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.
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.
“If you look closely you can actually pinpoint the exact moment his heart breaks in two”
Ha!
I didn’t honestly know Leibniz’ full name and was on mobile and didn’t want to make the effort to go google it and copy it.
But, now that I’m on a full-sized qwerty keyboard, his full name is “Gottfried Wilhelm Leibniz”.
Let’s add “the famous mathematician and philosopher” at least ;)
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”.