The way I estimate square roots, is by finding the closest lowest perfect square, then adding decimals to the number to determine the estimation. How do I estimate the square root of a number with decimals using this method using the number $3.51$ for example?
I know how to estimate for whole numbers using this method but I'm confused on the decimal numbers. When I used this same method on $3.51$, I tested $1.8^2$ and $1.9^2$ to see which one was closer. $1.9^2$ $(3.61)$ was closer to $3.51$ than $1.8^2$ $(3.24)$. But in my math book, the answer was that $1.8$ was the correct estimate.
How and why?
You are looking for the square root of $3.51$ and you know that it's somewhere between $1.8$ (square $3.24$) and $1.9$ (square $3.61$).
As the square of $1.9$ is the closest, you start from there:
$3.51 = (1.9 - \alpha)^2 = 1.9^2 - 3.8 \alpha + \alpha^2$
As $\alpha$ is quite small, let's drop it and continue the approximation:
$$3.51 \approx 3.61 - 3.8 \alpha$$ $$\alpha \approx \frac{3.61 - 3.51}{3.8}$$ ($3.8$ is close to $4$) $$\alpha \approx \frac{0.10}{4} \approx 0.025$$ Hence: $1.9 - 0.025 = 1.875$ is a better approximation than $1.9$.