## Approximating Square Root of Numbers by Averaging

There are several algorithms developed to manually calculate the square root of a number. My favorite is the method of averaging, which is probably the least complicated.  In this method, the number is divided by the nearest square root estimate, and then the quotient and the square root estimate are averaged. The result is the new estimate.

Sounds complicated? Not really.

Example: Approximate $\sqrt{24}$.

Note: Using a calculator, $\sqrt{24} \approx 4.8989794855$.

Solution

We know that $\sqrt{24}$ is between $4^2$ and $5^2$. Between the two, it is nearer to $5^2$. So, the first estimate is $5$. Next, we divide $24$ by the first estimate and “average” it with the quotient . » Read more

## What is i? What are imaginary numbers?

Squaring numbers always give a positive result or a $0$.  For example, $3^2 = 9$$0^2 = 0$$(-5)^2 = 25$.  Using this argument, it follows that we can always get the square root of $0$ or any positive number. Now, what about the square root of a negative number, say, $-4$?

Let’s see:  $2 \times 2 = 4$ and $-2 \times -2 =4$; therefore, no number exists that when multiplied by itself equals $-4$. In fact, it looks like we cannot find the square root of any negative number.

Now, what if — just what if — we invent square root of negative numbers? Nobody would stop us right? We start with $-1$.  Suppose $- 1$ has a square root, and we call it $i$. If $\sqrt{-1} = i$, then $i^2 = -1$.  If so, we can also answer the question that we have asked above: $\sqrt{-4} = \sqrt{4(-1)} = 2i$As a matter of fact, we can generalize this operation for any negative number.  If $a$ is positive, then $-a$ is a negative number. So, $\sqrt{-a} = \sqrt{a(-1)} = i\sqrt{a}$.

But what kind of number is $i$? Surely, it is not a real number since we cannot locate it on the real number line. And, since we have all the real numbers and $i$ is not a  real number, maybe, we should call $i$ an imaginary number.

The Real Number Line

Imaginary numbers were ‘invented’ (or discovered if you prefer) because mathematicians wanted to know if they could think of square root of negative numbers, particularly, the root of the equation $x^2 + 1 = 0$ (that is, $x^2 = - 1$ which is the same as  finding the $\sqrt{-1}$).   Just like the many ‘radical’ ideas in mathematics,   it was not widely accepted at first, but eventually its invention proved to be very useful and has opened a lot of new ideas in mathematics.

In the next post in this series, we are going to discuss about the operations on imaginary numbers. .