4

Originally, taking the square root of a negative number wasn't possible, but then Rafael Bombelli added an imaginary number axis to allow that. Why, then, is there no 3rd dimensional number line to allow division by 0? If there was, you could create a unit z, where z is 1/0. Therefore, 5/0 could be 5z, etc., and multiplying by its reciprocal is a 90° rotation to the real number axis, just like squaring i causes a 90° rotation to the real axis.

username65536
  • 51
  • 3
    what is $0 \times z$ in this system? – ComptonScattering Feb 20 '21 at 02:09
  • 3
    Adding a square root of $-1$ is less "destructive" than adding a multiplicative inverse of $0$ in a precise sense: we can do the former and still satisfy the basic properties of addition and multiplication, but doing the latter immediately breaks things. So it's not that there is no coherent number system in which division by zero is possible, but rather that there is no coherent number system in which division by zero is possible and the usual basic properties of addition and multiplication hold. – Noah Schweber Feb 20 '21 at 02:31
  • 4
    There is a different sort of algebra where division by $0$ is a defined sort of concept; Wheel theory: https://en.wikipedia.org/wiki/Wheel_theory - but notice this is very different to just "adding an extra axis". It's an entirely different algebraic structure – Riemann'sPointyNose Feb 20 '21 at 02:34
  • Usually questions like this are crank-y, but I like this one. +1 – Randall Feb 20 '21 at 02:58

2 Answers2

7

This is good, creative thinking about number systems. But here's the hitch: you have to follow through with the consequences of your definitions. If $z = 1/0$ as you suggest, then it must be that $z \cdot 0 = 1$.

Why is this a problem? Well, you are probably already familiar with the idea that $a \cdot 0 = 0$ for any number $a$. Incidentally, this is a consequence of the fact that $0$ is the additive identity and that your operations of multiplication and addition satisfying a distributive law. To wit,
$$ a \cdot 0 = a \cdot 0 + a \cdot 0 - a \cdot 0 = a \cdot (0 + 0) - a \cdot 0 = a \cdot 0 - a \cdot 0 = 0 $$ With $a = z$, your special reciprocal of $0$, you now have that $z \cdot 0 = 1$ and $z \cdot 0 = 0$. Thus, $1 = 0$.

Once you establish that $1 = 0$ in your number system, then any number is equal to any other number! Consider two numbers $b$ and $c$, and calculate: $$ b = b + 0 = b + 0 \cdot (c-b) = b + 1 \cdot (c-b) = c. $$

All your numbers collapse into a singularity. Oh noes!

Sammy Black
  • 28,409
3

Interesting question! Although I’m not sure Rafael Bombelli created the imaginary axis since his life largely predates the widespread use of Cartesian coordinate systems.

Before postulating a division by zero axis, it may help to revisit why x/0 has consensus as an undefined concept. It’s not just a void of mathematics, its use leads to contradictions within our other established axioms:

It follows from the properties of the number system we are using (that is, integers, rationals, reals, etc.), if b ≠ 0 then the equation a / b = c is equivalent to a = b × c. Assuming that a / 0 is a number c, then it must be that a = 0 × c = 0. However, the single number c would then have to be determined by the equation 0 = 0 × c, but every number satisfies this equation, so we cannot assign a numerical value to 0 / 0

Lots more contradictions in the wiki, over many different areas of math https://en.wikipedia.org/wiki/Division_by_zero

Which then comes around to, what are we hoping to postulate a new axis or number line to achieve? Exploratory math, sure go ahead, but I’d like to tickle your brain more on why you’d even want to do this.

Stephen R
  • 41