How is this definition of a constant divided by zero called?

Question

I divide a constant by zero. One example is the following: 2/0 My father told me he learned at school earlier that the result is "not defined". If I enter this arithmetic problem in Wolfram Alpha, I will get ∞ as result. How is this kind of definition called?

Answer 1

The quotient $x=b/a$ is defined as the unique solution to the equation $ax=b$ for $a$ and $b$ real numbers. Now if $a = 0$ and $b\neq 0$ then there is no $x$ for which $$ax=b.$$ This is because for every real number $x$, $$0\cdot x = 0 \neq b$$ thus there is no $x$ that would satisfy this equation, and we say $b/0$ is undefined.

Finally for the case of $a=b=0$ we can see that any real number $x$ satisfies $a\cdot x = b$, since both sides are zero. So in this case, any number $x$ can be chosen as the solution. Hence in this case as well, we cannot write $x=0/0$ since there is no unique number to pick for this situation.

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The interpretation that $2/0$ is $\infty$ arrises when you take limits. It's a shorthand of saying $\lim_{x\to0^+} 2/x=\infty$, but it is not strictly speaking $2/0$ which is undefined.