If we know that x – x = x / x – 1
where x - x = 0
and where x / x - 1 = 0
But we cannot prove x – x = x / x – 1
given that we cannot prove x - x = 0
or that x / x - 1 = 0
Then how can we know that x – x = x / x – 1 without proof.
Do we at least accept that x = x ?
Yes, and we know that x = x
where x is a certain known variable in x – x = x / x – 1
and where x is a another certain known variable in x - x, namely 0
Yes, and we know that x = x
where x is a certain known variable in x – x = x / x – 1
and where x is a another certain known variable in x = x, namely 0
Moderators, please delete this post and post 3
All “proofs”, and the “knowledge” arrived at thereby, must begin from our agreement on some axiom or other. (Even in this thread, we have effectively agreed to communicate in English using accepted rules of grammar.)
If we cannot agree to accept the axiom that x=x, then no proofs involving x can yield knowledge. In short, without unproveable axioms, proofs and knowledge are redundant. We must at least agree on some kind of axiomatic system in order to get anywhere (and even then, we can’t get everywhere!).
If we agree that x = x, what is the specific problem with x - x = 0 ?
There is an example of knowledge with proof.
If we know that 3 – 3 = 3 / 3 – 1
where 3 - 3 = 0
and where 3 / 3 - 1 = 0
And we can prove 3 – 3 = 3 / 3 – 1
given that we can prove 3 - 3 = 0
or that 3 / 3 - 1 = 0
Then how can we know that 3 – 3 = 3 / 3 – 1 with proof.
Please give me a proof that 3 – 3 = 3 / 3 – 1
therefore 0 = 0
where 0 = 0 is the axiom
and where 3 = 3 is the axiom
Axioms: i) x-x=0, and ii) x/x=1
Proof:
For x=3, 3-3=0 from i).
From ii), 3/3=1. Thence from i), 1-1=0
Therefore 3-3 = (3/3) - 1
First axiom: x = x
Second axiom: x - x = 0
Third axiom: x / x = 1
3 - 3 = 0 since 0 + 3 = 0
The first axiom is self-evident.
The second axiom is proven.
Please provide a proof for the third axiom where it follows that because 3 - 3 = 0 since 0 + 3 = 0
then 3 / 3 = 1 since ____________________
Axioms aren’t proven. Theses are proven as following from axioms.
Axioms aren’t proven. They are accepted or not. The symbol “/” is defined such that x/x=1, regardless of what x-x equals.
:smack:
9 / 3 = 3 since 3 * 3 = 9
Equation A is TRUE and has symmetry
9 * 3 = 3 since 3 / 3 = 9
Equation B is FALSE and has no symmetry
3 * 3 = 9
9 / 3 = 3
Equation A is TRUE and has symmetry
9 * 3 = 27
27 / 9 = 3
Equation B is TRUE and has symmetry
Why is the first Equation B FALSE and unsymmetrical? Why isn’t the first Equation B symmetrical? Does it follow that if there is no symmetry in an equation then that the equation is FALSE?
I reason that x = x is a self-evident axiom because it is symmetrical.
The definition of an axiom is something that is assumed to be true so they are not proved.
The axioms of algebra are taken to be:
- a+b=b+a commutative law for addition
- a·b=b·a commutative law for multiplication
- (a+b)+c=a+(b+c) associative law for addition
- (a·b)·c=a·(b·c) associative law for multiplication
- a·(b+c)=a·b+a·c distributive law
- a+0=a the number 0
- a+(-a)=0 opposite number
- a·1=a the number 1
- a·(1/a)=1 om a<>0 inverted number
Your first equation is true by the definition of equality. A quantity is equal to itself.
Your second equation is axiom #7 which is the definition of subtraction.
Your third equation is the definition for division.
(a-b)=(b-a) looks “symmetrical”. It is also false (unless a=b). I don’t think you’re using the word “symmetry” correctly.
And we can never define the first word in our dictionary unless we agree on the meaning of a few words that we can’t define.
I have the nagging suspicion that we’re only looking at the preamble to some puzzle Kozmik intends to reveal with a flourish (and possibly an exclamation of “Aha! Then…”) when the requested proofs are given and accepted; Am I grasping completely the wrong end of the stick, Kozmik, or do you have something up your sleeve that will appear to prove that x = 0 for any value of x?
On thinking things over I agree with Mangetout. And on thinking things over a little further, I think we can get even more basic than the axioms of algebra.
For example, how about this proof of x - x = 0?
Take group one of x objects - 1, 2, 3, … x
and group two of x objects - 1, 2, 3, … x
Now take one object from group one and one from group two, pair them up and throw them away. Do the same thing with the second object from each group, and so on. When we have thrown away all of the objects from group one we have also thrown away all of the objects in group two and so the remainder is zero.
So it would seem that we can prove that x - x = 0.
Your first statement in the OP is “If we know that x – x = x / x – 1”
I don’t think we can possibly know this.
Division is the process of finding out how many times you can repeatedly subtract the denominator from the numerator.
For example suppose I have this many objects in the numerator ||||||| and this many objects in the denominator |||||.
I line them up like this
|||||||
|||||
Pairing the first of the upper and lower I throw them away leaving
||||||
||||
When I continue the process I find that I can take ||||| from ||||||| one time and have a remainder of ||.
So dividing x/x-1 if I pair them up I find that I can subtract x-1 from x one time with a remainder of 1. That is it equals 1 + 1/x - 1. And according to the counting proof above x - x = 0 so x - x <> x/x - 1.
Am I completely missing your point?
Oops. I misread your x/x - 1 as x/(x - 1).
In any case I think my proof of x - x = 0 is valid, so we can in fact prove that x - x = 0.
And we can prove that x/x = 1 by repeated subtraction because we can subtract x from x exactly one time with 0 remainder.
So x/x = 1 and when we subtract 1 from that we get 0.
Doesn’t that work?
I think the really big issue here is the misunderstanding of “axiom”.
Kozmik, axioms are not proven. They are assumed. It does not make sense to describe them as “self evident”. They don’t have to be. You can just make them up!
Of course, if you make up the rules differently to everyone else and still claim to be doing standard real number algebra nobody else would be able to follow you. We have standard rules because those rules have been extremely useful in the real world.
An analogy to axioms could be the rules of a baseball game. When the umpire makes a call or a mathematician writes a step of a proof they do not have to prove the axiom or rule. They just have to apply it.
I hope my ideas aren’t too rusty (i.e. horribly wrong), but this is definately the impression I got from my years of college math.