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

##### 1 Answer

Apr 23, 2017

#### Explanation:

If

#sqrt(n) = a+b/(2a+b/(2a+b/(2a+b/(2a+...))))#

where

If

#sqrt(n) ~~ a+b/(2a) = a+(n-a^2)/(2a)#

#sqrt(102) ~~ 10+2/20 = 10.1#

#sqrt(99) ~~ 10-1/20 = 9.95#

Note also that:

#3pi ~~ 3*3.14 = 9.42#

#1.1 * 10^1 = 11#

So the correct order of the given numbers is:

#9.099, 3pi, sqrt(99), 10, sqrt(102), 1.1*10^1#