An algorithm is the set of calculations and procedures to work out an operation. For example, when we were children we were taught to sum up using a pencil and a piece of paper, we were told to align the numbers in columns starting from the right with the units of the same range below one another, to sum up the units, to write down the outcome below - if it is larger than 9, the extra digit is carried into the next column - and to add it to the tens...
Each step of an algorithm has a reason for being, but we can implement an algorithm and work out an operation correctly and not know why we have to do it that way. The same happens with the algorithm of the square root.

As a curiosity, we are going to analyse on which the algorithm of the square root is based. In order to do this, we start by revising the square of a binomial.

The square of an addition of two terms.
- The square of an addition is equal to the square of the first term plus two times the first term multiplied by the second, plus the square of the second term.

It is graphically presented in this window:

- The value of those numbers can be obtained by measuring the sides of the squares and rectangles of the shape, which are marked pink.

- The square of the first term is the surface of the big square - marked yellow.

- The double of the first term multiplied by the second term is the surface of the two equal rectangles - marked green and placed horizontally and vertically.

- The square of the second term is the surface of the small square - marked yellow.

The algorithm of the square root is based on the square of an addition of two terms.
We are going to analyse how to get the square of a perfect square number of four digits; its root will have two digits.
- We have to decompose the root in an addition of what the ten digit and the unit digit express. The digit on the left expresses tens, it is worth ten times its surface value.

- We do the square of the first term when finding the first digit of the root.
- We do the double of the first multiplied by the second and the square of the second at the same time. We can separate it easily:
- We do the double of the first multiplied by the second when taking down the double of the first digit and multiplying it by the second digit.
- We do the square of the second term when putting the second digit next to the previous and multiplying it by that digit.

Here we have used numbers with roots of two digits, but this practice also works for roots of more tan two digits. When practising the algorithm, the two terms of the root are the expressions of the digits of the root obtained until that moment and of what is left to work out.
When we take down the double of the part of the root obtained to multiply it, we find the double of the first multiplied by the second.