why cant you divide anything by 0?

Question

this sounds dumb, but let me explain. since multiplacation is the opposite of division you can say 5x6=30, you can also say 30/6=5. so, since 4x0=0, shouldnt 4/0=0?

Answer 1

Because any way we try to define division by zero, this definition won't fit with the other rules for multiplication and division.
If, for example, you say, since 4x0 = 0, 4/0 should be 0, you should also be able to go the other way.
Just like 4x2 = 8 implies that 8/2 = 4 and 8/4 = 2
in the same way, by your reasoning,
since 4x0 = 0, 0/0 = 4
But you could just as well put any number in place of the 4 and get, for example
since 10x0 = 0, 0/0 = 10
since 5 x 0 = 0, 0/0 = 5
since 2x0 = 0, 0/0 = 2
etc.
Since we can't put any definite value to these terms, we call it indetederminate and leave division by 0 undefined.
This is proably one of the most frequently asked questions here on the math forum. Fortunately, we're willing to answer it over and over. Nerds are like that.

Answer 2

Let's start with your example: 5x6=30 and 30/6=5. Replace 5, 6 and 30 with 4, 0, and 0, and you get 4x0=0 and 0/4=0. Those are also both true-- and both have no division by zero.

Now, if you allowed division by zero, from 4x0=0 you might say 0/0=4, but NOT 4/0=0. That's NOT the way you were manipulating the formulae with 5, 6 and 30...

So, what's wrong with 0/0=4? Well... wouldn't you also say 3x0=0 and 2x0=0? But, how could all of 0/0=2, 0/0=3 and 0/0=4 all be true?

But, it's not crazy to want to divide by zero-- at least not by something "almost zero". In calculus, derivatives are often explained (at least informally) in terms of division by an "infinitesimal" or "vanishingly small" quantity. And, there is a branch of mathematics called non-standard analysis (developed in the 1960s) where you can do just that.

Answer 3

No-one has yet told you what's wrong with your own reasoning.

Yes, they have. She-nerd did so while I was typing mine!

In the example you gave, you replaced 5 by 4, 6 by the first 0, and 30 by the second zero.
So when you compare with 30/6 = 5, you should be saying
0/0 = 4.

But of course instead of 4 you could have any other number at all, so if division by zero is permissible then you can prove that 0/0 is equal to any number you like, i.e. it's indeterminate.

Another way of looking at it is:
Say 4/0 = r

Then 4 = r*0 and we have 4 = 0.

As she-nerd says, division by zero just doesn't fit with the rules.

Answer 4

If we allowed division by 0, the whole mathematics foundation would crumble because we can prove 1 = 2 if we allowed division by 0. If 1 = 2, then 2 = 3, and numbers would lose their uniqueness as we know it, not to mention everything in math that relies on the natural numbers.

Let me prove 1 = 2, should division by 0 exist.

Let a = 1 and b = 1. Then,

a = b. From a = b, we can get two equations:
1) Multiply both sides by b, and we get ab = b^2
2) Square both sides, and we get a^2 = b^2

From ab = b^2, ab - b^2 = 0
From a^2 = b^2, a^2 - b^2 = 0

Since these are both equal to 0, they are equal

ab - b^2 = a^2 - b^2

Factoring both sides,

b(a - b) = (a - b) (a + b)

Now, let's cancel out (a - b) from both sides by dividing by (a - b). We are then left with

b = a + b

If we substitute back a and b (which are both equal to 1), we get

1 = 1 + 1
1 = 2

However, this is obviously FALSE. Our incorrect step was when we divided both sides by (a - b). (a - b) = (1 - 1) = 0, so in that step we were dividing by 0.

Answer 5

> ".. since 4x0=0, shouldnt 4/0=0?"

No. It should be 4 = 0/0. And that is correct, but 5 = 0/0 is also
correct. 0/0 is indeterminate!

4/0 is infinite or minus infinite. Therefore, it is also indeterminate, but not finite.

Answer 6

That may sound right but what about 0/0 that doesn't give you 4. It gives you infinity
Proof:{ (unknown #) n =0/0
multiply both sides by 0
0=0n
what number times 0 gives you 0,
every number times 0 gives you 0,
the anwer is ifinity}
And inifnity is not 4 so that is why you cant divide by 0 ( unless it is 0/0)

Answer 7

When you divide 4 by 4 you get 1. 4 / 2 = 2; 4/1 = 4; 4 / .5 = 8; 4/.25 = 16.... It's going to continue this way...'promise! The smaller the divisor, the greater the quotient. When you get to zero as a divisor, you have reached infinity as a quotient.

Answer 8

The answer could also be infinity. When you divide you are saying eg 12/3= how many lots of three are in twelve. So you are saying how many lots of 0s in 12. You could add up as many zeros as you want and you would never get to twelve. So it could be infinity. I asked my bloke this and he's very good at maths, science etc and he didn't say I was wrong! (But he did say it could also be 0). So will we ever know the truth?

Answer 9

I can't think of a single practical application for even wanting to divide by zero. I wouldn't dwell on it too long. There are much more important things in life.

Answer 10

You can divide things by 0. The answer is infinity.