Why can’t you invent an imaginary number for division by zero like you can for a square root of a negative?
In: Mathematics
Disclaimer: In math, we can look at all kinds of different structures which have all kinds of different properties. This leads to a lot of variety, but there are some limitations. One such limitation is, that a structure cannot have contradictory properties.
Example:
>This structure has a smallest number.
This statement is true for some structures like the natural numbers {0,1,2,…} and false for others, like the whole numbers {…, -2, -1, 0, 1, 2, …}.
>This structure does not have a smallest number.
This statement is true for some structures like the whole numbers {…, -2, -1, 0, 1, 2, …} and false for others, like the natural numbers {0,1,2,…}.
But it is not possible for a structure to simultaneously possess both properties, as one is the negation of the other. These properties are contradictory.
The answer to your question is, that division by zero is possible given the right structure (these would be pretty obscure, especially for a layman), but contradictory with the properties of our usual number systems.
To see that, let’s look at some of those properties!
Firstly, let’s look at “neutral elements”. When add/multiply these to another number, the outcome doesn’t change. The neutral element of addition is 0, for multiplication it’s 1.
Example:
>2 + 0 = 2 , 2 * 1 = 2 , in general n + 0 = 0 , n * 1 = n
Secondly, let’s look at multiplication by 0. I’m not going to go into too much detail because this post is already going to be very long, but basically multiplying anything by 0 will be 0.
Example:
> 3 * 0 = 0 , in general n * 0 = 0
Thirdly, let’s look at so called “inverse elements”. If we can combine two numbers to make the neutral element, these numbers are inverses of each other.
Example:
>2 + (-2) = 0 , 2 * (1/2) = 1 , in general n + (-n) = 0 , n * (1/n) = 1
You may have noticed that we haven’t talked about subtraction or division up to this point. In fact, we can simply define these two operations in terms of addition and multiplication using the notion of inverse elements.
> 4 – 2 is the same as 4 + (-2)
> 5 / 3 is the same as 5 * (1/3)
We’re mainly interested in multiplicatve inverse elements, which means we’re going to take a look at rational numbers because we need fractions.
What is a fraction?
Basically, a fraction is written *a / b* where *a* is an integer and *b* is a natural number bigger than zero with the property
> If you multiply this number by *b* it will be equal to *a*.
Now let’s take a look at division by 0. Recall that division is simply multiplication with the inverse. So imagine there was a fraction of the form *1 / 0*. Let’s look at the term
> 0 * (1 / 0) = ?
This is where we run into our contradiction, because we can apply two different rules to get different outcomes.
By the definition of a fraction, *1 / 0* is number becomes 1 when multiplied by 0, so the answer should be 1.
In contrast, anything multiplied by 0 is 0, so the answer should be 0.
So we can see that introducing this fraction *1 / 0* would lead to a contradiction, that’s why we explicitly forbid 0 to be the denominatior of a fraction in the definition.
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