How come there isn’t an imaginary system for dividing a number by 0, while there is one for taking the square root of a negative number?
In: Mathematics
Using i to represent the root of -1 has lead us to a lot of cool stuff. We have used it to find new formulas and we apply them to things that aren’t imaginary.
However in the case of dividing by zero, we haven’t found a way to make it consistent or useful.
Why are imaginary roots of negative numbers consistent while imaginary divisions by 0 are not? Let’s take a closer look:
If we have sqrt(-9) = 3i
Then 3i * 3i must be -9.
Multiplication is associative, meaning it works any order you it in.
3 * 3 = 9. Then 9 * i = 9i. Finally, 9i * i= -9
Or 3i * i = -3. Then -3 * 3 = -9
We can get the same answer in two ways because of how multiplication works.
Now let’s try that trick with dividing by 0.
Let’s say that 1 / 0 = q
And q * 0 gets us back to 1.
Then, 5 / 0 = 5q
So 5q * 0 = 5
If we do the q*0 part first it works fine and we get 5, but if we use the associative property of multiplication, things go wrong.
5 * 0 = 0. Then 0*q = 1.
We got 1 when we should have gotten 5. So the system is already broken.
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