Why does "making up" imaginary numbers work, but making up numbers for 1/0 etc does not?

Question

To be exact, we can label both $\sqrt{-1}$ as $i$ and $\frac{1}{0}$ as, say, $w$, but for some reason the latter "breaks" the logic, and introduces various uncertainties, while the former extends real numbers into a complex plane neatly, and forms a closed field. Can we predict which new concepts would fit into mathematics and which would simply break it?

Answer 1

How does giving a label to something that breaks mathematics change the fact that it breaks mathematics? Division by zero allows for contradicting transformations, such as yielding $1=2$. The existence of $\sqrt{-1}$ does not allow for such transformations. Giving or not giving something a label never changes its inherent implications.

Answer 2

Historically, I believe this comes from Cardano's formula for the cubic. In his formula, it is sometimes indicated to take the square root of a negative number. Before Cardano's work, it was just assumed an equation like $x^2+1$ had no solution. Using his formula though, even equations with all real solutions could involve taking the square root of a negative number. These weird quantities somehow lead to meaningful results. "Imaginary" numbers arose organically and their properties were teased out using the formula. They had definite additive and multiplicative behavior.

The multiplicative inverse of $0$ doesn't pop out in this way. By definition, the multiplicative inverse is the number by which you multiply 0 to get 1. 0 time anything is 0 though. There is no such element. The closest you can come to establishing a meaning is with limits, but treated rigorously, this still doesn't resolve the issue.

Answer 3

Who said introducing $i$ doesn't break something? The complex numbers have a lot of power over the real numbers, but you do lose out on other things that you might take for granted. For example, there is no natural total order on the complex numbers - if you have two real numbers $a$ and $b$, then we know for sure that either $a > b$, $a < b$ or $a = b$, and those relationships are preserved under a variety of useful functions (e.g. adding something to both sides, or multiplying them by a positive number, or taking the exponential). On the other hand, if $a$ and $b$ are complex then there isn't a standard comparison that naturally covers "bigger", "smaller" and "equal" with the same simplicity as $>$ on the real numbers.

The reason that lots of things don't break when you introduce complex numbers is because both the real and complex numbers both share a kind of structure called a field. To have a field, you need a set $F$ and two operations "$+$" and "$\cdot$" that follow specific rules:

1. If $x, y \in F$, then $x + y, x \cdot y \in F$ (i.e. the set is "closed" under these operations).
2. If $x, y \in F$, then $x + y = y + x$ and $x \cdot y = y \cdot x$.
3. If $x, y, z \in F$, then $(x + y) + z = x + (y + z)$ and $(x \cdot y) \cdot z = x \cdot (y \cdot z)$.
4. There is an element of $F$ labelled $0$ such that $x + 0 = x$ for all $x \in F$.
5. There is an element of $F$ labelled $1$ such that $x \cdot 1 = x$ for all $x \ in F$.
6. For every $x \in F$, there's an element of $F$ labelled $-x$ such that $x + -x = 0$.
7. For every $x \in F$, either $x = 0$ or there's an element of $F$ labelled $x^{-1}$ such that $x \cdot x^{-1} = 1$.
8. If $x, y, z \in F$, then $(x + y) \cdot z = (x \cdot z) + (y \cdot z)$.

You can construct a wide variety of fields, including the rational numbers, integers modulo a prime, etc., but as long as you have those 8 properties (called the "field axioms") there are a whole lot of other things you can prove must be true. Mathematicians like when this happens, because if you see something interesting in, say, the real numbers, then if you can prove that it happens solely because of the field axioms then you know that it also works in all those other fields.

As it happens, one of the things you can prove is that $0 \cdot x = 0$ for all $x \in F$. But that's a problem if you want to introduce a $\frac{1}{0}$ element to your field, because presumably the aim is to have $0 \cdot \frac{1}{0} = 1$, but we already know that $0 \cdot \frac{1}{0} = 0$, which means that the only way to keep everything working is for $1 = 0$, and if $1 = 0$ then that means that $x = 1 \cdot x = 0 \cdot x = 0$, so your field actually has only one element and depending on who you ask that's either a very useless field or it isn't actually a field at all.

So, can we predict whether adding something to a field breaks it? Well, if you can show that it doesn't play nice with the field axioms (like giving $0$ a multiplicative inverse), then absolutely. On the other hand, if the thing you introduce is consistent with the field axioms then at a bare minimum you know it doesn't break anything relating to the field structure, but for anything beyond that you need to work a bit harder. Extending the real numbers to the complex numbers doesn't break any of the field axioms because you can very easily check that addition and multiplication as they are commonly defined for complex numbers fit the field axioms just fine. There's an additional axiom that defines ordered fields, and proving that the complex numbers don't fit the bill for that takes a little bit of extra work.

Fields aren't the only structure out there, though. If you care more about geometry or topology then there are things like the real projective plane which adds a "point at infinity" to $\mathbb{R}^2$ and preserves plenty of nice properties about lines and curves and shapes, and there's an associated set of axioms for defining those kinds of structures. Or you can look at the proof for why $0 \cdot x = 0$ in a field and work out which of the field axioms are fundamental to it, and based on that you can try to work out what kind of structure "real numbers but with division by zero" can be, and hence what bits of the algebraic structure you can preserve.