Lesson 3: Compare and Order Real Numbers

Property

To approximate an irrational square root $\sqrt{n}$ to the nearest integer, find the two consecutive perfect squares that $n$ is between. If $a^2 < n < (a+1)^2$, then the value of $\sqrt{n}$ is strictly between the integers $a$ and $a+1$:

Examples

• Approximate $\sqrt{30}$ to the nearest integer.

The perfect squares closest to $30$ are $25$ and $36$.

Since $30$ is closer to $25$ than to $36$, the best integer approximation is $5$. So, $\sqrt{30} \approx 5$.

• Approximate $\sqrt{85}$ to the nearest integer.

The perfect squares closest to $85$ are $81$ and $100$.

Since $85$ is closer to $81$ than to $100$, the best integer approximation is $9$. So, $\sqrt{85} \approx 9$.

Property

For any two positive numbers $a$ and $b$, if $a > b$, then $-a < -b$.

Examples

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