The exact meaning of $1/0$?

Question

What is the exact meaning of $1/0$? Does that mean a number that is very large, a number that cannot be expressed as the one we know, infinite numbers of number so that giving one particular value is not feasible, or really undefined?

Just to check, consider functions $f_1(x) = 1/x$ and $f_2(x) = 1 - 1/x$. These two are "not defined" at $x = 0$. But their sum is. What should be interpreted from it? Can we say that $1/x$ was trying to get a value at $x = 0$, but $-1/x$ just cancelled it out and we get $1$? Or something else?

Answer 1

It's undefined. Remember that division is reverse multiplication, so $8 \div 2$ is asking "what number times 2 equals 8?". $8 \div 0$ is asking "what number times 0 equals 8?". However, every number times 0 is 0, so there is no number that when multiplied by 0 equals 8 (or any other nonzero number). Therefore, any nonzero number divided by 0 is undefined. $0 \div 0$ is indeterminate since it is asking "what number times 0 is 0?". Since every number times 0 is 0, we cannot "determine" a correct value.

Answer 2

Cogito ...

Taking a cue from the so-called imaginary number $i = \sqrt {-1}$, we could define $\frac{1}{0} = I$.

The value of a function like $f(x) = \frac{x + 3}{1 - x}$ at $x = 1$ would then be $f(1) = 4 \cdot \frac{1}{0} = 4I$

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