The "Proof" that 2=1

[Neil_Bryan]

Okay. I think you have seen this lately.

a=b
a²=ab
a²+b²=ab+b²
(a+b)(a-b)=b(a-b)
a+b=b
b+b=b
2b=b
2=1

I was just wondering: what is wrong with the theory? I know it's wrong, but I haven't found out what the heck is wrong with this sort-of "proof".

Can you please help? Math isn't up for debate so I posted it here.

[乳头的早餐谷物]

It looks like you're dividing by (a-b), and since a = b that's division by zero.

[otters]

Yeah. After that step, both a and b are undefined values, so I guess two a's equals one a in that case.

[Kablizzy]

[T3chno]

By the way, to go from (a²+b²) to (a+b)(a-b) is not possible. The result if a difference of squares so it should've been (a²-b²).

[Drathmoore]

Okay. I think you have seen this lately.

a=b
a²=ab
a²-b²=ab-b²
(a+b)(a-b)=b(a-b)
a+b=b
b+b=b
2b=b
2=1

It looks like you're dividing by (a-b), and since a = b that's division by zero.

To quote my Maths teacher here...

IT'S A MASSIVE CLUDGE!


Yes. Point is, as soon as you divide by 0, everything goes out of the window. You might as well say 2+2=5 under the same proof. Nothing makes sense anymore when you've divided by 0.

Quite mind-blowing, going through school, only to find that Maths isn't perfect. I mean, 0! for example...

[lord_day]

Assertion:
The smallest positive real number is 1.

Proof:
Let r be the smallest positive real number. Either r > 1 or r = 1 or r > 1

If r > 1 then 0 < sqrt(r) < r and r is not the smallest positive real number.
If r < 1 then 0 < r^2 < r and r is not the smallest positive real number.

Therefore r = 1.

[squibbles]

Another one that fucks up with Divide by 0.

x = 0 in this problem.
4x + 2x = 9x
Add shit.
6x = 9x
Divide by the constant (x)
6=9

[jinxed_07]

a+b=b
b+b=b
2b=b
a=b
a²=ab
a²+b²=ab+b²
only possible when both a and b are = to 0
Otherwise, every algebra teacher in the US is going to drive by your house

a²+b²=ab+b²
You added a 'b' and are missing a 'a' wtf!

[t̷s͢uk̕a͡t͜ư]

Assertion:
The smallest positive real number is 1.

Proof:
Let r be the smallest positive real number. Either r > 1 or r = 1 or r > 1

If r > 1 then 0 < sqrt(r) < r and r is not the smallest positive real number.
If r < 1 then 0 < r^2 < r and r is not the smallest positive real number.

Therefore r = 1.

That conclusion is a total non-sequitur. What you've actually shown is that there's no such thing as the smallest positive real number.

[999_Springs]

integrate(tan x)
= integrate(sec x sin x)
= [ - sec x cos x] - integrate( - cos x sec x tan x) {Integrating by parts with u = sec x and v' = sin x}
= -1 + integrate(tan x)

Therefore integrate(tan x) = -1 + integrate(tan x)
=> 0 = -1
=> 3 = 2
=> 5 = 2 + 2. QED.

[Edited to make it seem more plausible]

[Chase]

Now this children, is why you don't divide by 0. It makes everything you've just learnt over the past 11 years useless.

[Kablizzy]

Now this children, is why you don't divide by 0. It makes everything you've just learnt over the past 11 years useless.

OH GOD MY DRIVING!!

[Spawn of Yanni]

integrate(tan x; 0, pi)
= integrate(sec x sin x; 0, pi)
= [ - sec x cos x; 0, pi] - integrate( - cos x sec x tan x; 0, pi) {Integrating by parts with u = sec x and v' = sin x}
= -1 + integrate(tan x; 0, pi)

Therefore integrate(tan x; 0, pi) = -1 + integrate(tan x; 0, pi)
=> [ln |sec x|; 0, pi] = -1 + [ln |sec x|; 0, pi]
=> 0 - 0 = -1 + 0 - 0
=> 0 = -1
=> 3 = 2
=> 5 = 2 + 2. QED.

[ - sec x cos x; 0, pi] = -1

- sec x cos x evaluated from 0 to pi equals 0, not -1. It equals -1 evaluated from 0 to pi, which is not just -1, it's 0. Therefore the integral of tan x between 0 and pi equals the integral of tan x between 0 and pi and everything from that point on is invalid.

[999_Springs]

integrate(tan x; 0, pi)
= integrate(sec x sin x; 0, pi)
= [ - sec x cos x; 0, pi] - integrate( - cos x sec x tan x; 0, pi) {Integrating by parts with u = sec x and v' = sin x}
= -1 + integrate(tan x; 0, pi)

Therefore integrate(tan x; 0, pi) = -1 + integrate(tan x; 0, pi)
=> [ln |sec x|; 0, pi] = -1 + [ln |sec x|; 0, pi]
=> 0 - 0 = -1 + 0 - 0
=> 0 = -1
=> 3 = 2
=> 5 = 2 + 2. QED.

[ - sec x cos x; 0, pi] = -1

- sec x cos x evaluated from 0 to pi equals 0, not -1. It equals -1 evaluated from 0 to pi, which is not just -1, it's 0. Therefore the integral of tan x between 0 and pi equals the integral of tan x between 0 and pi and everything from that point on is invalid.

Why the heck did I put limits in!? I've edited them out to make it *look* more plausible.

Though, I do understand what's wrong with it.

[Spawn of Yanni]

With your new version, you've ignored the arbitrary constant brought about by indefinite integration. Which brings me on to

Quite mind-blowing, going through school, only to find that Maths isn't perfect. I mean, 0! for example...

What? If anything, every one of these invalid proofs just shows that maths works. 0! is just a matter of definition.

[t̷s͢uk̕a͡t͜ư]

Quite mind-blowing, going through school, only to find that Maths isn't perfect. I mean, 0! for example...

Math is broken. Physics is wrong. I expect planes to start falling from the sky once reality realizes this.

[SlappyMcGee]

Quite mind-blowing, going through school, only to find that Maths isn't perfect. I mean, 0! for example...

Math is broken. Physics is wrong. I expect planes to start falling from the sky once reality realizes this.

Incidentally, this is also proof that Y2k was equal to Y1k.

[otters~1]

Quite mind-blowing, going through school, only to find that Maths isn't perfect. I mean, 0! for example...

Math is broken. Physics is wrong. I expect planes to start falling from the sky once reality realizes this.

Incidentally, this is also proof that Y2k was equal to Y1k.

Yet despite humanity's superior knowledge, better technology, and even despite Stephen Hawking, the reaction to both was about the same. Of course, back in the day when the knights and peons and whatnot saw a comet, they went and whipped themselves, so maybe we've gotten somewhere.

[Mute Monk]

Quite mind-blowing, going through school, only to find that Maths isn't perfect. I mean, 0! for example...

Math is broken. Physics is wrong. I expect planes to start falling from the sky once reality realizes this.

Why should they fall from the sky, if physics is wrong? Why not explode, or cease to exist entirely? Our preconceptions run pretty deep.

[t̷s͢uk̕a͡t͜ư]

Quite mind-blowing, going through school, only to find that Maths isn't perfect. I mean, 0! for example...

Math is broken. Physics is wrong. I expect planes to start falling from the sky once reality realizes this.

Why should they fall from the sky, if physics is wrong? Why not explode, or cease to exist entirely? Our preconceptions run pretty deep.

I hope you realize how unreasonable this question is.

[Drathmoore]

Solution!

∞ = An infinately big number.
2∞ = 2 x an infinately big number.
Yet ∞ is infinately big, so...

∞ = 2∞
1 = 2


Reality:

Aleph Numbers, and that sort of stuff. Orders of Infinity, etc.

[999_Springs]

Solution!

∞ = An infinitely big number.
2∞ = 2 x an infinitely big number.
Yet ∞ is infinitely big, so...

∞ = 2∞
1 = 2 <==You can't "cancel" infinities like that.

What this shows is that: You Don't Mess With Infinity.

Here are some more to decipher:

exp(i*pi) + 1 = 0 (A well-known fact...)
=> exp(i*pi) = -1
=> exp(2i*pi) = 1
=> ln(exp(2i*pi)) = ln(1)
=> 2i*pi = 0
=> 1 = 0
=> 3 = 2
=> 5 = 2 + 2. QED.

Alternatively:

ln 2
= 1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 + 1/7 - 1/8...
= 1 - 1/2 - 1/4 + 1/3 - 1/6 - 1/8 + 1/5 - 1/10 - 1/12 + 1/7...
= (1 - 1/2) - 1/4 + (1/3 - 1/6) - 1/8 + (1/5 - 1/10) - 1/12 +...
= 1/2 - 1/4 + 1/6 - 1/8 + 1/10 - 1/12 +...
= 1/2 ln 2

Therefore ln 2 = 1/2 ln 2
=> 2 = 1
=> 3 = 2
=> 5 = 2 + 2. QED. Zap.

[Donfuy]

Solution!

∞ = An infinately big number.
2∞ = 2 x an infinately big number.
Yet ∞ is infinately big, so...

∞ = 2∞
1 = 2


Reality:

Aleph Numbers, and that sort of stuff. Orders of Infinity, etc.

Does infinity equal to infinity?


edit: hah, 999_springs beat me to it.

[Drathmoore]

Solution!

∞ = An infinately big number.
2∞ = 2 x an infinately big number.
Yet ∞ is infinately big, so...

∞ = 2∞
1 = 2


Reality:

Aleph Numbers, and that sort of stuff. Orders of Infinity, etc.

Does infinity equal to infinity?


edit: hah, 999_springs beat me to it.

It does and doesn't at the same time. Mainly a bad joke, really.