2=1
 

a = b

a² = ab

a²-b² = ab-b²

(a+b)(a-b) = b(a-b)

a+b = b

2a = a

2 = 1


[This message has been edited by Mayfuck (edited 03-06-2002).]

wtf?


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I dont' discrimanate, I hate everyone.

..Please tell me that isn't Boolean algebra...

GEORGE BOOLE MUST DIE.

a²-b² is equal to zero. Factoring that would require dividing. You can't divide numbers into zero.

Owned.

Damn, beat me to it.


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i wasted all my years, been chasing all my fears

Which reminds me, I have to study and I'm not doing it.

u + me = us.

we hope that you choke.

I think it's fine. I'm not the official on math, but I am a senior math major. Just because something isn't correct on the integers under usual addition and multiplication doesn't mean it isn't correct, as long as it follows a set of rules. Without defined rules, 1=1 might be bullshit. Not that we had well defined rules for this problem... I don't think I did a good job of explaining that.

It's you!

dude, going from line 2 to line 3 he divides by zero. im really not big into math at all, but i cant think of any example in any system where division by zero is allowed. i dunno, if it's your major maybe you've seen something i havent, but i honestly cant think of any explanation for that being legit

http://www.aspirin-foundation.com/Headache.jpg

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hello kitten...you'll never know what it's like to be me curled at your feet...

liz you're the smartest. http://www.netphoria.org/wwwboard/smile.gif

thanks julie, you're the coolest. http://www.netphoria.org/wwwboard/wink.gif

er, i'm only doing this to avoid washing dishes, but to end the debate i can tell you about the left and right cancellation laws of abstract algebra. if you've got a few minutes you should read it, it's not as bad as it looks.

in abstract algebra, you study groups, rings and fields. here is the definition of a group.

Definition (Group)
A group is a set G, closed under a binary operation *, such that the following axioms are satisfied:
G1: Associativity of *
For all a,b,c in G, we have (a*b)*c=a*(b*c)
G2: Identity e for *
e*x=x*e=x
G3: Inverse a' of a
Corresponding to each element a in G, there is an element a' in G such that a*a'=a'*a=e.

So, for example, the integers [...,-2,-1,0,1,2,...] under addition are a group.
G1: We all know that the integers under addition are associative, i.e.,
(1+2)+3=3+3=6
1+(2+3)=1+5=6
G2: 0 is the additive identity for everything; 2+0=2, 3+0=3 and so on.
G3: And for any element, say 2, there is an inverse, -2 in the case, such that 2+(-2)=0.

So now we can introduce a theorem.

Theorem
If G is a group with a binary operation *, the the left and right cancellation laws hold in G, that is, a*b=a*c implies b=c, and b*a=c*a implies b=c where a,b,c are elements of G.

Proof
Suppose a*c=a*c. Then by G3, there exists a', and a'*(a*b)=a'*(a*c).
By the associative law, (a'*a)*b=(a'*a)*c.
By the definition of a' in G3, a'*a=e, so e*b=e*c.
by the definition of e in G2, b=c.

Yeah, so the group that mayfuck gave may or may not be a group [I can check if you care but I'm sure no one has read this far anyway!]. But the point is that under the left and right cancellation laws you can "divide by 0" [but mind you you're really not "dividing by 0" since this is a special system...]

[This message has been edited by pink_ribbon_scars (edited 03-06-2002).]

i reaize that i'm a huge dork, but i just figured out that you wouldn't even need the left and right cancellation laws in this case. if (a-b) was a member of a group, than its inverse (a-b)' must be in the group, too.
So (a+b)(a-b)=b(a-b) can be multipled on the right by (a-b)' such that
(a+b)(a-b)(a-b)'=b(a-b)(a-b)'
so
(a+b)e=be
which means
(a+b)=b

take that niggaz

Wow.
Nifty.

i kind of get what you're saying, that cancellation in a group isnt really the same thing as dividing both sides by the same term, but i dunno, theres something that still seems off somewhere

ok, now i dont disagree, you're completely right that that piece is legit. but, wouldnt that mean that a = 0, and therefore b = 0? and if so...

theres now a division of 0 going from the second line to the third line. it should actually read

a+b=b
2a=a
2a - a = 0
a = 0

which makes sense (to me at least)

I'm with you actually, I really doubt that it's a group. I just wanted to show you that it is possible in some cases.

That sounds good, I think that original prolem is a proof that if a=b then a must be 0.
I think we nailed it. http://www.netphoria.org/wwwboard/smile.gif

lol, good, so now ill be able to sleep tonite http://www.netphoria.org/wwwboard/wink.gif
this is why i stay away from math. give me a nice qualitative science like biology anyday

to prove 1 doesn't = 2
if 1=2 then 1=0
proof
1*** = 2
there fore (if 1=2)
1+2=2
1+2-2=2-2
1=0

and
1+j=0
j***-1=0-1
there fore j=0 (because 0=1)
there fore every number = every other number
and it doesn't work.

I've always pondered something...How can any number truly be defined? How can we say something is 2. There are infinite amount of numbers in between 1 and 2 (1.5, 1.55, 1.555, etc), thus one can never actually make it to 2. Does that make sense? It's kinda like putting a definite number as to the distance to the end of the universe. Assuming the universe is infinite, we can place an arbitrary value (for example, EU can stand for the distance to the end of the universe) but this value will never be defined because there is an infinite amount of numbers in between.

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i wasted all my years, been chasing all my fears

***?

numbers is sexy/+

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there was a time when you let me know
what's really going on below but now you never show that to me do ya

http://www.netphoria.org/wwwboard/biggrin.gif

I'm not really gonna answer your question other than the existence of numbers and the definite borders between them are there for quantifiable purposes. It's really as simple as that. But the existence of numbers as an autonomous physical embodiment is something to think about. There's an idea that, in a post-Platonic sense of numbers, integers do not exist seperately as individual numbers, but rather a result of counting, because without counting one can conclude that numbers don't have any ontoligical verification. Furthermore, there is a theory that math itself consists only of concepts and theorems and postulates that are only realized through an array of finite steps and procedures, in a limitied space/time, by a finite being. It seems that only through this realization can we associate mathematics with the physical world and thus justifying its existence, because to ask if "1" exists, or if "3.4" exists on its own, ideas begin to get sketchy.


[This message has been edited by Mayfuck (edited 03-07-2002).]

well that made me laugh

have you heard of Zeno's Paradox? here's an example i got off the web:

A runner wants to run a certain distance - let us say 100 meters - in a finite time. But to reach the 100-meter mark, the runner must first reach the 50-meter mark, and to reach that, the runner must first run 25 meters. But to do that, he or she must first run 12.5 meters.

Since space is infinitely divisible, we can repeat these 'requirements' forever. Thus the runner has to reach an infinite number of 'midpoints' in a finite time. This is impossible, so the runner can never reach his goal. In general, anyone who wants to move from one point to another must meet these requirements, and so motion is impossible, and what we perceive as motion is merely an illusion.

and calculus deals with these limits. i don't know if it solves this problem, but it does negate its effect.

The only reason space is allegedly infinitely divisible is because we apply the abstract concept of mathematics to it.

For instance, the quantam leap, the phenomenon on electrons traveling from different electron shells of an atom. When they do that, they don't (appear to) exist anywhere between the two electron shells, which in a way helps prove that space is definitely divisible.

*head implodes*

holy shit lawson was just telling me about this like a week ago.

Let's talk about PI!

*awaits discussion*

Dorks.

I agree with both of you, Irrelevent and ******. I don't know if there is an answer really. Numbers are arbitrary.

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i wasted all my years, been chasing all my fears

There's a lengthy answer to Zeno's Paradox, but it has to do with the collapse of waves and quantum mechanics, and other stuff that goes over my head.

that's what i'm talking about!!


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dying of boredom...i'll try it all

this thread makes me all warm inside.

well, this would certainly be true if 'infinity', a much misunderstood word, was a definitive number. in reality, it is agreed that there are various types of infinities, some 'larger' than others, or diverging more rapidly than others.

in this paradox, you're essentially summing an infinite series of increasingly small distances which sum to 100m (50+25***2.5+etc). corresponding to this series is the time taken to traverse these distances. each of these specific times is finite, but there is not a hard and fast rule with regards to summing an infinite series of finite numbers. in some instances, the total diverges to infinity and in others it converges to a finite answer, as in this example.

Assuming that the runner maintains a constant pace throughout the race, and it takes him X seconds to reach the 50m mark, then the total time for the race is:

T = X + X/2 + X/4 + X/8 + X/16 + ... (TO INFINITY)
= sum(X/(2^(n-1)) ; where n corresponds to the nth number in the series

This can be proven to sum to 2X (i'm sure as hell not gonna go into that now). in essence, the subsequent terms added on to the total each time are not 'strong' enough to break free from a finite asymptotic constraint, and hence this may be considered a 'weak' infinity. (otherwise it would sum to infiity)

hence the time it takes the runner to reach the 50m mark is exactly half the total race time. this is a result which you'd expect.

yeah, it's weird, but then so is infinity. deal with it. technically, it's not a paradox either...it's just phrased in a manner to deliberately confuse people. a paradox holds up to scrutiny. and yes, i'm a bit bored.