Theory? Why is (nonzero) / (zero) "undefined"?

[AtWitsEnd]

I may be using the wrong word for this topic. When I first went through school I completed through Calculus II but that was a long time ago and I've forgotten a lot. I've recently been reviewing concepts as far back as basic mathematics and I'm finding a great interest in the theory and proofs behind information that I just intuitively caught on to in the past. For example, I've been fascinated by Bertrand Russell's explanation of something as simple as what a number is. I followed, with rapt attention, his explanation/proof that a number is basically a set of equivalent sets.

Do you think there would be any interest in a separate forum for these types of discussions or should I just ask my questions in the forum that the theory and/or proof applies to? Maybe I can give an example of a really simple question that's recently crossed my mind. It's about zero and division.

That zero divided by another non-zero number results in zero makes sense to me because, well, as I've recently read "dividing nothing into shares just means that each share has nothing." However, I'm not so sure I understand why dividing a number by zero is "undefined". If you divide a number into zero shares, why wouldn't that equation simply return the number you attempted to divide? After all, for example, 2 divided into zero shares, it would seem to me, is still 2. Any thoughts to help me along on that? Any recommended reading? Thanks.

 

[Deleted member 4993]

Re: Theory?

I may be using the wrong word for this topic. When I first went through school I completed through Calculus II but that was a long time ago and I've forgotten a lot. I've recently been reviewing concepts as far back as basic mathematics and I'm finding a great interest in the theory and proofs behind information that I just intuitively caught on to in the past. For example, I've been fascinated by Bertrand Russell's explanation of something as simple as what a number is. I followed, with rapt attention, his explanation/proof that a number is basically a set of equivalent sets. Do you think there would be any interest in a separate forum for these types of discussions or should I just ask my questions in the forum that the theory and/or proof applies to? Maybe I can give an example of a really simple question that's recently crossed my mind. It's about zero and division. That zero divided by another non-zero number results in zero makes sense to me because, well, as I've recently read "dividing nothing into shares just means that each share has nothing." However, I'm not so sure I understand why dividing a number by zero is "undefined". If you divide a number into zero shares, why wouldn't that equation simply return the number you attempted to divide? After all, for example, 2 divided into zero shares, it would seem to me, is still 2. Any thoughts to help me along on that? Any recommended reading? Thanks.

In that context then - what is definition of dividing?

How would you define division by a fraction?

Now think about order of numbers - and start walking toward 'zero' - making the divisor smaller and smaller (to the limit)...

 

[stapel]

First, define "division": a/b = c for b non-zero means that bc = a. For instance, 12/3 = 4, because (3)(4) = 12. This works just fine with dividing into zero, since, for instance, 0/3 = 0 because (3)(0) = 0. Everything "works".

If you "define" (give a meaning to) division by zero, then, for non-zero a, a/0 must equal b for some value b. Then, by definition of "division", we have (0)(b) = a, where a and b are non-zero. Will this work with the rest of mathematics?

Division of zero by itself is a whole 'nother ball of wax....

Eliz.

P.S. Foundations and proof theory courses are, I believe, generally covered after calculus for math-ed and math majors, so "Advanced Math" is probably a good category for this sort of discussion. Other opinions welcome.

 

[AtWitsEnd]

Oh, wow! Thank you both. I think I'm starting to catch on. Can you tell me if my reasoning is right?

If I've got my mind wrapped around this right, it appears to me that division is really a series of subtractions that move toward a limit (zero) while multiplication is really a series of additions that move toward infinity (which exists on both ends of the number line).

I started to get this point after combining Eliz. definition with Subhotosh suggestion to make the divisor smaller and smaller. What I noticed almost immediately, is that, for example 12/3 = 4 actually means the same as 12-3 four times times = 0. I tried with fractional denominators and found that 12/.5=24 actually means 12-.5 twenty-four times = 0. Since this made sence to me with positive numbers, I decided to try it with negative numbers. Perhaps someone can tell me if I'm reasoning right. Example: -3/.25=-12. This would be the same as -3 minus .25, negative 12 times. Since -3 - .25 is the same as -3 + -.25 this would appear to move you toward infinity but the indicator of -12 times reverses your direction and sends you back toward zero. I tried the same thing with -3/-.25=.12 and it got me back to zero as well.

I decided to try the same thing with a divisor that was larger than the dividend, for example 12/15 = .8 which actually means 12 - 15 point 8 times, or 12 - .8 fifteen times. It works. So, it would at least appear to me that I've got this right: division is a series of subtractions ending in a limit of zero.

So, now I get why division by zero is undefined. When you consider the idea that 2/0 = 2, that would give 2-0 two times = 2 and that doesn't move at all. I almost tripped myself up when I considered that 2-0 one time doesn't move either, but it's a valid operation. The difference between division and subtraction is that one is a series of movements that ends in zero while the other is a single movement headed toward zero.

so a/b=c means bc=a in this case would be 2/0=2 which means 0x2 = 2 and this won't work because these equations are supposed to give movement back and forth toward infinity and limit 0, and well, again, this gives no movement.


so, I was defining division all wrong. Did I get it?

 

[AtWitsEnd]

You know what, I just had another epiphany. Now I understand the symbol for infinity. Two round bulbs on either end that swell connected by a band in the center that tightens. I can't believe I'm having so much fun! Why didn't I see any of this when I was in High School?