Lesson 4: Comparing and Ordering Real Numbers

Property

For any two real numbers $a$ and $b$, exactly one of the following is true: $a < b$ ($a$ is less than $b$), $a > b$ ($a$ is greater than $b$), or $a = b$ ($a$ is equal to $b$). On a number line, the number to the right is always the greater number.

Examples

• To compare $-\frac{3}{4}$ and $-0.5$, we can convert the fraction to a decimal: $-\frac{3}{4} = -0.75$. Since $-0.75$ is to the left of $-0.5$ on a number line, we have $-\frac{3}{4} < -0.5$.
• To compare $\sqrt{5}$ and $2$, we can approximate the square root: $\sqrt{5} \approx 2.236$. Since $2.236 > 2$, we have $\sqrt{5} > 2$.
• To compare $-4$ and $-\frac{17}{4}$, we can convert the fraction to a decimal: $-\frac{17}{4} = -4.25$. Since $-4$ is to the right of $-4.25$ on a number line, we have $-4 > -\frac{17}{4}$.

Explanation

Comparing real numbers means determining which number is larger, smaller, or if they are equal. A number line is a useful tool for this, as the number farther to the right is always greater. To compare numbers in different forms, like fractions and decimals, it is often easiest to convert them to the same form, usually decimals. You can then use the inequality symbols ($<, >$) to state the relationship between them.

Property

Real numbers can be plotted on a number line where each point corresponds to exactly one real number.
To compare real numbers, convert them to decimal form and plot their approximate positions: rational numbers have terminating or repeating decimals, while irrational numbers have non-terminating, non-repeating decimals.

Examples