This applies to any numbers being divided by 0 really. In all honesty, wouldn’t 0 divided by anything just be 0? There was nothing there to divide in the first place, so why expect there to magically be a resulting number when you divide nothing?
In: Mathematics
0 can’t be divided by 0 because if you allow that the math really breaks down and you get nonsense results. I feel like the best way to explain what I mean is just to work through it and demonstrate said nonsense. Let’s suppose that 0/0 is defined and equal to some number. We don’t (yet) know specifically what number the result should be, so let’s call it *R* for now. In equation form, we can write:
* 0/0 = R
The first important thing to note is that, even though we don’t know the value of the variable *R*, we’re assuming it’s a number. That means it behaves in the exact same way “regular” numbers do. More specifically, it means that we can multiply any equation by 0/0 so long as we multiply both sides at the same time. Another important note is that when two quantities are equal, we can always freely choose which one to use in any given expression (e,g. we could always write 2 + 2 instead of 4). That gives us our next equation:
* (0/0) * (0/0) = R * R
Now let’s simplify each side of the equation. Since, for the purposes of this example, 0/0 is a valid fraction, we can multiply them in the usual manner of multiplying straight across the numerators and straight across the denominators. That yields:
* (0 * 0) / (0 * 0) = R^(2)
* 0/0 = R^(2)
* R = R^(2)
This tells us that *R* (i.e. 0/0) must either be 0 or 1. To help us out further, let’s return to the original equation and manipulate it in a different way, this time by *adding* 0/0 to both sides of the equation:
* 0/0 = R
* 0/0 + 0/0 = R + 0/0
* 0/0 + 0/0 = R + R
As before, since we’re assuming 0/0 is a number we can add the fractions in the usual manner. The fractions have the same denominator, so we can just add the numerators:
* 0/0 + 0/0 = R + R
* (0 + 0) / 0 = 2R
* 0/0 = 2R
* R = 2R
This tells us that if 0/0 is defined at all it must be the case that 0/0 = 0. So why is that a problem? Well, as part of assuming that 0/0 has a value we’ve also assumed that dividing by 0 in general is something we can do. And therefore, for some arbitrary number *x*:
* x / (0/0) = x/0
Here we’ll bring in another fundamental rule of arithmetic, that dividing by something is the same multiplying by the reciprocal (aka “one over”) of that thing. In equation form, for any three non-zero numbers *a*, *b*, and *c*:
* a / (b/c) = (ac) / b
The left-hand side of that equation looks like the equation from two lines ago, if we substitute in a = x, b = 0 and c = 0:
* x / (0/0) = (x * 0) / 0
We can simplify the numerator of the right-hand side:
* x / (0/0) = 0/0
And since we’re assuming that 0/0 = 0 we can substitute that in too:
* x/0 = 0/0
* x/0 = 0
This doesn’t bode well. We’ve shown that if 0 / 0 = 0 then it must also be true that x / 0 = 0 for every number *x* (since we only ever worked with it in the abstract, rather than any specific value). In other words, if dividing by zero as an operation is defined, the result must always be 0 no matter what the numerator is.
But we can even go one step further and drive the final nail in the coffin. Recall that you can always add 0 to anything without changing its value. The most general equation we can write using this is (for some arbitrary number *y*):
* y + 0 = y
We’re still assuming that x/0 = 0 for any number *x* so let’s plug that in:
* y + x/0 = y
Since we’re assuming that dividing by zero is a valid operation, we can multiply both sides of the equation by 0 to “clear” the fraction:
* (y + x/0) * 0 = y * 0
* (y * 0) + (x/0) * 0 = 0
* 0 + x = 0
* x = 0
And, there we go. If we assume that 0/0 is defined, we can follow that logic through and reach the insane nonsense conclusion that every single number in existence is 0. And certainly we know there are numbers other than 0 so our initial assumption must be false (i.e. 0/0 is not defined).
Siri has a fun answer to this one.
“Siri, what is zero divided by zero?”
Technically, you can…
Any number times 0 is 0, so 0 divided by 0 is any number. Just don’t expect your calculator to understand, as dividing by zero gives an error.
It can. 0/0 is different than 2/0.
0/0 is just indefinite not undefined.
0/0 can equal to 3, 0, -10 and all the real and complex numbers.
6/2 = 3
(6*0)/(2*0) = 0/0 which is no longer 3 (maybe it is). But it is clearly defined. So 0 can be divided by 0 it is a legitimate operation.
> There was nothing there to divide in the first place, so why expect there to magically be a resulting number when you divide nothing?
Imagine the situation. You made an advertisement to give two free microwaves to every customer that buy a new TV. So on the first day, three customers buy a new TVs, so you take the six microwaves you have in stock and give them to the customers:
+ You had 6 microwaves
+ You had 3 customers
+ You gave 2 microwave to each customers, which works because 6/3 = 2.
But the next day, you’re in trouble, because you don’t have any microwave remaining in stock. Fortunately no customer tries to buy a TV this day, so everything is fine in the end. The situation is:
+ You had 0 microwaves
+ You had 0 customers
+ You gave 2 microwave to each customers, which works because 0/0 “=” 2
Note the quotes around the “=”, because by the same reasoning, you could say that 0/0 “=” 42 (if the ad was “42 free microwaves for one TV bought”).
When we do maths, we want an answer that “always works”. Zero divided by zero doesn’t have a single answer that “always works”. But mathematicians are rarely satisfied with a “there is no answers”, so there is a lot of complex theorems about how to deal with situations where you actually need to answer “what is 0/0?”.
For example, a lot of them deal with case of “almost 0 / almost 0” by studifying the meaning of this “almost”.
Remember what dividing actually is, it is actually just a form of multiplication (well, the inverse of a multiplication). Just like how subtraction is just addition by a negative number, 5-2 is just a short way of saying 5+(-2). The reason this kind of thinking is important is that it easily explains why, say 5/0, doesn’t make sense. The relation between multiplication and division is that if you divide by a number, then multiply you should end up with the same number you started. Take the number 1, divide it by 2 then multiply with 2, you end back at 1. So we can say x*a/a = x. Of course this must be true, as a/a always equal 1 no matter the number.
Now, since we want to look at dividing by zero, we need to look at a=0. If we assume dividing by zero is possible, then x/0 must equal some sort of number. From our rule, if we multiply this number back with ‘a’ we should come back to where we started, x. The problem is, when you multiply by zero, you get zero. For example, 5/0 = ‘some number’. Now we take ‘some number’ and multiply by zero, which equal zero, not 5. As you can see, dividing by zero simply doesn’t follow our rules for what division is.
Do note that when we say you can’t divide by zero we actually mean it is not *defined.* We have defined mathematic rules to follow, but since the way we defined division clearly doesn’t work with zero, it simply mean the rules do not define that specific operation.
Lastly, for some fields of math we actually have defined it, so it can be used. It’s not some apocalyptic, ruins all of math kind of deal that some people like to call it – we simply haven’t defined it with the normal rules for division, but it’s “easily” solved by defining it. For example, for the Riemann Sphere, 1/0=infinity, and 1/infinite=0.
We like things that have one and only one answer. Dividing by zero can generate more than one answer.
Take 10x/5x. For all non zero x the answer is 2. So it sure looks like it should be 2 at zero.
Now take 4x/4x. For all non zero x the answer is 1. So it sure looks like it should be 1 at zero.
In fact, you can get all kinds of answers that look right. This ambiguity leads us to say that it cannot be done, as it is ill-defined.
Everything in math has a definition which is precise enough for a computer to understand and formally reason about.
Given a number x, the “multiplicative inverse of x” (written x^(-1)) is a number, y, such that x*y = 1. For example the multiplicative inverse of 2 is 0.5.
Given two numbers a and b, the “division of a by b” (written a/b) is defined to be a*b^(-1), where b^(-1) is the multiplicative inverse of b, assuming that it exists.
0 doesn’t have a multiplicative inverse, because there is no number y such that 0*y = 1. Because 0^(-1) doesn’t exist, division by 0 is not defined because according to the definition given above, division by b requires b^(-1) to exist.
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