Instead of thinking of division as a “proper” operator, think of it as shorthand for “x0 = -1". Since the multiplication operator already defines "x0 => 0” and “-1 != 0”, there literally is no answer by definition. So “undefined”.
For “0/0”, it’s a little different. Since “x*0 => 0”, literally every number satisfies “0/0”. Since we’re expecting a single number as the answer, we call the answer to this “undefined” too.
That’s probably accurate, but I’m no mathematician. Specifically if you confine yourself to the set of real numbers, it’s undefined.
If you confine yourself to the set of rational numbers, integers, or natural numbers, you could get a bunch more undefined equations. For example, if you’re only working with the set of natural numbers, sqrt(4) is defined, but sqrt(5) through sqrt(8) are undefined because there are no whole numbers between 2 and 3 (2^2 and 3^2). Similarly, sqrt(-1) is undefined in this case because -1 isn’t part of the set of natural numbers, so any expression you used a negative number in would be undefined.
A more intuitive example might be to try using a non-number in an expression. Sqrt(🐱) is undefined. And 🦆-🦢 is also undefined. The operators aren’t defined to include cats or birds.
-1/0 = Boom
But more literally, the answer is undefined.
Instead of thinking of division as a “proper” operator, think of it as shorthand for “x0 = -1". Since the multiplication operator already defines "x0 => 0” and “-1 != 0”, there literally is no answer by definition. So “undefined”.
For “0/0”, it’s a little different. Since “x*0 => 0”, literally every number satisfies “0/0”. Since we’re expecting a single number as the answer, we call the answer to this “undefined” too.
So without i, sqrt(-1) would be… Undefined?
That’s probably accurate, but I’m no mathematician. Specifically if you confine yourself to the set of real numbers, it’s undefined.
If you confine yourself to the set of rational numbers, integers, or natural numbers, you could get a bunch more undefined equations. For example, if you’re only working with the set of natural numbers, sqrt(4) is defined, but sqrt(5) through sqrt(8) are undefined because there are no whole numbers between 2 and 3 (2^2 and 3^2). Similarly, sqrt(-1) is undefined in this case because -1 isn’t part of the set of natural numbers, so any expression you used a negative number in would be undefined.
A more intuitive example might be to try using a non-number in an expression. Sqrt(🐱) is undefined. And 🦆-🦢 is also undefined. The operators aren’t defined to include cats or birds.