# Dividing a number by zero

Discussion in 'Physics & Math' started by chikis, Mar 10, 2013.

1. ### RJBeeryNatural PhilosopherValued Senior Member

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It's simple.

A = B (FINE)

Then multiply both sides by A, so we get A^2 = AB (FINE)

Subtract B^2 from both sides, so we A^2 - B^2 = AB - B^2 (FINE)

Divide both sides by (A-B), so we get A + B = B (NOT FINE!!)

3. ### Prof.Laymantotally internally reflectedRegistered Senior Member

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Right, zero is the only number that can satisfy all of the equations. So then just by saying that a=b, and then putting them into a quadradic formula, makes the value of "a" and "b" become zero. So then in quadradic formula, "a" cannot equal "b" and they cannot be zero, zero has the ability to make equations like a+b=b become true when any other number wouldn't be able to satisfy the equation.

5. ### Prof.Laymantotally internally reflectedRegistered Senior Member

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That is not fine. Since, a=b then you cannot put it into a quadradic formula. If you did then $a^{2} - b^{2}$ would equal zero, and then $ab - b^{2}$ would equal zero. So then at that time you just have 0 = 0. You then factor a zero from both sides and divide by zero. If you wanted to find the solution to "a" and "b", you could have stopped right when you found $a^{2} - b^{2} = ab - b^{2}$, the solution is 0 = 0.

7. ### RJBeeryNatural PhilosopherValued Senior Member

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...or 1=1, or 2=2, or 3=3, or -100=-100. It's almost like there exists a solution for every time the first number equals the second number!

The problem resides in the step that I pointed out. I don't know what you're talking about regarding the quadratic formula, etc.

8. ### Prof.Laymantotally internally reflectedRegistered Senior Member

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I may have said quadratic formula by mistake, the point is that in the equations $a^{2} - b^{2}$ and $ab - b^{2}$, if "a" and "b" are equal to each other then they could be no other value other than zero. That is why "a" and "b" cannot be equal to each other in this kind of situation.

Think about it, one squared minus one squared would be zero. One times one minus one squared would be zero. Zero doesn't play by the same rules as other numbers, so then it could only lead to wrong answers. A variable that is a zero would not act the same as any other variable. Both sides of the equation would already be nothing, so anything more than that would just be working on nothing even more still. It would just be a lot to do about nothing. I think that is why they put those mathmatical rules into place, if a = b, then $a^{2} - b^{2} = 0$ and $ab - b^{2} = 0$. Then neither of them could really say anything meaningful about something else.

9. ### rpennerFully WiredValued Senior Member

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Those are not "equations" those are "expressions."

$a^2 - b^2 \; = \; ab-b^2$ is an equation which is true if a = b OR if a = 0. Either condition is sufficient.

10. ### Prof.Laymantotally internally reflectedRegistered Senior Member

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Ah yes, that was the word I was looking for, it has really been a long time since I heard that one. I don't know what you mean by, "either condition is sufficient". I would think that it is the only values that could satisfy those conditions, but zero and a=b is said to not be valid in either of these expressions.

11. ### KitemanSARegistered Senior Member

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Prof., you are wrong. There is no need for "a" to equal zero. "a" can equal any number. It is only in the (a-b)/(a-b) step that things become undefined.

12. ### Prof.Laymantotally internally reflectedRegistered Senior Member

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"a" can equal any number, but like I said, zero and a=b are the only conditions where the equations would be valid, and those expressions are no longer valid when a=0 and b=0 and a=b.

They say any variables can be any number, but they can only be certain numbers in certain equations in order for them to be true. So all I did really is think what numbers the equations are true for, so then you can know what the value of "a" and "b" actually are even without constants in this case. The equations simply do not allow "a" and "b" to be any number when they are put together in this fashion. They could have been any number when it was said $a^{2} = ab$ and a = b, but then when it was put into the next expressions, the values of "a" and "b" could only be zero. The expression would limit the values of "a" and "b" since $a^{2} - b^{2} = 0$ and $ab - b^{2} = 0$, so then from then on "a" and "b" would have to be zero. They could no longer be any number once these expressions where set equal to each other.

13. ### PeteIt's not rocket surgeryRegistered Senior Member

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Remember the difference between an expression and an equation.
This is an expression:
$a^2 + b^2$
This expression can be evaluated for all values of a and b.

This is an equation. It equates two expressions:
$a^2 - b^2 \; = \; 0$
It is true for some values of a and b. In this case, for any value of a, there are two values of b that make the equation true.

No. The next equation is:
$a^2 - b^2 \; = \; ab-b^2$
In this equation, a and b can be anything, as long as a=b.
(considered in isolation, the equation is also true for all values of b when a=0, but it's already given that a=b.

The two expressions both evaluate to zero, but that does not imply that a=0 or b=0.

14. ### Prof.Laymantotally internally reflectedRegistered Senior Member

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It does if you factor it, you would be saying that zero times whatever is left is equal to zero. You could only take a zero out of zero, so then it couldn't be any number at that point. You wouldn't factor out a zero from one and say that it is still one. If you did anything with it from that point "a" and "b" would have to equal zero. The entire expression is equal to zero, so then any other operation would force it to become zero from that condition, since the expression itself is equal to zero.

Otherwise it would be like saying that you could factor zero out of a whole number, that is something that you cannot do.

15. ### PeteIt's not rocket surgeryRegistered Senior Member

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Layman,
One of the factors of the expressions on both sides of the equation is (a-b), which we know is zero.
Right?

And I'm sure you'll agree that if a=10 and b=10 (for example), then (a-b) is still zero.
Right?

And this will make both sides of the equation equal to zero.
Right?

I'm not sure exactly where your confusion is, but perhaps you'll see if you plug some numbers in.

Plug in a=10 and b=10 into $a^2-b^2 = ab-b^2$.
You'll end up with 0=0, which I'm sure you'll agree is a valid equation.

16. ### PeteIt's not rocket surgeryRegistered Senior Member

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10,167
I missed this from yesterday.
That's correct. This is why it's important to be clear what the limit expression is.
You can't just say "The limit of 0/0" because it doesn't tell you where those zeros came from.

The point is that different limit expressions that approach 0/0 have different answers. There is no single answer to "The limit of 0/0".

No, this example show that approaching a limit of 0/0 does not always cancel to 1. You need to know how the numerator and denominator approach zero. "The limit of 0/0" is just not clearly defined enough to give it a definite value. You need to spell out the limit expression in full. No shortcuts.

17. ### Captain KremmenAll aboard, me Hearties!Valued Senior Member

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What happens if you divide infinity by zero?

18. ### EmilValued Senior Member

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:spank: Division by zero is forbidden!

19. ### krash661[MK6] transitioning scifi to realityValued Senior Member

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it appears if you try to divide infinity by 0 it will still be infinity since there's nothing ( 0 ) to divide by

20. ### eramSciengineerValued Senior Member

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$\frac{0}{8}=0 \frac{8}{0}=8 \frac{0}{0}=R \frac{8}{8}=R (0)(8)=R$

where R is some real number. I may be wrong though.

I used 8 in place of ∞.

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22. ### originHeading towards oblivionValued Senior Member

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Ha, pretty funny.

I see where I got you confused now. So then it would follow that, $(10 + 10 ) (10 - 10 ) = 10 ( 10 - 10 )$ , by doing this would have factored out a zero out of each side of the equation. It would be the same as saying $(10 + 10) 0 = 10 ( 0 )$, but you cannot factor out a zero from a 10! So then "a" and "b" can no longer be 10, because the only time this would be true is when "a" and "b" is equal to zero. So then you would have $(0 + 0 ) (0 - 0 ) = 0 ( 0 - 0 )$, the value of "a" and "b" being zero is the only number that would be true by factoring out a zero.