Put quite clearly, this thread is NOT saying that the commutative property (in mathematics) is not real. I'm simply challenging anyone to prove it.
Let me phrase the question - Why should we accept the commutative property beyond the shadow of a doubt? Why should we think that it should be true of all numbers?
Just because it works a few times does not mean it is necessarily true for all numbers, so examples prove nothing.
My challenge - prove that a+b = b+a for any number a and b.
Well, when two sides are supposed to be equal to each other, you are supposed to come up with the same solution for each part.
A = 4
B = 6
4+6=6+4
If we subtract both 6's and fours, we each get 0=0. Does 0=0? Yes. If both numbers solved equal the same, then the solution is true. It can be the same for subtraction, multiplication and division, so long as you are consistent with both sides.
hmm this is a hard game... i dont think you can do it.. i mean the commutative property just seems so simple... there isnt magic numbers that add differently when you put them in different spots..
If we subtract both 6's and fours, we each get 0=0. Does 0=0? Yes. If both numbers solved equal the same, then the solution is true. It can be the same for subtraction, multiplication and division, so long as you are consistent with both sides.
Subtraction in itself assumes the commutative property - because you're taking 6 away, without the commutative property, it would matter which 6 you're taking away.
Subtraction in itself assumes the commutative property - because you're taking 6 away, without the commutative property, it would matter which 6 you're taking away.
Edit: Subtraction assumes the commutative property OR the axiom of choice.
Then let's just add them together. Because the answers are the same, it is assumed that the cumulative property will work. 10=10. Standard addition shows that since the two variables have the same numbers and both equal each other, there is no harm in switching them around. You could switch them around and the answers would be the same.
Then there's the part about experimentation. There's never any harm in checking your answers and trying Trial and Error. For this experiment, it does not matter which numbers you plug in. Saying 4+6 = 6+4 is exactly the same as "I will give you an orange and then an apple--giving you an apple and then an orange is the same value".
Then let's just add them together. Because the answers are the same, it is assumed that the cumulative property will work. 10=10.
The problem with this - based off of this, why can't I just say:
64/16 - cancel out the 6 on the top and bottom, and you get 4/1 = 4.
Examples don't prove anything.
For this experiment, it does not matter which numbers you plug in. Saying 4+6 = 6+4 is exactly the same as "I will give you an orange and then an apple--giving you an apple and then an orange is the same value".
You're misinterpreting it - I want to know why this general "rule" (that's what it sounds like from what you've described) should be applied to all sets of 2 numbers, real, imaginary, or complex?
64 divided by 16 is not the commutative property though. With this, it DOES matter which side is which, since one yields different solutions than the other.
The rule should be applied because it is a factor for solving complex equations and more so a guideline. If you know that two numbers paired with a positive tool (+, *) can be switched to come up with the same solution, it makes complex equations much simpler to solve.
Because it works! The addition and multiplication signs work so that it can. Go back to the analogy I gave you. It doesn't matter where the number is, so long as it is paired with the symbol and its compliment.
The commutative property is a definition, a description of a function. It's something that's implicitly held to be true. Even in math, we must start with items we assume to be true. So your question really has no basis, because it asks for a proof to a way we describe things, not an actual theorem.
Now, we can prove that a function is or is not commutative: see here for proofs that given functions are not commutative. Your misinterpretation, however, is that commutativity is a definition, not a theorem.
But how do we know beyond a shadow of a doubt that given any two complex numbers that you can add them different ways for the same answer?
We know this because the definition of addition is combination of objects into a larger collection. No matter how you combine two sets of things, their total will be the same.
What applies to Commutative Property? Only addition and multiplication. The Commutative Property essentialy says that 'order does not matter in addition or multiplication'. Therefore, the Commutative Property cannot apply to subtraction or division.
Proof:
8+2=2+8 Does 10=10? Yes, so Commutative Property applies to addition.
4x5=5x4 Does 20=20? Yes, therefore the Commutative Property applies to multiplication.
5-4=4-5 Does 1=(-1)? No, therefore the Commutative Property does not apply to subtraction, as 1=/=(-1)
10/5=5/10 Does 2=.5? No, therefore the Commutative Property does not apply to subtraction, as 2=/=.5.
Second:
The Commutative Property Formula is thus:
a+b=b+a
-OR-
axb=bxa
As we previously established, the Commutative Property only applies to Addition and Multiplication. Therefore, it is logical that we can assign any value to a and b, and as long as a and b's values remain consistent, and the operation does not change, than each side will equal each other once the problem is solved.
Proof:
a=5 b=9
5+9=9+5
14=14
TRUE
a=6 b=4
4x6=6x4
24=24
TRUE
Conclusion:
The Commutative Property works for all real numbers in a Positive tool (+,x). As each side yeilds the same anser, as long as the mode is not changed, and each side does yeild the same answer, than the Commutative Property has worked, and by working, it is prooved. __________ I would also like to point out that, if the Commutative Property doesn't work, in its long existence, don't you think someone would have pointed it out?
The Commutative Property works for all real numbers in a Positive tool (+,x). As each side yeilds the same anser, as long as the mode is not changed, and each side does yeild the same answer, than the Commutative Property has worked, and by working, it is prooved. __________ I would also like to point out that, if the Commutative Property doesn't work, in its long existence, don't you think someone would have pointed it out?
Maverick - we are looking for a deductive proof - not an inductive one. Observation is all fine, but many rules may work generally, but not necessarily all the time.
Can one actually prove that a+b = b+a without giving an example and assuming as few axioms as possible? I think so...but I'm interested in seeing as to what you come up with.
