Simplifying Heron's method output

Question

Working through a question on Heron's method I end up with an output of 1.7321428 and I'm asked to express the answer as a fraction. I'm given 4 choices and the correct choice is 97/56. How, other than just dividing 97 by 56, can I convert that decimal to that fraction? I've worked through the tutorials on converting decimals and I've put it in various online calculators but none of them return the fraction 97/56. What am I missing?

Answer 1

One approach is to try to find the continued fraction and stop when you get a very small remainder, the reciprocal of a very big number, and then not use that small remainder.

So here

Number       Integer part   Remainder   Reciprocal
1.7321428       1           0.7321428   1.365853765
1.365853765     1           0.365853765 2.733332537
2.733332537     2           0.733332537 1.363637845
1.363637845     1           0.363637845 2.7499888
2.7499888       2           0.7499888   1.333353245
1.333353245     1           0.333353245 2.999820809
2.999820809     2           0.999820809 1.000179224
1.000179224     1           0.000179224 5579.625006

So you get $$1+\dfrac{1}{1+ \dfrac{1}{2+ \dfrac{1}{1+ \dfrac{1}{2+ \dfrac{1}{1+\dfrac{1}{2+\dfrac{1}{1+0\text{-ish} } } }}}}} $$ Making that $0\text{-ish}$ exactly zero would give $\frac{97}{56}$

This approach will also give you rational approximations to irrationals. Try

Number       Integer part   Remainder   Reciprocal
3.141592654     3           0.141592654 7.062513306
7.062513306     7           0.062513306 15.99659441
15.99659441     15          0.996594407 1.003417231
1.003417231     1           0.003417231 292.6345909

which suggests $\pi = 3+\dfrac{1}{7+ \dfrac{1}{15+ \dfrac{1}{1+ 0\text{-ish} } } } \approx \dfrac{355}{113}\approx 3.14159292$ getting $7$ significant figures correct

If you start with $\sqrt{5}$ you never get very small remainders and just have to stop when you get bored