Math paradox?

@Noflaps said in #10:
> I don't really see dividing by zero as a "paradox," but I'm willing to read why it strikes you so, and I don't feel strongly that you can't possibly make an argument that it is.
>
> Perhaps because if we ASSUME it is valid it leads to a mathematical contradiction, which feels paradoxical -- but I wouldn't call that a "paradox" because it doesn't seem to be proper on its face to begin with. I think of paradoxes as apparently true things that lead to falsity, or apparently false things that lead to truth.
>
> I think of dividing by zero simply as an undefined activity. It leads to no answer.
>
> But, for fun, let's pretend it does: and call that answer "A" (for answer).
>
> Well, if we assume that, say, 8 / 0 = A
>
> Then we have, by simple algebra, 8 = 0 times A
>
> But 0 times ANYTHING is not 8. It is zero. "zero" examples of 8 cannot add to provide 8 -- and multiplication, after all, is merely repeated addition, made quicker for the fruitfully impatient.
>
> So if we assume that we can divide by 0, we have to throw algebra in the trash, or pretend that multiplying by zero yields something other than zero.
>
> So we must admit that division by zero cannot be done. Not exactly a paradox, but rather a wise community decision.
>
> Wise community decisions are increasingly rare, so it becomes harder to recognize them.

But here is the problem, if you say that you cannot divide by zero, it brings in many other problems.

Dividing and multiplying are essentially the same things, just inverses of each other. not being able to divide by zero implies an inability to multiply by zero. Since multiplying is the same as adding many times over, and dividing is subtracting many times over, then saying that you can't divide by zero implies an inability to add or subtract zero from something. Extrapolated enough, an argument could be made that using zero in any situation is actually unsolvable. This makes many functions of math void, and disallows you to ever come to a place where one side of an equation has a zero in it.

On the other hand, allowing division by zero has many other negative consequences. Although the exact answer of numbers divided by zero is debated, most mathematicians will agree that the answer is probably Infinity. for example: 1/0 = infinity (this doesn't quite work when you divide 0 by 0, but that's a completely different conversation). this would imply that the multiplication of 0 and infinity can equal any number (x/0 = infinity; therefore x = 0*infinity). using the additive quality of numbers, you can then add 0*infinity to both sides of an equation, and then come up with infinite answers to what was originally a one answer equation.

So most people just say that you can't divide by 0, because if you disregard something, other things can make more sense. If you want more proof on the division of numbers being infinity, then I can talk a little more about that.

@dstne , it is "dividing by zero" that brings problems, not saying that we "cannot" do so. Is that what you meant to type?

That's what my preceding post demonstrates. Dividing by zero leads inevitably to the false notion that zero times something DOESN'T equal zero.

But dividing by zero doesn't seem truthful or inevitable to BEGIN with, so I'm not surprised that it leads to falsity. That's why it doesn't seem to me like a "paradox," since I wouldn't have thought it true to begin with.

But I'll agree with you that it leads to contradiction. That's what my preceding post attempted to make clear.

Incidentally, I don't know any mathematicians who "debate" the answer obtained by dividing by zero -- because division by zero is impermissible -- it is an undefined operation, with no answer. Furthermore, "infinity" is not really a number. Indeed, there are an infinite number of different "infinities" ! It is shorthand for a process. The closest infinity comes to being "a number" is this: ONE of the infinities (aleph null) is the cardinality of the set of natural numbers. Another is the cardinality of the set of real numbers. Still another would be the cardinality of the power set of the real numbers.

But thank you for caring about math. I wish there were more like you. Many .many more.

And thank you for explaining your views. I am happy to be told when I've made a mistake, or even when somebody thinks I have -- since despite decades of effort, I can offer no guaranties or necessarily dependable representations about such matters, and still make plenty of mistakes! Like everybody who is not a glorious hippo.

Uh oh, I just planted a slightly-concealed logic problem in the text.

@Noflaps said in #12:
>

I guess the “paradox” that I am referring to is the fact that by allowing division by 0 causes problems, while not permitting division by 0 also causes problems. Of course, this is why most people simply say that you shouldn’t divide by 0 and that you shouldn’t question that decision, because if you do you get into some problems.

It’s just a weird little quirk about math that I think is interesting because it seems to be one of the fatal flaws that comes from math itself and not from logical fallacies. Of course, this assumes that the answer to x/0 is infinity (which there are a surprisingly large amount of proofs for). I just kind of like weird things, and theoretical mathematics always seemed to interest me.

I don't see the problems caused by not dividing by zero.

And the internet says many things, many of which are even true, but I do not agree that dividing by zero creates the number "infinity."

Indeed, "infinity" is simply not a real number -- in either sense of the word "real" (meaning either "actual" as opposed to "mythical," or instead referring to the name we give to that particular, never-ending set of numbers which contains, but adds on to, the rational numbers).

You say there are a "surprisingly large amount of proofs" for the notion that A / 0 = infinity. But it doesn't equal infinity. Dividing by zero is simply undefined. It doesn't produce any number at all. But don't believe me: check for yourself. We learn most by proving things to ourselves. And, as my relatives would agree, I am sometimes mistaken.

Could you provide us with one of the "proofs" to which you refer? I'd be curious to see it, genuinely. (Not by a link, but by typing).

Lastly, you wrote that "not being able to divide by zero implies an inability to multiply by zero. " But I don't agree with this at all. We can multiply by zero easily. We cannot divide by zero. I really don't think these things can be shown to be inconsistent. I know of no real proof that they somehow are.

But who knows what the internet says. Indeed, the internet is ready to teach me that Big Foot is right around the corner. And who knows? I've seen some pretty big tracks in the woods! If they hadn't been next to some discarded beer cans, I might have been tempted to embrace Big Foot's existence!

@Noflaps said in #14:
>

Well I personally believe in big foot (but that’s a completely different conversation)

Proofs: ok. So the most convincing proofs are geometric ones since that is how we prove most things, however since I can’t put a geometric model here, you’re gonna have to take my word for it. I don’t know the exact name, but it has to do with a circle, a line, the tangent of a line, and the meeting place of the one line and the y axis. I don’t remember exactly what it’s called but I’m sure you can find it online.

As far as logic proofs go, let’s go with any arbitrary number: Y. 1/Y is the equation we will deal with. Let’s say that Y = 10, then the equation would equal 0.1. Y got larger, the result would get smaller. If you followed this to its conclusion (well not really) at infinity, you would have to get an infinitely small number (0). Similarly, the smaller Y gets, the bigger the result is. As Y comes close to the value of 0, the result gets very big. When it hits 0, it must be very very very big, AKA, infinity.

Additionally, let us imagine that you are back in say third grade. Your teacher starts describing division, and it is explained that division is the amount of groups you can “give away” from something until you have nothing left. Well what if you have 5 oranges, but you want to give away groups of zero until you have nothing left. How many groups of zero can you give away before you have nothing left? Infinite. You will never give away all 5 of your oranges as groups of 0.

These are both logic based proofs, so I recommend looking into the circle proof for a more mathematical look at it.

Finally, with regards to why you can’t multiply if you can’t divide: here’s what I mean. Multiplication is the process of adding a certain amount of something some amount of times (X*Y = X + X +X... Y times). The same is true with division and subtraction. Division is simply subtracting a certain amount of something a certain amount of times (X/Y = X - (Y + Y + Y....)). So, if you can’t divide by 0, you theoretically should be unable to subtract 0 from something. Also, since subtraction and addition are the same thing, you shouldn’t be able to add 0, multiply by 0, or put anything to the 0th power, effectively rendering all equations with a 0 in them unsolvable.

Thanks for the response. But I can subtract 0 as many times as I need to. Indeed, I could subtract zero for the rest of my life, and if I died and reincarnated, I could subtract zero still more.

Here goes: ((((8 - 0) - 0)-0)-0) = 8

Rinse and repeat.

No matter how many times I subtract zero, I will get an answer.

That doesn't prove I can divide by zero. It just proves that I have too much time on my hands.

Be careful of saying things like "subtraction and addition are the same thing...." Subtracting is an inverse operation -- it is equivalent to adding the "additive inverse" of a number, not the number itself. And I can't think of ANY number that doesn't have an additive inverse. Keep that in mind.

Likewise, division is an inverse operation, and is equivalent to multiplying by the "multiplicative inverse" (or, in other words, "the reciprocal") of the divisor, not by the divisor itself.

But zero has no multiplicative inverse (or, in other words, no reciprocal).

We just can't overlook that and say that it "should" or "must" have one. The reciprocal of zero does not exist.

Zero is an interesting and unique number. It lives on the borderline between positive and negative. It is the additive identity. But it's not the loneliest number, since that number is the number 1 (according to a fine old song).

However, you MIGHT be able to convince me that Big Foot exits. After all, I can't really say, for sure, that Big Foot doesn't like beer!

@ronin3b said in #5:
> Math. Not even once!

Math. Not even: ones.

@Noflaps I didnt know u like maths so much? Are u a maths teacher? but kudos for xplaining.

@SlowBerserk said in #18:
> @Noflaps I didnt know u like maths so much? Are u a maths teacher? but kudos for xplaining.

lol I completely missed his response. I don’t think I want to write a whole response to this so, you can win this @Noflaps

I do not teach math for a living. But I once used math to make a living. We all use math, but it's not always obvious that we do.

Math prevents fantasy most of the time. Sometimes, it inspires fantasy.

But it's best to treat math as a friend. We should all get to know our friend better.

I love to explore math with serious admirers of math. Which should be everybody. But, unfortunately, and admittedly, is not.