Well, well, well. What do we have here?

Calculus Level 3

Carl learnt that 2 5 × 5 2 = 1 \dfrac{2}{5} \times \dfrac{5}{2} = 1 He also learnt that 0.2 0.5 × 50 20 = 1 \dfrac{0.2}{0.5} \times \dfrac{50}{20} = 1 Now, his teacher asked him : What is 5000 … 2000 … × 0.000 … 2 0.000 … 5 \dfrac{5 000 \dots }{2 000 \dots } \times \dfrac{0.000 \dots 2}{0.000 \dots 5} ?

What is the correct answer?

Note : The ellipses mean that the 0's go on forever

2 1.25 Undefined 1.5 1 2.25

Mohammad Farhat by Mohammad Farhat , 16, Singapore

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2 solutions

Mohammad Farhat
Oct 16, 2018

Relevant wiki: What is 0 divided by 0?

Since there is an infinite amount of 0 0 's it cannot end with any number (because we will never reach it). And thus the part of the equation is simplified to 0 0 \dfrac{0}{0} Since, we cannot divide by 0 0 , The answer is undefined

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Michael Mendrin
Oct 16, 2018

We can rewrite the expression as

5 2 2 1 0 m 5 1 0 n \dfrac{5}{2} \dfrac{\frac{2}{10^m}}{\frac{5}{10^n} }

If m = n m=n , then the answer is 1 1 . However, by "forever", it means that both m , n m, n are of indeterminate length, which means therefore so is the answer.
∞ − ∞ \infty - \infty has an indeterminate value, not 0 0

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