# What is the square root of 67?

##### 2 Answers

#### Explanation:

.........and thus

#### Explanation:

There are several methods you can use to find rational approximations.

Here's a method based on the Babylonian method...

To find the square root of a number

Then apply the following formulas repeatedly to get better approximations:

#{ (p_(i+1) = p_i^2+n q_i^2), (q_(i+1) = 2 p_i q_i) :}#

In our example, let

#{ (p_1 = p_0^2+n q_0^2 = 8^2+67*1^2 = 64+67 = 131), (q_1 = 2 p_0 q_0 = 2*8*1 = 16) :}#

#{ (p_2 = p_1^2 + n q_1^2 = 131^2+67*16^2 = 17161+17152 = 34313), (q_2 = 2 p_1 q_1 = 2*131*16 = 4192) :}#

If we stop here, we get:

#sqrt(67) ~~ 34313/4192 ~~ 8.185353#

which is accurate to