Can anyone do that 'proving 1 + 1 = not 2' thing.

[peter]

I seem to remember that using algebra it was possible to disprove that 1 + 1 = 2.

Can anyone show me that - and does it stack up [or is there like a 'hiccup' or something where the logic goes fuzzy]

[Hashi Lebwohl]

I am not coming up with that one off the top of my head but I can recall the classic that begins with a = b then

ab=a^2
ab - b^2 = a^2 - b^2
b(a - b) = (a + b)(a - b)
b = a + b
b = 2b
1 = 2
and if we carry this just one step farther we get
0 = 1

This allows us to divide by zero by substituting 1 in its place or claim that all numbers are the same number. Given any real number r we find that
0 = 1
0 * r = 1 * r
0 = r, so all numbers equal zero.

Clearly, the error is in the fourth step when we divided both sides by (a-b). Since our initial claim is a = b then we have divided by zero.

[Vraith]

I don't recall that one.
But I can prove that 0=1.

! means factorial [in case you don't know...I suspect you do].
0!=1 [definition]
1!=1
so
0!=1!
divide both sides by factorial.
0!/!=1!/!
The factorials cancel out so
0=1.

Shall I continue to pretend you are the math teacher I loved to play with in H.S.?
Her: This 3! is three factorial, which means 3x2x1.
Me: No it isn't, it is THREE!

Or...
Her: the area of a circles is pi r squared.
Me: NO, pie are round, fudge brownies are square.

[peter]

Vraaith! Were your intentions toward this math teacher entierly wholesome ( )

Nice one Guys. Hashi - your example I could follow and yes, see where the logic becomes suspect. Vraith - alas you give me too much credit; 'factorials' must have come into the class I skipped/fell asleep in. Does this example also have a clear point where things go awry [is that how you spell awry] or is the problem spread over the whole 'proof'.

I wondered whether the problems would ultimately turn out to be something like 'Zeno's Paradox' [the tortoise and the athlete one I think], where I believe the breakdown in logic occurs as a result of the way we actually use/understand language [quite possibly wrong here as well].

I'm a great lover of mathematical paradoxes [especially when they don't involve much maths! Mobius strips do it for me - how is it that when you cut one down the center you get another single one [with an extra turn?] but when you cut that one in the same way you get two interlocking ones.

Or that cube who's name escapes me [edit; Menger sponge]. Its made up of 27 smaller cubes and you remove the central one from each face and the central 'cube of the cube'. You repeat the process with each smaller cube that is left and keep doing so. With each step in the process the cube increases it's surface area, but decreases it's volume. Ultimately you finish up with an object of infinite surface area and zero volume! I love that! en.wikipedia.org/wiki/Menger_sponge

[deer of the dawn]

The only thing I know along that line is something about:

1/3=0.3333.....

3x0.333333.....=0.999999

Therefore 0.9999.... =1.

0.999...+0.999...=1.999...

Since 1.999... does not equal 2, 1+1 cannot =2.

Irrational numbers is where it all falls apart- not in reality, just in our being able to understand and measure it.

[I'm Murrin]

Except 1.999...recurring would indeed equal 2, for the same reason 0.999... = 1. (That is, 2 - 1.999... = 0.000...)

Most of the things along these lines work by dividing by zero at some point, often obscuring it through algebra.

[Vraith]

Vraaith! Were your intentions toward this math teacher entierly wholesome ( )

Nice one Guys. Hashi - your example I could follow and yes, see where the logic becomes suspect. Vraith - alas you give me too much credit; 'factorials' must have come into the class I skipped/fell asleep in. Does this example also have a clear point where things go awry [is that how you spell awry] or is the problem spread over the whole 'proof'.

On the first ... I, though over 40 [and thus the exception that proves your rule] am not a scoundrel so will remain silent.

On the last...yea, there's a problem. The ! is a an instruction/operation, not a value, so you can't "divide both sides."
It's like saying 3+2=2+3, divide both sides by the plus sign so it cancels out.

There are a number of ways to deal with the Zeno.
One of which is that the terms/distinctions he uses are false. He treats space, time, and motion as if they are discrete/separable entities/values.
But they aren't.
[another is that the sum of an infinite sequence has been proven to be 1.
1/2+1/4+1/8 etc..=1]
That might relate to Deer's thing too...but there's a simpler one for that.
1/3=.333repeating isn't true really. 1/3 is APPROXIMATELY .333repeating...[which I mention not to solve the problem, or enlighten anyone with something they probably know already, but because it made me think:::]:::
HEY...that's actually a better tale than Zeno's since space/time/motion are left out. .333etc gets infinitely closer to 1/3, but never quite catches it.
Though I admit the title "Decimal and the Fraction" isn't nearly as exciting as "Achilles and the Tortoise." Which is saying a lot, since Achilles fighting BATTLES is pretty damn boring even in 3D and surround-sound film...who cares about him trying to get ingredients for soup?

[Avatar]

HEY...that's actually a better tale than Zeno's since space/time/motion are left out. .333etc gets infinitely closer to 1/3, but never quite catches it.

You're right.

--A

[Gadget nee Jemcheeta]

Like! +1!

[Gadget nee Jemcheeta]

And for my 2000th Post, Av I love your sig hahaha

[Avatar]

Hahaha, a JemCheeta sighting and a milestone all in one go.

To the Mile-High thread...

(And welcome back. )

--A