[Industrialblog,
February 12, 2007]
Thinking out loud about a math question
It's well-known that if you divide by zero, the answer is "undefined".
I just had a thought. Isn't dividing by zero the same as dividing zero times? Thus, dividing zero times is the same as not dividing it at all. That is, the answer is the numerator. No, I am wrong, and I know it. But I can't exactly figure out why. I need someone to complete my thinking.
I think the answer is something like this: By saying I divide it "zero" times, I am making a mistake. You don't divide by one and make one cut in a pie. To divide by one, I make a pie into one equal part. That is, it's the whole pie, a correct answer. If I divide by two, I end up with two equal parts. But if I divide by zero, I am saying there are zero equal parts. So would any series of cuts that divides the pie into unequal parts be an answer?
"Why is this pie divided into seven unequal parts?"
"Because I divided it by zero."
Or is this the answer: It's undefined because there are an infinite number of answers — any way of dividing the pie into unequal parts ends up being the equivalent of "dividing by zero?" That is, to say the pie was divided by zero simply doesn't tell you enough about how it's divided — the equation written as "1/0=?" simply tells you that the pie has in fact been divided, and that the resulting parts are unequal.
I guess what I'm getting it is I've always assumed dividing by zero was some kind of unknown, unknowable thing, some great mystery, just one of those ineffable things that we'll have to find the answer to in the afterlife. Actually, it just means the answer to what happened to the pie cannot be defined in an equation — NOT that there may not be other means of expressing what happened to the pie.
Is that it or am I wrong? Does everyone already know this? Or have I had another penetrating glimpse into the blatantly obvious?
I just had a thought. Isn't dividing by zero the same as dividing zero times? Thus, dividing zero times is the same as not dividing it at all. That is, the answer is the numerator. No, I am wrong, and I know it. But I can't exactly figure out why. I need someone to complete my thinking.
I think the answer is something like this: By saying I divide it "zero" times, I am making a mistake. You don't divide by one and make one cut in a pie. To divide by one, I make a pie into one equal part. That is, it's the whole pie, a correct answer. If I divide by two, I end up with two equal parts. But if I divide by zero, I am saying there are zero equal parts. So would any series of cuts that divides the pie into unequal parts be an answer?
"Why is this pie divided into seven unequal parts?"
"Because I divided it by zero."
Or is this the answer: It's undefined because there are an infinite number of answers — any way of dividing the pie into unequal parts ends up being the equivalent of "dividing by zero?" That is, to say the pie was divided by zero simply doesn't tell you enough about how it's divided — the equation written as "1/0=?" simply tells you that the pie has in fact been divided, and that the resulting parts are unequal.
I guess what I'm getting it is I've always assumed dividing by zero was some kind of unknown, unknowable thing, some great mystery, just one of those ineffable things that we'll have to find the answer to in the afterlife. Actually, it just means the answer to what happened to the pie cannot be defined in an equation — NOT that there may not be other means of expressing what happened to the pie.
Is that it or am I wrong? Does everyone already know this? Or have I had another penetrating glimpse into the blatantly obvious?
One thing that may be worth thinking about is the sequence
1/1 = 1, 1/(1/2) = 2, 1/(1/4) = 4, ... as you keep on decreasing the denominator, you are getting closer to dividing by 0, and the answer is approaching positive infinity. From this perspective, what you are doing is blowing that pie up into so many pieces that there aren't finitely few of them anymore.
But this is non-nonsensical, because it's equally valid to approach division by zero from the other side.
1/(-1) = -1, 1/(-1/2) = -2, 1/(-1/4) = -4, etc, as you keep increasing the denominator (making it less negative), you are getting closer to dividing by 0, and the answer is approaching negative infinity.
"Oh, be glad that you don't divide by zero,
That's one thing you never do in algebree,
For if you were to divide by zero,
You'd be sent into infinity..."
Incidentally, it's not that the result of dividing by zero is "undefined", it's that there's no definition for division if the denominator is zero. That is, if the denominator is zero, no one can tell you how to compute the result; there are simply no instructions to follow to get an answer. It's like if you were weighing pigs and suddenly came across the feeling of being full after eating a lot of chocolate. You can ask around all you want, but no one can tell you how to put that on a scale.