Represent real numbers on a number line

Each real number corresponds to a unique point on the number line, whose left-to-right order represents numerical order; this includes positive and negative numbers, zero, opposites, rational numbers, and irrational numbers. Fractions, terminating or repeating decimals, and radicals such as $\sqrt{2}$ can be located exactly when appropriate or approximated to a stated precision, while recognizing that irrational decimals do not terminate or repeat. The scope excludes complex numbers and formal constructions of the real-number system.

Detailed Explanation: 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 $1.41$ .

Learn by doing: Represent real numbers on a number line

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Number Line - Real Numbers (Fractions, Roots and Constants), Select Dot

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