Lesson 4: Approximating Square Roots

Property

Together, rational and irrational numbers make up the real numbers. A rational number can be written as the ratio of two integers, $\frac{a}{b}$ where $b \neq 0$, and its decimal form either stops or repeats. An irrational number cannot be written as a ratio of two integers, and its decimal form never stops and never repeats. When a positive integer is not a perfect square, its square root is an irrational number.

Examples

• The numbers $5$, $-\frac{3}{8}$, and $0.333...$ are rational because they can be written as fractions ($\frac{5}{1}$, $-\frac{3}{8}$, $\frac{1}{3}$) and their decimals terminate or repeat. $\sqrt{81}$ is also rational because $9^2 = 81$, so $\sqrt{81} = 9$.
• The number $\sqrt{50}$ is irrational because $50$ is not a perfect square, so its decimal form goes on forever without repeating.
• $\sqrt{2}$, $\sqrt{7}$, and $\sqrt{15}$ are all irrational numbers because the numbers under the radical are not perfect squares.

Explanation

Property

To get a numerical estimate of a root, we look for perfect powers closest to the radicand. For example, to estimate $\sqrt{11}$, we see 11 is between the perfect squares 9 and 16. Its square root will be between 3 and 4. For a decimal approximation, a calculator is used. The symbol for an approximation is $\approx$.

Examples

• To estimate $\sqrt{50}$, we know $7^2 = 49$ and $8^2 = 64$. Since $49 < 50 < 64$, we can state that $7 < \sqrt{50} < 8$.

• To estimate $\sqrt[3]{70}$, we know $4^3 = 64$ and $5^3 = 125$. Since $64 < 70 < 125$, we can state that $4 < \sqrt[3]{70} < 5$.