Why can’t you invent an imaginary number for division by zero like you can for a square root of a negative?
In: Mathematics
Mathematicians like when things work nicely and in almost every case, allowing division by 0 will make things not work nicely.
For example, one of the nice things about numbers is that we can multiply numbers in whatever order we want and still get the same result. For example (2 * 5) * 4 = 2 * (5 * 4) and 2 * 3 = 3 * 2. (Mathematicians call these “associativity” and “commutativity” of multiplication.)
Another nice property is that a(x + y) = ax + ay. For example, 2*3 + 2*4 is the same as 2(3 + 4). (Mathematicians call this “distributivity” of multiplication over addition.)
Let’s see what happens if we allow division by 0. Let’s just make a new thing called X and we will define that 0 * X = X * 0 = 1. So 1 / 0 = X.
Then (3 * 0) * X = 0 * X = 1. Except that 3 * (0 * X) = 3 * 1 = 3. But 3 is not equal to 1!
So this breaks one of the rules we like about multiplication. Maybe this is no biggie, maybe we can just let it slide. Let’s keep exploring.
Let’s check if distributivity still holds. How about X * (0 + 0)?
X * (0 + 0) = X * 0 = 1, but X * 0 + X * 0 = 1 + 1 = 2. Another problem… we lost distributivity.
If you keep exploring, you’ll likely find even more problems with assuming that X = 1/0 exists.
We now have two options here: First, we can allow division by 0, in which case we would have to abandon a bunch of things that work nicely with arithmetic. Or, second, we could just say that X does not exist. The second option is almost always the best option.
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