Skill: Represent real numbers on a number line

Every real number matches exactly one point on a number line. Numbers increase as you move from left to right:

• Negative numbers are left of $0$.
• Positive numbers are right of $0$.
• Opposite numbers are the same distance from $0$ but on opposite sides.
• Fractions and decimals can be placed between whole-number marks.
• Irrational numbers, such as $\sqrt{2}$, can be placed using an approximation.

Example: Plot $-\sqrt{2}$, $-\frac{2}{3}$, $0$, $\frac{2}{3}$, and $\sqrt{2}$ on a number line.

Step 1: Approximate numbers when needed

The fraction $\frac{2}{3}$ is repeating:

$$\frac{2}{3}=0.666\ldots \approx 0.67$$

Also,

$$\sqrt{2}\approx 1.41$$

Therefore,

$$-\frac{2}{3}\approx -0.67 \qquad \text{and} \qquad -\sqrt{2}\approx -1.41$$

Step 2: Place the numbers from left to right

The numbers in increasing order are

$$-\sqrt{2},\quad -\frac{2}{3},\quad 0,\quad \frac{2}{3},\quad \sqrt{2}$$

because their approximations are

$$-1.41,\quad -0.67,\quad 0,\quad 0.67,\quad 1.41$$

A sketch of the number line is:

       -1.5       -1       -0.5        0        0.5        1        1.5
---------|----------|----------|---------|----------|---------|--------->
       -√2       -2/3                  0        2/3                 √2
      ≈ -1.41    ≈ -0.67                         ≈ 0.67             ≈ 1.41

Notice that $\sqrt{2}$ and $-\sqrt{2}$ are opposites: they are the same distance from $0$, on opposite sides. The decimal for $\sqrt{2}$ does not terminate or repeat, so its location is shown by an approximation such as $1.41$.