Lesson 3: Fractions and Decimals on the Number Line

Property

To represent the rational number system by points on a line, first draw a horizontal line and mark a point as the origin, denoted by $0$. Mark off a succession of equally spaced points to the right of $0$ as $1, 2, 3, \ldots$ and to the left as $-1, -2, -3, \ldots$.
To place a fraction like $p/q$, divide the unit interval into $q$ equal parts.
The point representing $p/q$ is found by taking $p$ of these parts, to the right of the origin if $p/q$ is positive, and to the left if negative.

Examples

• The fraction $-\frac{3}{4}$ is located by dividing the segment from $0$ to $-1$ into four equal parts and marking the point three parts to the left of the origin.
• The fraction $\frac{2}{5}$ is located by dividing the segment from $0$ to $1$ into five equal parts and marking the end of the second part.
• The fraction $-\frac{1}{2}$ is located at the midpoint of the interval between $0$ and $-1$.

Explanation

A number line gives every rational number a unique address. Integers are evenly spaced markers, and fractions are the points in between, located by dividing the spaces into equal parts. This helps us visualize the value and order of numbers.

Property

To place decimals and mixed numbers on a number line, convert them to equivalent forms that show their position clearly.
For decimals, identify which two consecutive integers the decimal falls between, then divide that unit interval into equal parts based on the decimal place value.
For mixed numbers like $a\frac{p}{q}$, locate the whole number part $a$, then add the fractional part $\frac{p}{q}$ by dividing the unit interval into $q$ equal parts and moving $p$ parts in the appropriate direction.

Examples