Question

What is the square root of 3.4?

Answer 1

`sqrt(3.4) = sqrt(85)/5 ~~ 1.8439`

Explanation:

If `a, b >= 0` then `sqrt(a/b) = sqrt(a)/sqrt(b)`

If `a >= 0` then `sqrt(a^2) = a`

So:

`sqrt(3.4) = sqrt(85/25) = sqrt(85)/sqrt(25) = sqrt(85)/5`

Here I have chosen to express `3.4` as `85/25` to get the smallest whole value for the denominator.

I could also have written `sqrt(3.4) = sqrt(17/5) = sqrt(17)/sqrt(5)`

If I wanted to calculate an approximation for `sqrt(3.4)` by hand then I would probably have used this instead:

`sqrt(3.4) = sqrt(340/100) = sqrt(340)/10`

Then I would work out an approximation for `sqrt(340)` and divide by `10`.

For example: `18^2 = 324` and `19^2 = 361`, so:

`18 = sqrt(324) < sqrt(340) < sqrt(361) = 19`

Using a Newton Raphson type method to find `sqrt(340)`, I might chose `18.5 = 37/2` as my first approximation `a_0`, then make better approximations using the formula:

`a_(i+1) = (a_i^2 + n)/(2a_i)`

where `n = 340`.

Then

`a_1 = (37^2 + 340 * 2^2) / (2 * 37 * 2) = (1369+1360)/148`

`= 2729 / 148 ~~ 18.439`

So `sqrt(340) ~~ 18.439` and `sqrt(3.4) ~~ 1.8439`