Section 7.4: 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

When a positive integer is not a perfect square, its square root is an irrational number. An irrational number cannot be written as the ratio of two integers, and its decimal form does not terminate or repeat.

Examples

Property

To determine if a number is rational or irrational, examine its decimal form.

• If the decimal form of a number stops or repeats, the number is rational.
• If the decimal form of a number does not stop and does not repeat, the number is irrational.

Examples

• The number $0.777...$ or $0.\overline{7}$ is rational because its decimal form repeats.
• The number $\sqrt{64}$ is rational because it simplifies to the integer $8$.
• The number $4.1234567...$ is irrational because the ellipsis indicates it continues without stopping or repeating.

Explanation

A simple test for classifying a number is to look at its decimal form. If the decimal terminates (ends) or has a repeating block of digits, it's rational. If it goes on forever with no pattern, it's irrational.