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How to prove 1 = 1

Posted on 2003-11-15
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What's the way to prove that 1 = 1

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Question by:maimran
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21 Comments
 
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Expert Comment

by:GwynforWeb
ID: 9757592
1 = 1 is an axiom ie assumption of arithmetic
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Author Comment

by:maimran
ID: 9757905
Some one wants to know how this assuption is supposed.
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Expert Comment

by:jares
ID: 9758116
As GwynforWeb said, some sentences propsitions are called axioms. An axiom is a proposition that is not susceptible of proof or disproof; its truth is assumed to be self-evident.

in other words, axioms cannot be proven nor disproven, they are facts . Every science has axioms, even reigions have something like axioms - eg. that God exists (for christians).
 
Another important point is that axioms are like groundwork of a building - if they were disproven, whole structure of a science built on this axiom would fall.

There were attempts to prove that 1=2, but most of them have one big catch - dividing by zero...
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by:GwynforWeb
ID: 9758711
Jares is correct, and well expressed.
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grg99 earned 20 total points
ID: 9758771
It's hard to prove it, as any proof is going to involve techniques that already ASSUME a lot more than that.
For example, we could use alittle algebra and rearrange the terms so we have :    1 - 1 = 0, but that already assumes we believe in the correctness of algebratic operations, and the orrectness of arithmetic, and that anything minus an equal amount equals "zero".

One possible approach is to say:  If we are going to be able to prove anything at all, we have to make a few assumptions, such as the location of a symbol on a page does not change its value, and a symbol is always numerically equivalent to itself.  If we can't make those assumptions, then nothing else we write is going to manke any sense, so we might as well just ASSUME that A = A, for all A, assuming A is a constant.   There's probably a logical proof of this in Principia Mathematica.

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Expert Comment

by:Thewizrd
ID: 9761493
A=A (Reflective Prop. of Math)
1 is a constant by Definition (Integer = Constant)
Let A equal the value of the constant '1'
therefore, A=A or 1=1
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Expert Comment

by:jares
ID: 9778105
Thewizrd is 100% correct.
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Expert Comment

by:dkloeck
ID: 9788620
1=1 can not be proved, 1 is 1 by definition, its an axiom

u can also elaborate an algebra that sais that
1!=1
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Expert Comment

by:Thewizrd
ID: 9792501
I know it can not be proven and is an axiom but I think maimran wants the logic behind the axiom, which I have provided.
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by:jares
ID: 9796229
> dkloeck: u can also elaborate an algebra that sais

Well, you can, but what sense has algebra that states that constant isn't equal to itself? It'll mean that anything is equal to anything!
 
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Expert Comment

by:Centeris
ID: 9805499
i'll give you one proof that you may want to look on. this is the story of euclid and his 5th geometric axiom. euclid, when made the book called "the elements" proposed 5 axioms and several theorems. the 5th axiom (which is something supposed to be so obviously true that it doesn't need proof) says something like "give a point and a line, only one line passing through the given point can be parallel to the given line." now, considering the first four axioms (i.e. "all right angles are equal to themselves") the 5th axiom is pretty long so a lot of mathematicians suspected that this axiom is actually a theorem (a mathematical truth proven by axioms for a certain domain). so they tried proving the 5th axiom using the first four. nobody succeeded in doing so and some of those math fanatics just gave up the fight and accepted that it may just have been a weird axiom, but it really is an axiom. however, some people are just too persistent. this is their "battle plan":

1. try to prove the opposite of the 5th axiom, that is:
  a. assume that NO parallel line can be drawn through the point that is parallel to the line, and
  b. assume that at least TWO parallel lines can be drawn through the point and parallel to it.

2. disprove assumptions a & b using the first four axioms
3. using a "double negative" attack, if all other possibilities were exhausted and proven to be false, the remaining (which is the original 5th axiom) must be true.

sounds like a master plan, right? the problem was, after doing #1, they were unable to do #2, which is understandibly the hardest part. as a matter of fact, these assumptions gave rise to two very important geometries, which are spherical geometry and hyperbolic geometry. they were unable to succeed in their mission, but it produced good results just the same.the end.

now, if you're thinking on how to prove an entire axiom, i am telling you two things:
a. you may want to try to assume that 1=any number greater than 1 and/or 1=any number less than 1 WITHOUT reflexivity then disprove both assumptions. you got yourself a proof. that is, if you can find something else to use withot reflexivity.
b. good luck on your search. this may sound sarcastic, but those mathematicians faced severe criticism by the math community before being able to come up with their discovery. come to think of it, who knows? you may be able to find that proof and topple down the foundation of mathematics just like einstein did with physics!

for anything it's worth,
Centeris

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Expert Comment

by:jares
ID: 9805934
Thewizrd:
>I think maimran wants the logic behind the axiom

What logic do U mean? there's no logic behind axioms...

Just a thought:
two guys had both one dollar note in their pockets. Both of them have spent whole dollar. now they have both no money in their pockets.
So, they had both certain number of one-dollar notes (one), now they have none.

Conclusion: they had equal number of one-dollar notes.  
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Expert Comment

by:Centeris
ID: 9807713
for the kind comment of jares:
how do we know if the dollar one guy had  has the same value with the very dollar the other guy had in his pocket? because a dollar is a dollar? isn't the assumption the conclusion that we want? you know what i mean.

but what he said is true. the axiom IS the logic and supposedly, it can no longer be broken down to smaller principles.

peace out!
Centeris
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Expert Comment

by:jares
ID: 9817062
I used this only as an example.
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by:Centeris
ID: 9817323
i know :-)
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Expert Comment

by:SunBow
ID: 9838580
> how this assuption is supposed.

A thing is itself,
no matter who is talking about it, or how they are refering to it
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Expert Comment

by:Centeris
ID: 9841283
truth is objective. truth is absolute. regardless of convention. but the interpretation is relative. the axiom of reflexivity lies on the truth.
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by:nigel5
ID: 10194854
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by:jares
ID: 10200679
@ nigel5 :

> divide by (a^2-a*b) ...

this is is invalid operation:

if a=b, then a^2=a*b.
result of substracting of 2 equal number is zero.
we can't divide by zero, so this is the reason why you get such result

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Expert Comment

by:nigel5
ID: 10202390
Yeah, but it confuses a lot of simpler folk for a while... including me :)
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Expert Comment

by:jares
ID: 10225860
Well, a couple of years ago, it confused me too...  
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