Skill: Compare and order real numbers

To compare and order real numbers, rewrite them in forms that are easy to compare, such as decimals. Then place them from left to right on a number line.

• Numbers farther left are smaller.
• Any negative number is less than any positive number.
• Among negative numbers, the number with the greater distance from $0$ is smaller. For example, $-5<-2$.
• Use approximations for square roots and other irrational numbers.

Example: Order the numbers from least to greatest:

$$-2.4,\quad -\sqrt{5},\quad -\frac{7}{3},\quad 0.\overline{3},\quad \sqrt{2},\quad 1.5$$

Step 1: Rewrite or approximate the numbers.

$$-\frac{7}{3}=-2.\overline{3}\approx -2.333$$ $$0.\overline{3}=\frac13\approx 0.333$$ $$-\sqrt5\approx -2.236$$ $$\sqrt2\approx 1.414$$

Now the numbers are approximately:

$$-2.4,\quad -2.236,\quad -2.333,\quad 0.333,\quad 1.414,\quad 1.5$$

Step 2: Order the negative numbers.

For negative numbers, the more negative number is smaller:

$$-2.4<-2.333<-2.236$$

So,

$$-2.4<-\frac73<-\sqrt5$$

Step 3: Order the positive numbers.

$$0.333<1.414<1.5$$

So,

$$0.\overline3<\sqrt2<1.5$$

Step 4: Combine the negative and positive numbers.

All negative numbers come before all positive numbers:

$$\boxed{-2.4<-\frac73<-\sqrt5<0.\overline3<\sqrt2<1.5}$$

When using approximations, keep enough decimal places to make the comparison clear.