Represent the rational numbers , and on a single number line.

Question

Homi · Accepted Answer

Understand the Question

To represent rational numbers on a number line, identify which consecutive integers each number lies between:
- $-\dfrac{5}{4} = -1.25$ lies between $-2$ and $-1$ .
- $\dfrac{2}{3} \approx 0.67$ lies between $0$ and $1$ .
- $1\dfrac{1}{2} = \dfrac{3}{2} = 1.5$ lies between $1$ and $2$ .
Divide each relevant unit interval into equal parts matching the denominator (or place by fractional distance) and plot the points.

Step 1 · Determine the Values and Intervals

Convert each rational number to find its exact or decimal value:

$\dfrac{2}{3} = 0.666\dots \approx 0.67$

$-\dfrac{5}{4} = -1.25$

\begin{aligned} 1\dfrac{1}{2} &= \dfrac{3}{2} \\[0.6em] &= 1.5 \end{aligned}

Ordering from least to greatest: $-1.25 < 0.67 < 1.5 \implies -\dfrac{5}{4} < \dfrac{2}{3} < 1\dfrac{1}{2}$

Step 2 · Plot on the Number Line

Diagram 1

$-\dfrac{5}{4}$ : Located between $-2$ and $-1$ , one-quarter of the unit distance to the left of $-1$ .
$\dfrac{2}{3}$ : Located between $0$ and $1$ , two-thirds of the unit distance to the right of $0$ .
$1\dfrac{1}{2}$ : Located between $1$ and $2$ , exactly halfway between them.

Answer

The positions on the number line are:

Common Mistakes

Negative Direction Error: Plotting $-\dfrac{5}{4} = -1.25$ between $0$ and $-1$ instead of between $-1$ and $-2$ . Remember that negative values move further to the left of $-1$ .
Incorrect Interval Division: For $\dfrac{2}{3}$ , divide the interval from $0$ to $1$ into $3$ equal parts and mark the $2^{\text{nd}}$ tick mark.

Question 1