How do you order the following from least to greatest without a calculator sqrt102, 10, 3pi, sqrt99, 1.1times10^1, 9.099?

Apr 23, 2017

$9.099 , 3 \pi , \sqrt{99} , 10 , \sqrt{102} , 1.1 \cdot {10}^{1}$

Explanation:

If $a$ is an approximation to $\sqrt{n}$ then:

$\sqrt{n} = a + \frac{b}{2 a + \frac{b}{2 a + \frac{b}{2 a + \frac{b}{2 a + \ldots}}}}$

where $b = n - {a}^{2}$

If $a$ is a good approximation to $\sqrt{n}$ then a better one will be:

$\sqrt{n} \approx a + \frac{b}{2 a} = a + \frac{n - {a}^{2}}{2 a}$

$102$ and $99$ are both close to $100 = {10}^{2}$, so using $a = 10$ we find:

$\sqrt{102} \approx 10 + \frac{2}{20} = 10.1$

$\sqrt{99} \approx 10 - \frac{1}{20} = 9.95$

Note also that:

$3 \pi \approx 3 \cdot 3.14 = 9.42$

$1.1 \cdot {10}^{1} = 11$

So the correct order of the given numbers is:

$9.099 , 3 \pi , \sqrt{99} , 10 , \sqrt{102} , 1.1 \cdot {10}^{1}$