Is zero divisible by zero?
Is zero divisible by zero?
Seems like it should and the result should be one. Does mathematics agree with me on that?
There are several ways of approaching that particular question. And none are simple, actually.
First, just to frame why 0/0 is so weird, consider 1/0. Asking "what's 1/0" is like asking "what number when multiplied by 0 equals 1?" There's no answer because any number multiplied by zero is zero and no number multiplied by zero is one.
So now on to 0/0. "What's 0/0" is like asking "what number when multiplied by zero gives zero?" And the answer is "all of them." 1 times 0 equals 0, so 1 is an answer. But also 2 times 0 is 0. And so is pi. And 8,675,309.
So, you could say that 0/0 doesn't have a single answer, but rather an infinite number of answers. That's one way to deal with 0/0.
Another way is with "limits". They're a concept usually first introduced in calculus. Speaking a bit vaguely (though it's definitely worth learning about if you're curious, and it seems you are), limits are about dealing with "holes" in equasions.
Consider the equasion y=x/x. With only one exception, x/x is always 1, right? (5/5=1, 1,000,000,000/1,000,000,000=1, 0.00001/0.00001=1, etc.) But of course 0/0 is a weird situation for the reasons above.
So limits were invented (by Isaac Newton and a guy named Leibniz) to ask the question "if we got x really close to zero but not exactly zero and kept getting closer and closer to zero, what number would we approach?" And the answer is 1. (The way we say that is "the limit as x approaches zero of x divided by x is one.")
Sometimes there's still weirdness, though. If we look at y=x/|x| (where "|x|" means "the absolute value of x" which basically means to remove any negative sign -- so if x is -3, |x| is positive 3) when x is positive, x/|x| is positive 1. When x is negative, x/|x| is negative 1. When x is 0, x/|x| still simplifies to 0/0, so it's still helpful to our original problem. But when we approach x=0 from the negative side, we get "the limit as x approaches 0 from the negative side of x/|x| is -1" and "the limit as x approaches 0 from the positive side is (positive) 1". So what gives?
Well, the way mathematicians deal with that is just to acknowledge that math is complex and always keep in mind that limits can differ depending which direction you approach them from. They'll generally consider for their particular application whether approaching from the left or right is more useful. (Or maybe it's beneficial to keep track of how the equasion works out for both answers.)
I'm sure there are other ways of dealing with 0/0 that I'm not directly aware of and haven't mentioned here.
So, to wrap up, there are some questions in mathematics (like "what's 0/0?") that don't have a single simple answer. Mathematicians have come up with lots of clever ways to deal with a lot of these cases and which one helps you solve one particular problem may be different than which one helps you solve a different problem. And sometimes "there's no right answer" is more helpful than using clever tricks. Sometimes the problem can also be restated or the solution worked out in a different way specifically to avoid running into a 0/0.
It's definitely unfortunate that they don't teach some of the weirdness of mathematics in school. But something I haven't even mentioned yet is that all of what I've said above assumes a particular "formal system." And the rules can be quite vastly different if you just tweak a rule here or there. There's not technically a reason why you couldn't work in a system which was just like Peano Arithmetic (conventional integer arithmetic) except that 0/0 was by definition ("axiomatically" -- kindof "because I said so") 1. (Or 42, or -10,000, or whatever.) That could have some weird implications for your formal system as a whole (and those implications might render that whole formal system in practice useless, maybe), or maybe not. Who knows! (Probably someone does, but I don't.) (Edit: looks like howrar knows and it does indeed kindof fuck up the whole formal system. Good to know!)
One spot where mathematicians have just invented new axioms to deal with weirdness is for square roots of negative numbers. The square root of 1 is 1 (or -1), but there's no number you can multiply by itself to get -1.
...right?
Well, mathematicians just invented something and called it "i" (which stands for "imagionary") and said "this 'i' thing is a thing that exists in our formal system and it's the answer to 'what's the square root of negative one' just because we say so and let's see if this lets us solve problems we couldn't solve before." And it totally did. The invention(/discovery?) of imagionary numbers was a huge step forwards in mathematics with applications in lots of practical fields. Physics comes to mind in particular.
a guy named Leibniz
“If you look closely you can actually pinpoint the exact moment his heart breaks in two”
This was enjoyable to read. Nice flow and storytelling, especially in the first half. Thanks!
A very useful answer. 👍
This was awesome. Thank you
You've made this mistake a couple of times throughout your comment, the correct spelling is "equation".
It does not. If you enforce 0/0=1, then you end up in a situation where you can prove any two numbers are equal to each other and you end up with a useless system, so we do not allow for that.
e.g. 0=0*2 -> 0/0 = (0/0)*2 -> 1=1*2 -> 1=2
If you get into calculus though, you'll have ways to deal with this to some extent using limits.
Dividing by zero is literally a prospect that breaks the algebraic rules. The general high school way to think about this is:
I have no pizzas, and no friends to split them amongst. How many does each one get?
It really doesn't matter whether infinity, zero, or anything in between in this context, which is why it is undefined.
x/x = 1
0/x = 0
x/0 = ±infinity
When x=0, it is all three of these rules.
0/0 is an indeterminate form and could equal anything depending on the specific zeros Involved.
I need you to explain that one. Specific zeros? Aren’t they all just equal to zero?
Indeterminate forms come from limits. It's not the question you asked, and I think this answer was a little off the mark because of it. For the sake of shared knowledge, I will explain anyways:
When looking at a limit, it's important to note that you aren't working with zero (or infinity, or any number you are studying the limit of), what you are working with are numbers approaching the limit. For example, for (x+1)/(x), the expression has no equivalent value at x=0, as 1/0 does not exist. We can see why if we use the limit as x approaches zero. The numerator will approach 1, and the denominator approaches 0. The numerator has little impact on the value of the expression, but the denominator... dominates the value, for the pun. And, while we can't evaluate at 0, we can put really small numbers in there and see what happens- and what happens is the expression becomes incredibly large. I'm sure that if you don't see where this is going, you can go to Desmos or some other graphing calculator and try it for yourself.
As far as the indeterminate form- 0/0 is always undefined, at least in most mathematics. However, if you were to look at equations :
- y = x/x
- y= x2/x
- y= x/x2
you'll see the curves behaving differently around x=0. The first makes 0/0 look like 1, the second makes 0/0 look like 0, and the last will make 0/0 look like infinity*. Once again, note, however: 0/0 does not exist, and there is discontinuity on all of these curves at x=0.
*Edit: or negative infinity, I forgot that this limit doesn't exist. Even though the limit doesn't exist, it is still a useful example.
It's a calculus thing. We can only give the expression a value if we know the functions giving us a zero value that are being devided. For example if we were dividing the function (X) by the function (X^2) at zero our we would get infinity (Wikipedia has a pretty good page on indeterminate forms).
You could also think of it like multiplying both the numerator and denominator of a fraction by 0. This should preserve the fractions value, but multiplying by 0 essentially erases both values so we can no longer know what the fraction equals unless we know how both values came to be 0.
Zero divided by zero is undefined. In that it literally does not meet the definition of division (from a mathematical perspective.)
This is a bit tricky because the reason that 0/0 is undefined is separate from why any other number divided by zero is undefined.
If I divide 6 by 0, there's no number I can multiply by zero to get back to 6. Since I can't get back to the 6, this is undefined.
If I divide 6 by 2, I get 3. And I can multiply 2 by 3 to get 6. Now it's genuinely important that there is no other number I can multiply 2 by to get 6. There has to be a single unique result for both the division and the going back via multiplication.
Now, if we assume 0/0 = 1, that is fine. And I can multiply 1 times 0 to get back to 0. Checks out so far. However, 1 isn't unique in getting back to 0. If I try 5 x 0, I get 0. Which, by the rules of division should mean that 0/0 = 5. Which clearly it wouldn't.
So zero divided by zero is undefined because there is an infinite amount of numbers that would get me back to zero.
You can think of it like this:
If a / b = c, then c • b = a
So if 0 / 0 = 1, then 1 • 0 = 0
Which is true. It feels right at first.
But what about other numbers?
If 0 / 0 = 7, then 7 • 0 = 0
That also works. But if every number works, then which one is the correct one?
The question boils down to:
Find x, where x • 0 = 0
Now it might be more clear that the question doesn't really make sense, so no answer will make sense.
We can still solve the equation by making a set of all possible solutions which conveniently is a set of all real numbers.
x is in a superposition :D
On a serious note, can quantum physics help?
Can nothing go into nothing once or not at all because there's nothing to put somewhere and nowhere for it to go? 🤔
How many nothings are in nothing? I guess one nothing.
Mathematically, zero is nothing.
You can't have one of nothing.
There's no such thing as a single nothing. All of the nothings are all the same nothing, there are infinite amounts of nothing but no nothing at all.
Of course there is a limit to nothing because there is something, in a more real and universal sense, but mathematically speaking there is no nothing.
So whenever you divide or multiply anything by nothing all that you get is nothing. If you multiply nothing by itself you end up with nothing. If you divide nothing by itself you get nothing.
When I was a kid I got really fascinated with Zen Koans.
Koans being the kind of self-evident riddles that are most famously popularized by the question, "if a tree falls in the woods, and no one is around to hear it, does it make a sound?"
And ultimately I ended up creating my own Zen Koan by accident.
That Koan being, "What does nothing look like?"
I did everything in my power to visualize what nothing would look like. I imagined infinite black spaces with nothing around and nothing in it, a void of eternal darkness. Vast mental landscapes devoid of heat or cold for light or dark or sound or wind or air, but it just wasn't "nothing", and I didn't know why.
I tried and I tried and I tried but I couldn't help but feel like I had failed to understand.
Then one day I was standing on a hill, the wind was blowing through my hair, the Sun was shining on me, it was spring outside and the birds were singing and I was trying to visualize what nothingness would look like when I realized the reason why I couldn't visualize nothingness.
I couldn't visualize nothingness because I was looking at it.
If there was nothingness I wouldn't be there to see it.
It was at that moment a space in my brain opened up for the concept of nothing and I understood.
You can't divide nothing by nothing because there's nothing to divide with. You can multiply something by nothing because there is something to multiply with, yes, but you can't divide something by nothing because there's nothing to divide with.