Lesson 2-4: Estimate Irrational Numbers

Property

To estimate $\sqrt{x}$, find the two consecutive whole numbers it is between. Find perfect squares $a^2$ and $(a+1)^2$ such that $a^2 < x < (a+1)^2$. Then, the root is between $a$ and $a+1$.

Examples

• $\sqrt{15}$ is between which two whole numbers? Since $9 < 15 < 16$, then $\sqrt{9} < \sqrt{15} < \sqrt{16}$, so it's between 3 and 4.

• $\sqrt{50}$ is between which two whole numbers? Since $49 < 50 < 64$, then $\sqrt{49} < \sqrt{50} < \sqrt{64}$, so it's between 7 and 8.

Property

To estimate the square root of a non-perfect square, first find the two perfect squares it is between. The integer whose square is closer to your number is your estimate.

Examples

• To estimate $\sqrt{50}$, notice 50 is between $49(7^2)$ and $64(8^2)$. Since 50 is closer to 49, $\sqrt{50} \approx 7$.
• $\sqrt{37}$ is between $\sqrt{36}=6$ and $\sqrt{49}=7$. Because 37 is closer to 36, we estimate $\sqrt{37} \approx 6$.
• $\sqrt{40}$ is between the whole numbers 6 and 7, since $6^2=36$ and $7^2=49$.

Explanation

This is like being a number line detective! You trap the tricky, non-perfect square root between two “friendly” perfect squares. Whichever perfect neighbor it’s cozier with gives you the closest whole number guess for its value. It's a great trick for quick estimations!

Property

To get a more precise approximation of an irrational square root, use trial and error. After finding the two consecutive integers the square root is between, test decimal values within that range by squaring them. Continue to test values with more decimal places to squeeze the gap and narrow down the range.

Examples

• To approximate $\sqrt{30}$ to one decimal place: We know it is between $5$ and $6$. Let's test decimals between $5$ and $6$:

$5.4^2 = 29.16$
$5.5^2 = 30.25$
Since $30$ is much closer to $30.25$ than $29.16$, $\sqrt{30}$ is approximately $5.5$.

• To compare $\sqrt{5}$ and $2.3$, we square both numbers to see their true size:

$(\sqrt{5})^2 = 5$
$2.3^2 = 5.29$
Since $5 < 5.29$, we know that $\sqrt{5} < 2.3$.

Explanation

Since you cannot write down the exact decimal value of an irrational number, you have to trap it! By testing decimals and squaring them, you are squeezing the gap between rational numbers to find an approximation that is as accurate as you need it to be.