Why can’t you invent an imaginary number for division by zero like you can for a square root of a negative?
In: Mathematics
There are two ways to tackle this question: symbolically and physically. Let’s go with the physical first.
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Consider what division represents, using the expression `6 / 2 = 3` as a guide.
>You have six apples. Dividing them into two groups leaves you with three apples per group.
Similarly, you can extend this to division by fractions: `6 / .5 = 12`.
>You have six apples. Dividing them into half a group means that one whole group would have twelve apples.
So far, so good, right?
But: `6 / 0 = ?`.
Let’s divide six apples into zero groups. How many apples per group? …Well, there are zero groups, so…you can’t answer the question.
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Now, symbolically. Let’s do what you suggest, and invent a new number to represent the multiplicative inverse of zero — the number such that z = 0^-1 .
This means that 0z = 1.
But we know that 0z = 0.
By defining a number to be the multiplicative inverse of 0, we end up attempting to assert that 1 = 0, which we know to be false. Therefore, there can be no number for division by zero.
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