Square roots

You will not be able to avoid square roots for long. Other than solving for perfect squares, there is no simple equation for calculating the decimal form of a square root. But you can use a “square root engine” to calculate the decimal form of a square root to as many digits as you want. The square root engine takes a good estimate for a square root and gives you a better estimate every time. Take that better estimate and plug it back into the engine, and you will get an even better estimate. Repeat this process until you have as many significant digits as you need.

This process works quickly when you start with a good first estimate of the square root. The table on the right is handy to get a good first estimate. Learn your perfect squares well: 1² = 1, 2² = 4, 3² = 9, 4² = 16, 5² = 25, 6² = 36, 7² = 49, etc. It is easy to calculate the squares of half-integers, as a pattern can be seen in the table. For 1.5², take 1, add 1, and add 0.25. For 2.5², take 4, add 2, and add 0.25. For 3.5², take 9, add 3, and add 0.25. Now that you know the pattern, you can use it to expand the table as much as you want. For now, we want to calculate √½. We look for ½ = 0.5 in the right hand column and see that it is between 0.25 and 1. Therefore, we choose 0.7 for an estimate of √½; this is our good first estimate.

It is time to meet the square root engine.

Simplify the left-hand side of the equation and you get the right-hand side. But notice that √x is alone on the right hand side. That is what the engine produces: a better estimate of √x. The engine is the left-hand side of the equation; it works by taking an estimate of √x and making a better √x. The engine improves the estimate you put into it.

We now recall our first good estimate of the square root of ½:

Check how close we are to the real square root:

The estimate squared has one digit of accuracy. Now plug the good first estimate into the engine.

Now the estimate squared has four digits of accuracy. You can see that the next estimate is indeed better. If you want more precision, plug this better estimate back into the engine.

Now the estimate squared has eight digits of accuracy. The precision keeps getting better.

Suppose you want to estimate the square root of a number that is close to a perfect square. For example, suppose you want to estimate the square root of 4.07.

$$\sqrt{4.07}=\sqrt{4\left(1+\frac{.07}{4}\right)}=2\sqrt{1+\frac{.07}{4}}=2\sqrt{1+.0175}$$ The following Taylor Series can be useful: $$\sqrt{1+x}=1+\frac{1}{2}x-\frac{1}{8}{x}^2+\frac{1}{16}{x}^3-\frac{5}{64}{x}^4+\mathrm{..}.$$