Question

find the square root of .1 by 3 decimal

Answer 1

First let us derive a simple formula for the square root of a number.

Let N be the number. Let us make a neat round figure guess for the square root, say a.

Divide N by a to get b.

The average of a and b is closer to the required root.

(a+b)/2

Divide N by this average to get N/((a+b)/2).

Further, the average of this and (a+b)/2 is still closer to the required root.

Simplifying the above strategy we get

Square root of N= (a+N/a)/4 + a*N/(a^2+N)

In the given question, N = 0.1.

Let a= 0.3

Square root = (0.3+0.1/0.3)/4 + 0.3*0.1/(0.09+0.1)

= 0.31622807

This is quite close to the root as per calculator, 0.316227766.

Another example:

To find Square root of 87

N= 87

Guess = a=10

Square root=

(10+8.7)/4 + 10*87/(100+87)

= 9.327406417.

so come back to answer.
=0.316/1.73

Answer 2

1/3.0=10/30


=√(10/30 )

=3.1/5.4