What’s stopping mathematicians from defining a number for 1 ÷ 0, like what they did with √-1?
In: Mathematics
Nothing, and they have. The most common definitions are the Real Projective Line, and Riemann Sphere, which defines z/0 = ∞ for all non-zero real or complex numbers z.
There is also a less well known structure called a Wheel that fully defines division by adding 2 elements: ∞ = z/0 and ⊥ = 0/0.
As others have alluded to, these structures are not nuce to work with in general, which is why we don’t use them as much. In contrast, the complex numbers turn out to be suprisingly nice to work with in general, with few downsides, so we pretty much always assume they are available.
The concept of “limits” as used in calculus is a more precise way of treating this. 1÷0 is neither positive nor negative infinity, BUT the limit of 1/x as x approaches 0 is +infinity from the right, and – infinity from the left.
It turns out that infinities don’t behave with the same kinds of properties as numbers in general. They are best treated in conventional math as a value you can *approach* but never *equal*.
On the other hand, when you define i, and derive all the rules of how imaginary and complex numbers behave… What logically follows is a very self consistent system of mathematical rules.
It *is* defined when working in geometry with complex numbers and it does have useful properties. There is something called the Riemann sphere, in which you have all complex numbers (which you already know about based on the end of your question and one more number ∞, where we have rules such as
> z/0 = ∞ and z/∞ = 0 when z is any nonzero complex number
> z ± ∞ = ∞ and ∞ ± ∞ = ∞ when z is any complex number.
A related video: https://www.youtube.com/watch?v=hhI8fVxvmaw
because it causes contradictions and mathematicians don’t want people to realize that their system is flawed.
as it turns out 0 isnt actually a number.
tldr: it’s not useful, because it leads to logical contradictions that force you to abandon extremely basic principles of what it means to be a number.
Declaring *i* to be a value such that *i^2 = -1* turns out to not break anything when you go to do algebra/arithmetic, and it turns out to have a bunch of very useful properties. Here by “break anything” I mean does it lead to logical contradictions when you apply familiar rules for how equations behave and such, and it’s somewhat surprising that everything works out so nicely when you do this.
If we try to do the same thing with 1/0, things break pretty much immediately. Let’s see how. Call that value f, so we have the definition `f=1/0`. The definition of multiplication/division then implies that `f*0=1`, and then from there the definition of zero implies that `0=1`. Uh-oh.
So what does that mean in terms of implications for mathematics? Nothing, really. What actually happened in the previous paragraph is I defined a *new* number system that works like the real numbers but has an extra element whose multiplicative inverse is zero. And then I ended up showing that oopsies, that number system actually collapses in on itself; the only value in it is zero and nothing meaningful can be done with it. Once you have *one* logical contradiction in a system, all bets are off; nothing is true and everything is permitted.
If you do the same thing with defining a *new* number system that works like the real numbers but has an extra element whose square plus one is zero, you can follow a similar process of applying known rules to see what happens and figure out how such a number system might look. And this time rather than the whole thing blowing up in your face, you get *extra* stuff that wasn’t there before and new useful properties that weren’t accessible before.
All that is to say “why does one work and the other doesn’t” really just boils down to checking what happens when you take an existing set of axioms (statements that are assumed as foundational truths) and add a new statement you declare to be true. If the new statement contradicts the existing ones in some way, the result is useless. If the new statement is a logical consequence of the existing ones in some way, the result is unchanged. If neither the new statement nor its negation contradicts the existing ones, congrats, you’ve found a new, larger logical structure with more stuff in it. Do some math and poke around to see how it works; maybe it’s useful for something.
√-1 is consistent with other maths rules, while 1/0 isn’t.
Just consider 1/0=?, we can rearrange to make 0x?=1. But by the definition of multiply anything multiply by 0 must be 0. So that would break multiply.
i x i = -1 works and doesn’t break anything.
You can’t have such a number and still have arithmetic with the rules we’re familiar with. More specifically, with real or complex numbers multiplication is associative, which means that for any 3 numbers a,b and c it holds
(a•b)•c=a•(b•c)
This eule is extremely fundamental. Now suppose we defined some number x to be equal to be 1/0. Then we should have that 0•x=1. But then, since zero times any real number is zero, we see that
(2•0)•x=0•x=1 while 2•(0•x)=2•1, so associativity fails.
So a number system that includes 1/0 will be very different from the way we’re used to numbers behaving. On the other hand, complex numbers behave very similar to real numbers, in the sense that they satisfy (essentially) all the same laws
I can’t see a comment that has explained it this way: but the answer is you can’t do that *consistently*.
It is obviously quite true that you can define anything in any way you like. But if you say defined a fictitious number “infinity” and said 1/0 = infinity, then you immediately get into difficulties as follows:
1/0 = infinity
1 = infinity * 0 (multiply both sides by zero)
1 = 0 (any number times zero is zero)
If 1=0 that is going to unravel most of your mathematics. Of course you can say “new rule: infinity * 0 = 1” but then you will get into more trouble later on.
If you play around with this you will find that either you will change the rules so much that division is not really division in the usual sense and that you 1/0 doesn’t fit into the normal number system, or you will have to give up with too many inconsistencies.
The complex numbers are different. If you add the following assumptions (1) there is a – let’s call it a number – “i” and (2) i squared is -1, but you do not change any of the other rules of multiplication, division, addition or subtraction, you get a consistent system. Everything continues to work as before. You can solve equations in the same way. You do loose ordering (you can’t use > or < usefully any more – complex numbers are better thought of as a 2D plane) but you still get to do a great deal of mathematics.
The reason dividing by zero doesn’t work is that division is an inverse of multiplication. Multiplication by zero squashes everything down to one number (zero). You can’t invert that to get back your original number.
I’m going to take a stab actually going for an ELI5 answer… Which is to say it’ll be mathematically wrongish but hopefully useful for a 5 year old to understand generally.
> Divide by 0 doesn’t have a name because it’s not just one weird number. It’s different weird numbers depending on when you ask. It’s like asking “what color is a rainbow?” and expecting a single response. The best answer is “it depends where you look” it’s not that rainbows don’t have colors, they’re full of color, but there is no single color to describe what color a rainbow looks like.
At least one programming language I used defined it as the largest number the computer could handle. This makes sense as if you graph out y=1/x, you can clearly see it goes towards infinity
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