Okay, say I wanted to factor a 0 out of 10. So then I could just say that 10 x 0 = 0? That 0 and 10 would be the two factors? Then I could just multiply them back together to get 10? So if I had (10 + 10 ) = 10 I could just factor out a zero out of both sides? Then I would have 0 ( 20 ) = 10 ( 0 ) Then I would get 0 = 0 Wow Pete, I think you helped me solve this problem! Finally figured it out!
The expression states "as a approaches zero", which does not imply equalling zero, therefore the expression is not evaluated for a=0.
Could you give an example how you would factor from zero? I don't remember this being a valid operation in algebra.
The limit is where the hole would be in the equation. So by finding the limit, you know where on that line a=0, and that location is 1. So then you know the value of a/a when a = 0.
You're looking at it. 0 = 10 x 0 No, you know the limiting value that a/a approaches as a approaches zero. This is an important distinction.
The question is absurd. Once a term of an equation is found to be zero, it vanishes from the equation: x = 3 + 1.234 * y Upon discovering y=0: x=3.
What is absurd is having to argue that you cannot factor zero's out of equations. There are no mathmatical principals that explain how to do this. So of course no one would know how that is supposed to be done. You didn't even factor a zero anywhere out of this equation. Saying that you can factor out zero's is absurd! This is obviously trolling.
Try it with a^2 - b^2 = 0, (a+b) = 10 0 = 10 x 0 \(a^2 - b^2 = (a+b)(a-b)\) We've factorized 10 from 0. We've factorized (a+b) from (a^2 - b^2)
I'm not arguing that. "Factoring zeroes" is meaningless. Because it's meaningless. It's not supposed to be done. The factor was y. It was found y=0. Once that was obvious the term vanishes. The rest is meaningless. Then why are you harping on it? Then get with the program. The trolling will certainly end.
You can also rearrange the terms as a=a, meaning it's an identity, and true for all a. Now all it states is 0=0.
This argument supports that axiomatically zero is not a number on the number line. Because it is undefined when used like the numbers in your example. Undefined means not axiomatically consistent when treated like the numbers that are on the number line. In arithmetic it is a placeholder. Where no action is warranted. Multiplying by zero is apriori non-action anyway. Dividing by zero is also a non-action. So no axiomatic treatment necessary except to say no action apriori. Undefined and no action means zero is not a number. It is a placeholder or no action symbol in arithmetic. Saying division of zero by zero is undefined is another fancy way of saying apriori zero doesn't behave like a number on the number line. In mathematics and physics zero is a boundary or singularity condition symbol indicating transition from one number system or scale or value system to another. Like in the split number line where zero is origin symbol for both sides of the split number line into negative and positive numbers. If zero was a actual number instead of only origin symbol on the split number line, then it must be both negative and positive at the same time! But it isn't because both the negative and positive numbers on split number line originate from zero boundary or transition singularity condition, not from a zero number both negative and positive simultaneously. I will listen intently to counter arguments to what I posted now and before about this zero and undefined things. No attacks on me, just on my posted naive understandings simply stated please.
If you're looking for a good laugh, look in the mirror at person who can't repeat a posted statement without twisting it into direct opposition to what was stated. As I said, "factorization by zero" (factoring out a zero from a term of an expression) is meaningless. Your inability to connect physical interpretation with mathematical abstraction is the only meaningful sense that attaches to what you're saying.
I must have gotten you confused with Pete, because you where backing him up and this is what Pete has been saying. That is the whole point, if there is a factorization of zero, then the variable of the other factor will become meaningless. It will start to act like it is a zero itself. You will get equations where only zero could be a correct possible solution for that variable. So all I am saying is that if you factor out a zero and "a" is left as the other factor, you could end up with something like a + a = a. The only number that then fits this description of "a" is then zero. 0 + 0 = 0, it is the only value that could be the value of "a" so that the equation is true.
