When you factor an expression you divide them into more parts. Say 10/2 = 5, so then 5 x 2 = 10. You would have divided 10 into two different parts, 5 and 2. They are divided into different parts so then to undo that division you would multiply them together. Multiplication is the inverse of division. That would be why when you factor an expression, they are multiplied together. This is really basic stuff, I don't know what all that other garbage proves. Zero is the only number that has zero as a factor, both sides where equal to zero, the only number that you could factor out a zero from is zero since it is the only number that has zero as a factor. Did I mention that the only number that has zero as a factor is zero? Do you know of any other numbers that can be a product of a factor of zero? Lets try 1, does 1 have a factor of zero in it? Can we factor out a zero out of 1? Lets see 1/0 = undefined darn didn't work. Lets try 2, does 2 have a factor of zero? 2/0 = undefined , How about 3 anything we could multiple by zero in order to get 3? 3/0= undefined, oh well I give up, have a nice day.
No, you could not. That would mean dividing by zero. Think again, Layman. 10 x 0 = 0 0 = 10 x 0 Simple arithmetic. 0/0 is undefined. Limits that approach 0/0 can be anything. When you're stuck in a hole, stop digging.
And you never will, unless you open your eyes and learn a little. Your broken record is not relevant, remember?
The definition is the other way around really, ie 1/3 is the number which is defined by the equation 3X = 1. What number is equal to 1 when I multiply it by 3? 1/3. Hence the obvious notation we have for fractions. 1/0 does not exist because it is not required that ALL numbers have a multiplicative inverse, ie XY=1 defines Y = 1/X need not apply for all X, specifically X=0 is not required nor allowed. This is formalised in the notion of integral domains. For numbers if XY = 0 then either X or Y or both must be 0. For more elaborate constructs that may not be true, such as for matrices.
If you start factoring out zeros out of equations, and then start getting the wrong answers, don't say I didn't warn ya. I don't know of one counter example where factoring out a zero then gives correct values.
But there's the thing, you can't factor zero whether it's in an equation or otherwise. The counterexample is the arithmetic impossibility of factoring zero. It isn't allowed, is undefined, so can't possibly give correct values, or any value. The trivial ring is the set {0} with addition and multiplication. Since 0 + 0 is allowed, so is the inverse 0 - 0; and 'trivially' these have identical values, namely 0. Likewise 0x0 is allowed, but NOT 0/0. It's a ring because it's closed under addition and multiplication, but division isn't defined on {0}.
According to Layman, you can change 0 to 0 x 0, but not 0 x a, where a is not zero. This means according to Layman's rules, b + b = (1+1)b = 2b is not valid if b is zero. Go figure.
If you ever factored a zero out of anything you would just be setting it to zero, it would be like saying that you can set any side of an equation or any variable equal to zero any time you want, you just can't do that. Then you would have 0=0 in the equation, but 0=0 as an equation could never be used to solve for anything else. You would just be saying that everything in the equation is zero.
Obviously, the factor can't be zero unless the initial expression is zero. We've agreed on that over and over again, so I don't know why you keep repeating it. Your mistake is insisting that if one factor is zero, then all factors must be zero.
I have thought that way through at least six courses of algebra and aced it every time. It has never done me wrong. I think you finally got my point. I don't think I could have explained it better myself. And I think I remember one of my instructors saying that is not a valid operation when doing limits. That the arithmetic of being able to say 0 x a and "a" can be anything is wrong way of going about the problems. Then almost every expression says that it is not valid when equal to zero, go figure.
I'm not a mathematician, so all I can say with confidence is that nothing seems to be happening here.
Woo... algebra! You must be so smart. Please Register or Log in to view the hidden image! Perhaps you should work on your arithmetic. 0 = 10 x 0 What, your point that "if one factor is zero, then all factors must be zero"? It's wrong. Or do you mean the never-in-dispute and inexplicably belaboured point that "the factor can't be zero unless the initial expression is zero"? I don't think anyone has tried to factor a zero from any non-zero expression in this thread, so it's irrelevant. Your vague recollection of a poorly defined rule is decidedly unconvincing. Perhaps if you formed a coherent sentence we could understand what you're trying to say.
Yes, that is why I stressed what factors of numbers actually are. Zero is not a factor of ten. The only factors of ten are one, two, five, and ten. Since zero is not one of those factors then it cannot be factored out of it. You say 0 = 10 x 0, both sides of that equation are not equal to ten, they are both equal to zero. So then you haven't factored ten you actually factored zero. You cannot factor a zero out of ten and then have both sides of the equation still be equal to ten.
Right, we factored zero, not ten. Why would anyone think otherwise? Right, both sides are equal to zero, not ten. Nobody said both sides were equal to ten. Nobody factored a zero out of ten.
Okay, now it seems we are getting somewhere. So then since the only number that has a factor of zero is zero, then the number you are factoring is zero. Say like when we had b ( a - b ) and a = b, so then ( a - b ) = 0. Out of all reals on the number line, the only number that keeps the same value or that has a factor of zero in it is zero. So then b = 0 since 0 x 0 = 0 and 0 x n =/= n. If you factored out a zero from "n" then "n" would have to be zero, becaues any number times zero is zero. It is the only number that possesses that factor in it. If you factored zero from any other number then the factors wouldn't have the same value, the product of the factors would be zero. The product of the numbers is the same value of the number that you factored.
Layman, you're saying nothing new. You've posted this same thing a dozen times, I've agreed with it every time. It's irrelevant. Nobody is factoring zero out of anything except zero. Look: 0 = 10 x 0 Both sides are equal to zero. Right? So, the equation is correct.
It is relevant in saying that b=0. Both sides are equal to zero, but you could never factor out a zero out of all real numbers other than zero. So then that forces the other variable in the expression to act as though it is zero. Zero is the only real number that wouldn't change values from factoring a zero out of it. Since you factored a zero out of an expression, it forces the other variables in that expression to act like zero. So then you could end up getting things like a + b = b, it acts like zero, 0 + 0 = 0, if a=b and a=0 and b=0, then then I could say, oh then the equation is right. 0 + 0 = 0. The only reason why I think it is an issue because by your logic, you would find the wrong limit. In order to find the correct limit you would have to know that a wrong step was already taken before you took that limit. So I don't agree that taking the limit out of those equations gives the wrong answer and limits are wrong, I am saying that you took the wrong steps before taking the limit and that is why you found the wrong limit. You would just be bad at finding limits. Just think about the number line, and all those numbers. How many of them you could split into different products. The only real number on the number line that has zero as a product is zero, so then you can assume that the other product is also zero. It doesn't matter that 0= 10 x 0, the only number that has zero as a product of one of its factors is zero out of all real numbers on the number line.
As agreed. Repeatedly. Wrong. 1 x 0 = 0 0 = 1 x 0 Wrong. 2 x 0 = 0 0 = 2 x 0 Wrong. All numbers (real, imaginary, complex) have zero as a product. 10 x 0 = 0 You contradict yourself.
