1. Rational Numbers
Definition
A number that can be written in the form p/q, where
- p and q are integers
- q ≠ 0
পৰিমেয় সংখ্যা হ’ল সেই সংখ্যা যাক p/q আকাৰত লিখিব পাৰি,
য’ত p আৰু q পূৰ্ণ সংখ্যা আৰু q ≠ 0।
Examples / উদাহৰণ
1/2, −3/5, 0, 7, −9
(7 = 7/1, 0 = 0/1)
🔹 2. Types of Numbers (Number System)
| English | Assamese | Examples |
|---|---|---|
| Natural Numbers | স্বাভাৱিক সংখ্যা | 1, 2, 3, … |
| Whole Numbers | পূৰ্ণ সংখ্যা | 0, 1, 2, 3, … |
| Integers | পূৰ্ণাংক | … −2, −1, 0, 1, 2 … |
| Rational Numbers | পৰিমেয় সংখ্যা | 1/3, −5/7, 4 |
📌 All integers are rational numbers, but all rational numbers are not integers.
📌 সকলো পূৰ্ণাংক পৰিমেয় সংখ্যা, কিন্তু সকলো পৰিমেয় সংখ্যা পূৰ্ণাংক নহয়।
🔹 3. Representation on Number Line
- Rational numbers can be represented on the number line.
- More negative → left side
- More positive → right side
পৰিমেয় সংখ্যা সংখ্যা ৰেখাত দেখুৱাব পাৰি।
🔹 4. Properties of Rational Numbers
✔ Closure Property
- Addition, subtraction, multiplication → closed
- Division → not closed (division by zero not allowed)
✔ Commutative Property
- Addition & multiplication are commutative
- Subtraction & division are not
✔ Associative Property
- Addition & multiplication are associative
- Subtraction & division are not
🔹 5. Additive Identity
0 is the additive identity.
For any rational number a:
a + 0 = a
শূন্য হ’ল যোগাত্মক একক।
🔹 6. Additive Inverse
For a rational number a, its additive inverse is −a.
Example:
5/7 → −5/7
−3/4 → 3/4
a + (−a) = 0
🔹 7. Multiplicative Identity
1 is the multiplicative identity.
a × 1 = a
১ হ’ল গুণাত্মক একক।
🔹 8. Multiplicative Inverse (Reciprocal)
For a non-zero rational number a/b,
its multiplicative inverse = b/a
Example:
3/5 → 5/3
−7/2 → −2/7
a × (1/a) = 1
📌 Zero has no multiplicative inverse.
📌 শূন্যৰ কোনো গুণাত্মক বিপৰীত নাই।
🔹 9. Finding Rational Numbers Between Two Numbers
Method:
- Convert both numbers into same denominator
- Write numbers between them
Example:
Between 3/5 and 4/5
→ 30/50 and 40/50
→ 31/50, 32/50, 33/50 …
🔹 10. Word Problems
- Convert mixed fractions into improper fractions
- Use multiplication or division carefully
- Simplify the answer
🔹 11. Important Exam Points ⭐
✔ 0 is a rational number
✔ Rational numbers are infinite
✔ Reciprocal of 1 is 1
✔ Reciprocal of −1 is −1
✔ Division by zero is not defined
🔹 12. Common Mistakes to Avoid ❌
❌ Forgetting q ≠ 0
❌ Wrong sign while multiplication
❌ Assuming zero has reciprocal