Lesson 2-5: Compare and Order Real Numbers

Property

To compare and order real numbers, convert each number to its decimal form and compare digit by digit from left to right.
For any two real numbers $a$ and $b$: $a < b$ if the decimal representation of $a$ is less than the decimal representation of $b$.

Examples

Property

To order a mixed set of rational and irrational numbers visually, approximate the irrational numbers as decimals and convert any fractions. Then, plot all numbers on a number line to determine their relative positions. Numbers always increase in value from left to right on the number line.

Examples

• Order from least to greatest: 2.5, $\sqrt{7}$, $\frac{8}{3}$

Since $\sqrt{7} \approx 2.646$ and $\frac{8}{3} \approx 2.667$, the relative order is $2.5 < \sqrt{7} < \frac{8}{3}$.

• Order from least to greatest: $\sqrt{15}$, 4, 3.9, $\sqrt{14}$

Since $\sqrt{15} \approx 3.873$ and $\sqrt{14} \approx 3.742$, the sequence is $\sqrt{14} < \sqrt{15} < 3.9 < 4$.

Explanation