Is division by zero actually infinity?

Question

When I was in Middle School I asked why you couldn't divide by zero. I was told the generic answer that ... you just can't. In high school I was told that it is undefined. I was told by an engineer that zero divided by zero was actually infinity. At the time he was trying to explain that the closer and closer you get to zero the higher and higher the number got but I couldn't "get" it then.

Now having taken Calc 1 I understand limits and I guess what he was trying to explain was (and excuse my syntax, as the text only formula is not conductive to mathematical formula writing) The limit as x approaches 0 of 0/X is infinity. Is this what he was getting at by saying that zero / zero is infinity or was he just making a false assumption that since both the right and left side of the function approach infinity that zero / zero was infinity?

I appreciate your help!!!

Okay ... my question was misleading... not saying dividing ANYTHING by zero is infinity, specifically diving ZERO by ZERO .... still ... undefined?

Answer 1

Ok. zero divided by zero is indeterminate.

Quoted from Wikipedia:
{quote}
The most common example is 0/0. In the expression x/x, as x approaches 0, the limit is 1; but in the expression x2/x, as x approaches 0, the limit is 0. In each case, however, if the limits of the numerator and denominator are evaluated and plugged into the division operation, the solution is 0/0. So (roughly speaking) 0/0 might mean 1, or it might mean 0 (and in fact, by appropriate examples, it can be made to mean anything); so it is indeterminate.

Note that an indeterminate form is not the same thing as an expression that is undefined.
{end quote}

So he was wrong in saying it was infinity. It is not neccessarily undefined either. Check wikipedia's indeterminate form for more info.

Answer 2

Anything divided by zero is undefined because remember, when doing division you can always reverse it with multiplication, that is if A/B = C, the A = B X C, but this does not hold true if you define something divided by zero as infinity. For example, lets look at 5/0 = infinity, they 5 = 0 X infinity.

That is why dividing by zero is undefined, not infinity.

Answer 3

Let's look at four variations here.

1/1 = 1

This one has no problems.


0/1 = 0

This one is also OK.


1/0

We have a problem here.

1/0 is said to be undefined because division is defined in terms of
multiplication. a/b = x is defined to mean that b*x = a. There is
no x such that 0*x = 1, since 0*x = 0 for all x. Thus 1/0 does not
exist, or is not defined, or is undefined.


0/0

The expression is called "indeterminate form." This can have a large range of different values.

0/0 = 1, since 0 = 1*0
0/0 = 2, since 0 = 2*0
0/0 = 3, since 0 = 3*0
...
0/0 = 1000, since 0 = 1000*0
...
etc

In fact, there is no meaningful solution. The system is not closed under addition, subtraction, multiplication, or division.

Answer 4

The strict answer is no.When you divide by zero, let's say 6/0, you're actually asking the question: how many zeros are there in the number 6. Well, it's obvious that no matter how many times you add up zero to itself, you'll never get the number six. This is why the divide by function approaches, but never reaches infinity.

The definition of zero being what it is, we can safely say that if you add up an infinite number of zeros, you will still not get 6. So, it can't be done at all.

And... how many zeros are there in zero? The answer is, indeed, not only infinite, but it can literally be any number you want it to be. Are there one zeroes in zero? You bet. Ten zeroes in zero? Sure. Infinite? Absolutely!

Answer 5

You want a real tangible way to understand this? Go back to first grade math. No, really!

Take an apple and divide it into four. Take another and divide it in half. Now take another and divide by nothing.....

Divide it by noth...erghhghghg

divide by n...aklajfjlkfjuoeiur

Oh, wait, you can't really do that can you? Why? Because nothing is nothing. Zero is not a number. It has no value.

The only way to get nothing from something is to subtract (remove) the something and then you have nothing, or don't have something.

By the way, there is no such thing as negative numbers either, but that's another story. And that's exactly why time travel is not possible.... OOPS, there I go again! I LOVE math!

by the way, zero divided by zero is not infitity, It is a meaningless expression because you cannot divide nothing into anything. Nothing cannot be divided or multiplied by anything because there is nothing there to multiply or divide to begin with. you can't subtract from nothing because there is nothing there to remove! the only thing you can do with nothing is add to it. If you have a bag with nothing in it you can add as many apples to it as the bag will hold, but you cannot take any apples out of an empty bag.

Negative numbers (a misnomer) are nothing more than a concept. The negative symbol we put in front of numbers is really just a way of saying the apples are somewhere other than this bag. For instance:

Let's say you have ten dollars in the bank and you write a check to Joe's store for twenty dollars. The bank sends you a letter telling you you have negative ten dollars in your account. Well, no, you don't! You don't have anything there because you can't have less than nothing. Joe has ten real dollars in his till that belongs to the bank and you have to pay the bank back because you essentially borrowed ten dollars from the bank because they had to cover the ten dollars you didn't have! The negative ten dollar balance is nothing more than the bank's way of reminding themselves and you that you owe them ten dollars. Acually it's probably more like thirty because they have probably added (or subtracted would be more accurate) a twenty dollar fee from your balance for doing this foolish thing in the first place.

And infinity is a concept even more non-existant than zero (is that possible?). There is no such thing as infinity unless the universe is not finite...

Okay, that's enough, I'm done. Are you really confused now? if so, sorry.

Answer 6

Part of the reason is that "infinity" is not a real number. So saying that 0/0 is "infinity" is another way of saying it is not real.

In Calc 2 or 3 you will discover that regarding limits, 0/0 is not definitively undefined. L'Hospital's Rule is what you need to see.

On the other hand, my favorite explanation to give my pre-calc and college algebra students, when we start to discuss domains of rational functions, is "Division by zero makes baby Jesus cry."

Answer 7

Division by zero is simply undefined, not allowed, a paradox in mathematics and so on.

The terms 'infinity' and 'infinite' belong in your imagination - but the number zero is very real.

It is simply because anything - ANYTHING - multiplied by zero - NOTHING - is zero!

Think about it:
If a number divided by zero is infinity, then any number divided by zero is infinity and zero multiplied by infinity is any number you want it to be - and that, my friend, just isn't logical. And if mathematics is anything, it's must be logical.

And that's the way is in a nutshell. You don't want to upset the "divide by zero" police, now do you?

Answer 8

Yes, my brother explained the same thing to me (the limit thing). I wouldn't say division by 0 is infinity -- it really is undefined. But the limit as x tends to 0 of 0/x, is infinity.

[Think about it, how many times can 0 go into anything?]

Answer 9

this can get dicey once you initiate entering into more suitable mathematical properties. maximum easily- branch is sequential subtraction. (20/5= 4... 20 minus 5 is 15, minus 5 back is 10, minus 5 back is 5, minus 5 back is 0...4 "subtractions"). Now take 20/0. you may subtract 0 a limiteless volume of situations from 20, and 20 will in no way get any smaller. therefore, dividing a huge decision by using 0 is more beneficial or less equivalent to infinity.

Answer 10

I don't like your professor's statement. You flat out can't divide by zero, because division is the inverse of multiplication.

6/3=2 b/c 2*3 = 6

But if you try this with 1/0=x, the corresponding mult. statement is x*0=1, which is impossible.