Class 8 Mathematics — Rational Numbers | Interactive Visual Revision Notes

📝 Section 1 — Classroom Notes

🌎 Real-World Connection: Think about a thermometer on a cold day. When the temperature is −5°C it is below zero — a negative rational number. When you cut a pizza into 8 equal slices and eat 3, you have eaten 3/8 — a positive rational number. Rational numbers are everywhere in daily life: bank accounts, temperatures, cooking measures, and cricket averages!

🔢 Topic 1: What Are Rational Numbers?

A rational number is written as p/q.
Here p and q are integers.
The value of q must not be zero.
Examples: 3/4, −5/2, 7, 0
Every integer is a rational number.

🔢 Topic 2: Positive and Negative Rationals

If p/q is greater than zero, it is positive.
Examples: 4/5, 7/3, 2/9
If p/q is less than zero, it is negative.
Examples: −3/4, −8/5, −1/6
Zero is neither positive nor negative.

🔢 Topic 3: Rational Numbers on Number Line

Draw a number line with equal gaps.
Mark integers first: −2, −1, 0, 1, 2.
Divide each gap into equal parts.
For 3/4, divide gap into 4 parts.
Count 3 parts from zero to the right.

🔢 Topic 4: Comparing Rational Numbers

To compare a/b and c/d, cross multiply.
Find a × d and b × c.
If a × d > b × c, then a/b > c/d.
If a × d < b × c, then a/b < c/d.
If both are equal, the fractions are equal.

🔢 Topic 5: Between Any Two Rationals

Between any two rational numbers, there are infinite rationals.
Find the average of two rationals.
That average is always in between them.
You can repeat this again and again.
So there is no end to rationals between two.

🔢 Topic 6: Addition of Rational Numbers

If denominators are same, add the tops.
Example: 2/7 + 3/7 = 5/7.
If denominators are different, find LCM.
Make denominators equal using LCM.
Then add the numerators.

🔢 Topic 7: Subtraction of Rational Numbers

To subtract, add the additive inverse.
Additive inverse of a/b is −a/b.
Example: 5/8 − 3/8 = 2/8 = 1/4.
For different denominators, find LCM first.
Then subtract the numerators.

🔢 Topic 8: Multiplication of Rational Numbers

Multiply numerator by numerator.
Multiply denominator by denominator.
Example: 2/3 × 4/5 = 8/15.
A negative times a positive = negative.
A negative times a negative = positive.

🔢 Topic 9: Division of Rational Numbers

To divide, multiply by the reciprocal.
Reciprocal of a/b is b/a.
Example: 3/5 ÷ 2/7 = 3/5 × 7/2.
That gives us 21/10.
Division by zero is not allowed.

🔢 Topic 10: Properties of Rational Numbers

Closure: add or multiply two rationals, you get a rational.
Commutative: a/b + c/d = c/d + a/b.
Associative: (a + b) + c = a + (b + c).
Zero is the additive identity (a + 0 = a).
One is the multiplicative identity (a × 1 = a).

a/b + c/d = c/d + a/b (Commutative Law)

🎨 Section 2 — Easy Illustrations

Picture 1: Number Line with Rational Numbers

Positive rationals go right of zero. Negative rationals go left of zero.

Picture 2: Comparing Two Fractions (Cross Multiply)

Cross multiply and compare. The bigger product shows the bigger fraction.

Picture 3: Addition of Fractions (Bar Model)

Red = 1/4 and Blue = 2/4. Together they fill 3 out of 4 parts.

Picture 4: Multiplication (Area Model)

The purple overlap shows 2/3 × 3/4 = 6/12 = 1/2.

Picture 5: Properties of Rational Numbers

Six important properties every rational number follows.

✏️ Section 3 — Worked Examples

Example 1: Find 5 Rational Numbers Between 1/3 and 1/2

Question: Find 5 rational numbers between 1/3 and 1/2.

1 Make the denominators equal.

2 LCM of 3 and 2 is 6.

3 1/3 = 2/6 and 1/2 = 3/6.

4 Only one number (no 5). Multiply top and bottom by 6.

5 2/6 = 12/36 and 3/6 = 18/36.

6 Pick 5 numbers between 12 and 18.

7 They are: 13/36, 14/36, 15/36, 16/36, 17/36.

13/36, 14/36, 15/36, 16/36, 17/36

Tip: multiply to get more gaps between the two fractions.

Example 2: Simplify (−3/7) + (5/14) − (1/2)

Question: Simplify (−3/7) + (5/14) − (1/2).

1 Find LCM of 7, 14, and 2. LCM = 14.

2 −3/7 = −3 × 2 / 14 = −6/14.

3 5/14 stays as 5/14.

4 1/2 = 1 × 7 / 14 = 7/14.

5 Now add: −6/14 + 5/14 − 7/14.

6 = (−6 + 5 − 7) / 14 = −8/14.

7 Simplify: −8/14 = −4/7.

−4/7

Example 3: Verify the Distributive Property

Question: Verify that (−2/5) × (3/7 + 1/3) = (−2/5 × 3/7) + (−2/5 × 1/3).

1 Left side: First solve 3/7 + 1/3.

2 LCM of 7 and 3 = 21. So 3/7 = 9/21, 1/3 = 7/21.

3 9/21 + 7/21 = 16/21.

4 −2/5 × 16/21 = −32/105.

5 Right side: −2/5 × 3/7 = −6/35.

6 −2/5 × 1/3 = −2/15.

7 LCM of 35 and 15 = 105. −6/35 = −18/105, −2/15 = −14/105.

8 −18/105 + (−14/105) = −32/105.

Left = −32/105 = Right. Verified!

Distributive property: a × (b + c) = (a × b) + (a × c)

📊 Section 4 — Cheat Sheet

Term	What It Means	Key Fact
Rational Number	Written as p/q	q must not be 0
Positive Rational	Greater than zero	4/5, 7/3
Negative Rational	Less than zero	−3/4, −1/6
Number Line	Shows position of numbers	Left = smaller
Cross Multiply	Way to compare fractions	a/b vs c/d: compare ad, bc
Additive Inverse	Number + its inverse = 0	Inverse of 3/5 is −3/5
Reciprocal	Flip the fraction	Reciprocal of 3/5 is 5/3
Closure	Result stays rational	For + and ×
Commutative	Order does not matter	a + b = b + a
Associative	Grouping does not matter	(a+b)+c = a+(b+c)
Distributive	Multiply over addition	a(b+c) = ab + ac
Identity	Does not change value	0 for +, 1 for ×

📄 Section 5 — Take-Home Handout

📝 A. Fill in the Blanks

1. A rational number is written as p / _________.
2. The additive identity is _________.
3. The reciprocal of 4/9 is _________.
4. −3/5 + 3/5 = _________.
5. Between two rationals there are _________ rationals.

✓ B. True or False

1. Zero is a rational number. _______
2. Every integer is a rational number. _______
3. Subtraction is commutative for rationals. _______
4. The reciprocal of 1 is 1. _______
5. Division by zero is allowed. _______

✏️ C. Solve These

1. Find 4 rational numbers between −1/3 and 1/3.

Show your work here

2. Simplify: (−5/6) + (3/4) − (1/12).

Show your work here

🔗 D. Match the Columns

Column A	Column B
1. Additive identity	a. −2/3
2. Reciprocal of 3/2	b. 0
3. Additive inverse of 2/3	c. Infinite
4. Rationals between 1/5 and 2/5	d. 2/3

🎨 E. Draw the Number Line

Draw a number line and mark these rational numbers: −3/4, −1/4, 0, 1/2, 5/4.

Draw your number line here

✅ Detailed Solutions — Section 5 Handout

A. Fill in the Blanks — Answers

1. A rational number is written as p / q.

2. The additive identity is 0. (Any number + 0 = the same number.)

3. The reciprocal of 4/9 is 9/4. (Flip numerator and denominator.)

4. −3/5 + 3/5 = 0. (Additive inverse property: a/b + (−a/b) = 0.)

5. Between two rationals there are infinitely many rationals.

B. True or False — Answers

1. Zero is a rational number. — True. (0 = 0/1, so p=0, q=1.)

2. Every integer is a rational number. — True. (Any integer n = n/1.)

3. Subtraction is commutative for rationals. — False. (3/5 − 1/5 = 2/5 but 1/5 − 3/5 = −2/5. Not equal.)

4. The reciprocal of 1 is 1. — True. (1/1 flipped = 1/1 = 1.)

5. Division by zero is allowed. — False. (q ≠ 0 is the rule for rational numbers.)

C1. Find 4 rational numbers between −1/3 and 1/3

1 Multiply numerator and denominator by 6 to widen the gap.

2 −1/3 = −6/18 and 1/3 = 6/18.

3 Integers between −6 and 6 (exclusive): −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5.

4 Pick any 4: −4/18, −2/18, 2/18, 4/18 = −2/9, −1/9, 1/9, 2/9.

−2/9, −1/9, 1/9, 2/9 (many other answers are also correct)

C2. Simplify: (−5/6) + (3/4) − (1/12)

1 Find LCM of 6, 4, and 12. LCM = 12.

2 −5/6 = −5 × 2 / 12 = −10/12.

3 3/4 = 3 × 3 / 12 = 9/12.

4 1/12 stays as 1/12.

5 Now: −10/12 + 9/12 − 1/12 = (−10 + 9 − 1)/12 = −2/12.

6 Simplify: −2/12 = −1/6. (Divide top and bottom by 2.)

−1/6

D. Match the Columns — Answers

1. Additive identity → b. 0

2. Reciprocal of 3/2 → d. 2/3 (flip 3/2 to get 2/3)

3. Additive inverse of 2/3 → a. −2/3 (change sign)

4. Rationals between 1/5 and 2/5 → c. Infinite

🃏 Section 6 — Revision Flashcards

👉

🔢

What is a rational number?

A number that can be written as p/q where p and q are integers and q is not zero.

🔎

How do you compare two rational numbers?

Use cross multiplication. Compare a × d with b × c for a/b and c/d. The bigger product shows the bigger fraction.

➕

How do you add rational numbers with different denominators?

Find the LCM of the denominators. Make them equal. Then add the numerators.

➗

How do you divide rational numbers?

Multiply by the reciprocal. Flip the second fraction and multiply. Example: 3/5 ÷ 2/7 = 3/5 × 7/2 = 21/10.

❓

What is the additive inverse of a rational number?

The number you add to get zero. Additive inverse of a/b is −a/b. Example: inverse of 3/4 is −3/4.

💡

What is the commutative property?

The order does not matter. a + b = b + a and a × b = b × a. Works for addition and multiplication.

🔧

What is the distributive property?

a × (b + c) = (a × b) + (a × c). Multiplication spreads over addition.

🎯

How many rationals exist between any two rational numbers?

Infinitely many! You can always find more by making the denominator bigger or by taking the average.

Class 8 Mathematics — Rational Numbers

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🔢 Class 8 Mathematics — Rational Numbers Interactive Visual Revision Notes Aligned with the Class 8 Mathematics syllabus. Independently created study material. Not affiliated with or endorsed by NCERT.

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