eli5: Why haven’t mathematicians invented a symbol for x/0 like they have pi and imaginary numbers?
In: 277
Because there is no one value that x/0 could possibly be equal to. We know this because of a thing called limits.
The limit is the mathematical idea of the value that an expression approaches as some terms approach some value. It’s useful for assigning numerical values to expressions that wouldn’t normally have one.
So, let’s try to find a value for x/0 using limits. Let’s assume x=1, because we should just be able to multiply 1/0 by x to get the value we want. 1/0 can be thought of as the limit of 1/z as z approaches 0. So let’s plug in smaller and smaller values for z and see where the solutions tend towards.
1/0.1=10, 1/0.01=100, 1/0.001=1000, and so on. You can see that the value just keeps getting larger and larger, it’s tending towards infinity. Now, in general usage, infinity isn’t treated as a numerical value. But that’s fine, we could just redefine infinity and give it the numerical value 1/0 if we thought it would be useful. But there is one problem. For a limit to exist, the expression must approach it from both sides, so to speak. So let’s start taking negative values for z closer and closer to 0.
1/-0.1=-10, 1/-0.01=-100, 1/-0.001=-1000, and so on. Now, as z goes to 0, 1/z is getting smaller and smaller, tending towards _negative_ infinity. The limit is different for different directions of approach, meaning the limit doesn’t exist.
Some people have attempted to reconcile this by saying “maybe negative infinity and positive infinity are the same thing”, but they still run into problems down the line. It just doesn’t work.
Pi is simply a number, 22/7, when written in decimal simply never ends. And is used so often in so many fields that a shorthand reference to it is useful. Quite a few other constants/numbers have shorthand notations.
Imaginary numbers are a bit more complex. There was a lot of debate on whether or not they were meaningful, but we’ve found quite a few actual systems that seem to operate by them. You can’t accurately describe an electromagnetic field without imaginary numbers and some quantum/dynamic systems are most accurately described with “i”. You could write (The square of -1) in its place but its used often enough a shorthand is useful.
There is no meaning to x/0, how many of x would fit into no containers isn’t a reasonable question. Nor has any analogues been found for it in the world. It always generates a failed equation when used, so there isn’t any use for it in math or science. So it doesn’t need a shorthand.
Math breaks down when you divide by zero. There are wonderful “proofs” you can do where sneakily dividing by zero leads to things like 1=2.
Replacing 1/0 with a symbol like i doesn’t help- the same breakdowns in logic happen regardless.
Once logic is broken, you can “prove” anything at all, even contradicting statements. That makes math not useful anymore.
As an addition of the already mentioned points if you construct a modulo ring with a non prime, you can have the product of two numbers which gives 0. E.g. For Z_6 you can have 2*3=0 since 2*3 =6 mod 6 =0.
Because x/0 doesn’t work consistently with the mathematics of finite numbers, basically – and there’s no point in giving a symbol to something that you’re not going to use. Dividing by zero breaks arithmetic and leads to invalid answers. Things like negative numbers and i (and, indeed, zero) may have seemed ludicrous when they were first proposed – but they’re well behaved and don’t mess the place up, so we give them house room. (They also turn out to be very useful, to boot.)
Pi is different, in that it’s a very specific number. It crops up all over the place – but it’s irrational, so the only practical way to write it down is to give it a name/symbol.
Because it doesn’t make sense. How can you split 5 apples into zero groups?
Pi and imaginary numbers have a mathematical value, and coresponding symbols and variables to represent those values. X/0 does not have a value to define. It cannot be anything other than undefined.
X/0 being undefined is different from i being imaginary. because the square root of a negative number is not undefined, its just impossible to represent with real numbers. So we defined a symbol to represent this non-real number. i very much is defined.
X/0 has more to do with philosophy than mathematics. We can describe why x/0 is undefined with mathematics, but beyond that it cannot have useful implication. You could say that x/0 is a symbol to represent an undefined value.
This is the premise of the Riemann sphere — created by one of the most famous historical mathematicians. There are many other analogues, but that’s a great place to first get into the topic.
Because π and I can participate in mathematical expressions in a way that x/0 can’t. You can’t use the infinity symbol in expressions, because It’s just a tendency, not a value.
In computing there is [NaN](https://en.wikipedia.org/wiki/NaN) (Not a Number) for results that are undefined or unrepresentable. It is there really to signal and propagate error state. There are number of ways to [get NaN](https://en.wikipedia.org/wiki/NaN#Operations_generating_NaN) including x/0, but once you have an NaN all further operations take that NaN will keep returning NaN.
While useful for computing as far I know it’s not particularly useful for mathematicians?
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