Rational Numbers - Representation of rational numbers on number line and comparison

Welcome, young mathematicians! Today, we are going to master two fundamental skills in our study of Rational Numbers: how to plot them on a number line and how to compare two or more rational numbers.

By the end of this lesson, you’ll be able to visualize these numbers effortlessly and figure out which one is larger without any confusion!

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Quick Recap: What is a Rational Number?

A rational number is any number that can be written in the form $\frac{p}{q}$, where:

$p$ and $q$ are integers.

$q \neq 0$ (the denominator can never be zero).

Examples: $\frac{3}{4}$, $-\frac{5}{7}$, $0$ (since $0 = \frac{0}{1}$), and $4$ (since $4 = \frac{4}{1}$).

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Part 1: Representing Rational Numbers on a Number Line

Real-World Analogy: The Footstep Path

Imagine standing at a starting post marked 0.

Taking steps to your right represents positive movement (+).

Taking steps to your left represents negative movement (-).

If one full step equals 1 whole unit (meter), a rational number like $\frac{1}{2}$ simply means dividing that 1-meter gap into 2 equal parts and taking 1 small step!

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Step-by-Step Guide to Plotting

Step 1: Determine the Direction (Sign)

If the number is positive (e.g., $\frac{3}{5}$), it lies to the right of 0.

If the number is negative (e.g., $-\frac{3}{5}$), it lies to the left of 0.

Step 2: Determine the Region (Proper vs. Improper Fraction)

Proper Fractions (Numerator < Denominator, e.g., $\frac{2}{3}$ or $-\frac{2}{3}$):

Always lie between 0 and 1 (if positive) or between 0 and -1 (if negative).

Improper Fractions (Numerator > Denominator, e.g., $\frac{7}{3}$ or $-\frac{7}{3}$):

Convert to a mixed fraction first!

Example: $\frac{7}{3} = 2\frac{1}{3}$. This tells you the number lies between 2 and 3.

Step 3: Divide and Mark

Look at the denominator ($q$): Divide each unit space into $q$ equal parts.

Look at the numerator ($p$): Count $p$ parts from 0 (or from the whole number part) in the correct direction.

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Let's See Examples!

Example A: Represent $\frac{3}{4}$ on a Number Line

Direction: Positive $\rightarrow$ Right of 0.

Region: Proper fraction ($\frac{3}{4}$) $\rightarrow$ Lies between 0 and 1.

Divide: The denominator is 4, so divide the space between 0 and 1 into 4 equal parts.

Mark: Count 3 parts to the right from 0.

```

0 3/4 1

-------|---|---|---|---|------->

0 1/4 2/4 3/4 4/4 (1)

```

Example B: Represent $-\frac{5}{3}$ on a Number Line

Convert to Mixed Fraction: $-\frac{5}{3} = -1\frac{2}{3}$.

Direction: Negative $\rightarrow$ Left of 0.

Region: Between -1 and -2.

Divide: Denominator is 3, so divide the segment between -1 and -2 into 3 equal parts.

Mark: Count 2 parts to the left from -1.

```

-2 -5/3 -1 0

<----------|---|---|---|---|-----------|------->

-2 -1⅔ -1⅓ -1 0

```

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Part 2: Comparing Rational Numbers

Comparing rational numbers means deciding which number is greater ($>$), smaller ($<$), or if they are equal ($=$).

Golden Rules of Comparison

Positive vs. Negative: Every positive rational number is always greater than zero and any negative rational number.

$$\text{Positive} > 0 > \text{Negative}$$

Number Line Rule: On a number line, the number that lies further to the RIGHT is always GREATER.

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Method 1: Rational Numbers with Same Denominators

When denominators are positive and identical, simply compare their numerators.

Example 1: Compare $\frac{3}{7}$ and $\frac{5}{7}$.

Denominators are the same ($7$).

Compare numerators: $3 < 5$.

Therefore, $\frac{3}{7} < \frac{5}{7}$.

Example 2: Compare $-\frac{4}{9}$ and $-\frac{2}{9}$.

Denominators are the same ($9$).

Compare numerators: $-4$ and $-2$. Since $-2$ is to the right of $-4$ on a number line, $-4 < -2$.

Therefore, $-\frac{4}{9} < -\frac{2}{9}$.

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Method 2: Rational Numbers with Different Denominators

When denominators are different, we make them the same using the LCM (Least Common Multiple) method.

Steps:

Make sure both denominators are positive (if negative, shift the minus sign to the numerator, e.g., $\frac{3}{-4} = \frac{-3}{4}$).

Find the LCM of the denominators.

Convert each rational number into an equivalent rational number with the LCM as the common denominator.

Compare the numerators!

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Worked Example: Compare $-\frac{3}{4}$ and $-\frac{5}{6}$

Check Denominators: Both denominators ($4$ and $6$) are positive.

Find LCM: LCM of 4 and 6 is 12.

Convert to Equivalent Fractions:

$-\frac{3}{4} = \frac{-3 \times 3}{4 \times 3} = -\frac{9}{12}$

$-\frac{5}{6} = \frac{-5 \times 2}{6 \times 2} = -\frac{10}{12}$

Compare Numerators:

Compare $-9$ and $-10$. Since $-9 > -10$, we have:

$$-\frac{9}{12} > -\frac{10}{12}$$

Final Answer:

$$-\frac{3}{4} > -\frac{5}{6}$$

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Summary Checklist

Situation	Action
Plotting Proper Fraction ($\frac{a}{b}$)	Divide space between 0 and 1 into $b$ parts; count $a$ steps.
Plotting Improper Fraction	Convert to mixed fraction $W\frac{a}{b}$; plot between $W$ and $W+1$.
Comparing Positive & Negative	Positive is always greater.
Comparing Different Denominators	Take LCM of denominators $\rightarrow$ make equivalent fractions $\rightarrow$ compare numerators.

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Practice Corner

Try solving these questions yourself first, then check the detailed step-by-step solutions below!

Question 1

Represent the following rational numbers on a single number line:

a) $\frac{2}{5}$

b) $-\frac{7}{5}$

Question 2

Which of the two rational numbers is greater?

$$\frac{-4}{5} \quad \text{or} \quad \frac{5}{-7}$$

Question 3

Arrange the following rational numbers in ascending order (smallest to largest):

$$-\frac{3}{4}, \quad \frac{1}{2}, \quad -\frac{5}{8}, \quad 0$$

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Solutions

Solution 1:

For $\frac{2}{5}$:

It is positive, so it goes to the right of 0.

Since $2 < 5$, it lies between 0 and 1.

Divide the space between 0 and 1 into 5 equal parts and mark the 2nd point from 0.

For $-\frac{7}{5}$:

Convert to mixed fraction: $-\frac{7}{5} = -1\frac{2}{5}$.

It is negative, so it lies between -1 and -2.

Divide the space between -1 and -2 into 5 equal parts and mark the 2nd point to the left of -1.

Number Line Diagram:

```

-7/5 (-1⅖) 2/5

<---|---|--•--|---|---|---|---|---|---|---|---|---|--•--|---|---|--->

-2 -1 0 1

```

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Solution 2:

Standard Form:

Rewrite $\frac{5}{-7}$ with a positive denominator: $\frac{-5}{7}$.

Find LCM of Denominators:

LCM of $5$ and $7$ is $35$.

Convert to Equivalent Rational Numbers:

$\frac{-4}{5} = \frac{-4 \times 7}{5 \times 7} = \frac{-28}{35}$

$\frac{-5}{7} = \frac{-5 \times 5}{7 \times 5} = \frac{-25}{35}$

Compare Numerators:

$-28$ and $-25$. On a number line, $-25$ is to the right of $-28$, so $-25 > -28$.

$$\frac{-25}{35} > \frac{-28}{35} \implies \frac{5}{-7} > \frac{-4}{5}$$

Answer: $\frac{5}{-7}$ is greater.

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Solution 3:

Given numbers: $-\frac{3}{4}, \frac{1}{2}, -\frac{5}{8}, 0$

Identify Positives, Negatives, and Zero:

Negative numbers: $-\frac{3}{4}, -\frac{5}{8}$

Zero: $0$

Positive number: $\frac{1}{2}$

We know that: $\text{Negative numbers} < 0 < \text{Positive numbers}$.

So, $\frac{1}{2}$ will be the largest, and $0$ will be the second largest.

Compare Negative Numbers ($-\frac{3}{4}$ and $-\frac{5}{8}$):

Find LCM of $4$ and $8$, which is $8$.

$-\frac{3}{4} = \frac{-3 \times 2}{4 \times 2} = -\frac{6}{8}$

$-\frac{5}{8} = -\frac{5}{8}$

Compare numerators: $-6 < -5$, so $-\frac{6}{8} < -\frac{5}{8} \implies -\frac{3}{4} < -\frac{5}{8}$.

Combine in Ascending Order:

$$-\frac{3}{4} < -\frac{5}{8} < 0 < \frac{1}{2}$$

Answer: The ascending order is $-\frac{3}{4}, -\frac{5}{8}, 0, \frac{1}{2}$.