Square root is common function in mathematics. It has a wide range of applications from the field of mathematics to physics. Sometimes it gets hard to calculate square root of a number, especially the one which are not actually square of a number. Often the method we employ are to tedious work with decimals. Here is a guide to find square root or rather their approximates.
Let be the number whose square root we need to calculate. Let can be written as where the largest perfect square less than and be any positive real number. Then,
Approximate the square root of 968.
Let us first find the perfect square less than . To do this we would be comparing with perfect squares that are easy to figure out like .
Now try the square of another number greater than like and .
Now applying my formula,
You can cross check by squaring the answer .
That is indeed a good approximation.
Here is the proof of the formula mentioned above.
The proof of the formula lies inside the formula itself. The following formula is based on assumption that roots between two perfect squares are uniformly distributed.
To understand it better we can use a graph-
We know that integers exist between a perfect square and .
be any real number that exists between and . Also, .
will exist between and .
Clearly the difference between and is , now roots can be uniformly distributed between and , if each root occupies space.
The number will occupy space. Hence the formula turns out to be
This section deals with conventional method of square root-finding. This method is more accurate, however it is a tedious one too.