What’s stopping mathematicians from defining a number for 1 ÷ 0, like what they did with √-1?
In: Mathematics
A reason for doing so. The rules of mathematics can change based on what you are doing, but you still need an actual reason to do something that makes sense. If some mathematics was developed where it made sense to define this value, and it added value to the mathematics to do so, you can bet they absolutely would define it for that branch of mathematics. Even then, it doesn’t mean that this definition would carry over to things like basic arithmetic.
Let us suppose we make a representation for the logical “number” that we would define, which is infinity. This behavior can be seen by the smaller the divisor as it approaches zero, the larger the quotient. Until the result is a number so big it is indescribable and meaningless.
What then is the value of 2 ÷ 0? Twice as much infinity?
So what is stopping divide-by-zero being assigned any number or symbol is the land of nonsense you then enter by continuing the thought experiment.
Because √-1 gives a consistent result. Depending on how you approach it ÷ 0 can give you any number depending on how you approach it
They can do it, but it doesn’t really have any useful properties and you can’t do a lot with it. The main reason why mathematicians still use i for the square root of minus one is because i is useful in a lot of equations that have real world applications.
To the extent that we want or need to do math that involves dividing by zero we can use limits and calculus. This lets us analyze these equations in a logical way that yields consistent results.
First, there is only kind of a definition for √-1. It’s not actually solvable to anything. Like √2 can be represented as a decimal but √-1 just gets called i to allow to solve other √-x.
The second is because there isn’t a way to represent dividing by zero. It also doesn’t need to exist to solve any kind of problem. Think of buckets. If I have 10 apples and 5 buckets I can put 2 apples in each bucket. But it i have no buckets, there is no answer that resolves how many apples are in each bucket. There are no buckets.
Division is about sets of things.
4/8 is 4 things spread evenly into 8 sets.
10/2 is 10 things spread into 2 sets.
1/0 is 1 thing in no sets.
That can’t be defined.
You can’t have no sets of 1 object.
This has been done. Those number systems are called wheels. If you have heard of fields and rings, and wheel is kind of like those.
These number systems have positive and negative zero, so that 1/-0 = -inf.
The reason you don’t see them often is because they don’t describe reality well. For any real world calculation you need to do, you can do it without diving by 0. Because of that, these number systems are more of a curiosity for mathematicians than something you’d encounter in education.
Mathematicians have done exactly this, it’s just rare to encounter it outside of specialized fields.
In math, we set up a system of rules (formally, “axioms”), then work out the consequences. Importantly, there is no one best/correct set of rules to start from. But we prefer the simplest set of rules that let us handle whatever scenario we are thinking about, and prefer rules that lead to fewer inconsistencies.
Thus, when you are just getting started learning about roots and exponents we say “negative numbers don’t have square roots” and “you cannot divide a number by zero.” But then we get to quadratic equations and realize that being able to assign some value to the square root of a negative number would be convenient. So we come up with some new rules that let us do that, while still keep inconsistencies to a minimum.
We can do the same for 1/0, but the scenarios where we want to do so are rarer so you may not have run into it. And there in fact several different choices of rules that get made depending on exactly what we want to do.
In floating point math (what computers mostly use), we invent the “Not a Number” value and say that is what 1/0 is. In other contexts, we might say 1/0 “equals” positive infinity, and -1/0 is negative infinity. And in still other settings we invent hyperreal numbers which (sort of) give a value to 1/0 (https://en.wikipedia.org/wiki/Hyperreal_number).
You are looking for a number that, when multiplied by 0, equals 1. No real number does that, so you need to define a new system of numbers, and we want that system to be self-consistent. What other properties does this number (call it *q*) have? Can I add it to itself or another number and get another number? Does 0 × *q* equal *q* × 0? If *p* is the number that, when multiplied by 0, equals 2, what does *p* × *q* equal? Are *q* and *p* different numbers? If not, does *p* = 2*q*? It turns out it is tough to answer these questions in a satsifactory way.
division (a/b) answers the question “suppose I have b*x=a. what is x?” Well if b is 0, x cant be anything, since 0*anything is 0, and a is not 0.
and it is a key property of multiplication that anything*0 **IS** 0 so you cant redefine that property. that just leaves you in an impossible position where the answer to 0*x=a is unanswerable (unless a is also 0). even i*0=0
This brings us to the interesting 0/0, which is undefined. if you look at the equation 0*x=0, x could be ANYTHING and it would be true, so 0/0 is simultaneously ALL numbers.
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