# How can 3/1 ≡ 1/(1/3), when left side features merely integers, but right side features a repetend?

On one hand, I know that algebraically, $\dfrac{3}1 ≡ \dfrac{1}{\color{red}{1/3}}$.

On the other hand, they differ in practice, not least because $\color{red}{1/3}$ contains 3 as the repetend. For example, if a scrap of physical material must have a 3:1 ratio and a length of 3 m, then I shall make the width 1 m.

But presuppose a length of 1 m. Then a 3:1 ratio is impossible to accomplish, because it would require a width of $\color{red}{1/3=0.\overline{3}}$. But it's impossible to measure and cut anything physical at a repetend!

Doesn't this physical impossibility due to the reptend belie, or undermine, the theoretical "equality" that $\dfrac{3}1 ≡ \dfrac{1}{\color{red}{1/3}}$? How can this physical impossibility due to the repetend be reconciled with equality?

## 2 answers

... it's impossible to measure and cut anything physical at a repetend.

This is incorrect. Given a segment with $1$ unit length, one can easily construct, "physically", a segment with a length of $1/3$ units. In fact, given any line segment, there are many ways to *trisect* it. See for instance this article or this YouTube video. See also this Wikipedia article on $\sqrt{2}$; one can even get a line segment with the length of an irrational number!

Your confusion essentially boils down to why $0.\overline{9}=1$. Notice that on the right you have the integer $1$, while on the left, you have a repeating decimal which continues infinitely.

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