Is the fraction
$$\frac{1}{\frac{1}{0}}$$
undefined?
I know that division by zero is usually prohibited, but since dividing a number by a fraction yields the same result as multiplying the number by the fraction's reciprocal, you could argue that
$$\frac{1}{\frac{1}{0}} = (1)\left(\frac{0}{1}\right) = 0$$
Is that manipulation permissible in this case? Why or why not?
Answer
Another way to think about this is order of operations:
$$
\frac{1}{\frac{1}{0}}=1/(1/0)
$$
I always compute what's inside the parenthesis first, which gives me undefined, and I have to stop there.
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