When working with continued fraction expansions, I sometimes have to calculate the integer part of irrationals quickly without a calculator, what would be an effective way to do this?
For example, how would one quickly and accurately find the integer part of:
i) $\frac{1}{3}\sqrt{5}$
ii) $\frac{1}{4}\sqrt{11}$
iii) $\frac{1}{8}\sqrt{13}$
iv) $\frac{1}{2}(\sqrt{11}+3)$
I can only use working out on paper and I would like a consistent method to use for an upcoming test
Example: for $\sqrt{11}$ I know that $11$ is between $9$ and $16$ so the integer part would clearly be $3$
The same method cannot be as easily applied for when we are working with a fraction of this irrational
The method works for your examples :
$$2\lt\sqrt 5\lt 3\Rightarrow 0\lt\frac 23\lt\frac{\sqrt 5}{3}\lt \frac 33=1$$ $$3\lt \sqrt{11}\lt 4\Rightarrow 0\lt \frac 34\lt\frac{\sqrt{11}}{4}\lt\frac 44=1$$ $$3\lt\sqrt{13}\lt 4\Rightarrow 0\lt\frac 38\lt\frac{\sqrt{13}}{8}\lt \frac 48\lt 1$$ $$3\lt\sqrt{11}\lt 4\Rightarrow 6\lt\sqrt{11}+3\lt 7\Rightarrow 3\lt\frac{\sqrt{11}+3}{2}\lt\frac 72=3.5$$