Rational Numbers on the Number Line

• Represent positive and negative rational numbers on a horizontal number line.
• Compare rational numbers by their position: further right means greater value.
• Find rational numbers between any two given rational numbers.
• Use the number line to visualise addition and subtraction of rationals.

1. Building the number line for rationals

   • Draw a horizontal line with arrows at both ends.
   • Mark 0 (origin). Positive rationals go to the right; negative rationals to the left.
   • Unit length between consecutive integers can be subdivided into equal parts for denominators 2, 3, 4, etc.

2. Plotting p/q

   • If q > 0: from 0, move |p| units of length 1/q in the direction of the sign of p.
   • Example: 3/4 is three jumps of 1/4 to the right of 0; −5/3 is five jumps of 1/3 to the left (past −1).

3. Comparing rational numbers

   • On a number line, the number further to the right is greater.
   • Convert to common denominator or decimal to compare without a diagram:
     −3/4 vs −1/2 → −3/4 = −0.75 and −1/2 = −0.50, so −1/2 > −3/4.

4. Between any two rational numbers — There are infinitely many rational numbers between any two distinct rationals.
   Method: find the mean (average): (a + b)/2 lies strictly between a and b.
   Example: between 1/3 and 1/2, mean = (1/3 + 1/2)/2 = (2/6 + 3/6)/2 = (5/6)/2 = 5/12.

5. Density of rational numbers — No matter how close two rationals are, you can always find another rational between them. This is a key NCERT Class 7 idea.

Represent −7/4 on the number line.

Step 1 — note that −7/4 = −1¾, between −2 and −1
Step 2 — divide each unit into 4 equal parts (quarters)
Step 3 — from 0, move 7 quarter-units to the left
         OR from −1, move 3 quarter-units further left to reach −7/4
Answer: point at −1.75 on the number line

Find a rational number between 2/5 and 3/5.

Method 1 — mean: (2/5 + 3/5)/2 = (5/5)/2 = 1/2
Check: 2/5 = 0.4 < 0.5 < 0.6 = 3/5 ✓

Method 2 — equivalent fractions with larger denominator:
         2/5 = 4/10 and 3/5 = 6/10, so 5/10 = 1/2 lies between them.
Answer: 1/2 (many other answers possible, e.g. 11/20)

• Which is greater: −5/8 or −3/4? (−5/8, because −0.625 > −0.75)
• Plot 1/3 and 2/3 on a number line divided into thirds.
• Find one rational between 1/4 and 1/2. (e.g. 3/8)
• Is −1 a rational number? Where does it lie relative to −3/2? (Yes; −1 > −3/2, so −1 is to the right)

Open the Practice tab for graded questions on rational numbers and the number line.