortof similar to this, which involved dividing a strike of factorial pairs by another string of factorial pairs.

The key to it lay in the fact that (n+1)! = (n+1)n!  So, uh...