Derivation of a Square Root

The derivation of a square root was taught in high school years ago. Here is a concise derivation compliments of Roman Golik.

1. Select the target number, and partition it into columns of two digits from right to left.

2. Determine the largest root contained in the value of the digits in the leftmost column. This is the first integer of the final answer.

3. Square the root, and subtract this product from the value of the digits in the leftmost column. Concatenate the next column’s digits. (If no column exists, then concatenate two zeros.) Call this concatenation M.

4. Now double the current answer; call this C. Concatenate this product to the largest possible integer T, such that the value of the concatenation multiplied by T is less than or equal to value of M.  T is the next integer of the final answer. This product is subtracted from M, and the remainder becomes the new M.

5. Repeat step #4 until M equals zero, or the desired precision is reached.

   **    A decimal point is inserted in the final answer at the point just prior to the first
    concatenation of double zeros.