Squares and Square Roots - Estimating Square Roots

Perfect squares are numbers like $1, 4, 9, 16, 25...$ which are obtained by multiplying an integer by itself. For non-perfect squares, we must estimate their square root as it will not be a whole number.

To estimate a square root, identify the two consecutive perfect squares between which the given number lies. For example, to estimate $\sqrt{20}$, we note that it lies between $4^2 = 16$ and $5^2 = 25$.

On a horizontal number line, the estimated square root $\sqrt{n}$ is positioned between the integers representing the square roots of the bounding perfect squares. If $n$ is closer to the smaller perfect square, the estimate will be closer to the lower integer.

The 'Proximity Rule' helps refine the estimate. If a number is almost equal to a perfect square, its square root is almost equal to the root of that square. For instance, $\sqrt{99}$ is very close to $\sqrt{100}$, so it is approximately $9.9$.

Visually, if you imagine a square with an area of $30$ units, its side length $\sqrt{30}$ would be slightly larger than a square of area $25$ (side $5$) and smaller than a square of area $36$ (side $6$).

You can further narrow down an estimate by checking the square of the midpoint. If we know $\sqrt{n}$ is between $6$ and $7$, we check $6.5^2 = 42.25$. If $n < 42.25$, then $\sqrt{n}$ is between $6$ and $6.5$.

Estimation is particularly useful in real-life geometry problems where exact measurements are not required, such as finding the approximate length of a diagonal or the side of a square plot.