| --- |
| source: ncert |
| topic: numbers |
| class_level: class_8 |
| chapter: ch1 |
| difficulty: beginner |
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| |
| Class 8 | Ch1: Rational Numbers |
| Rational number: p/q where p,q integers and q≠0. Includes integers, fractions, terminating and repeating decimals. |
| PROPERTIES OF RATIONAL NUMBERS: |
| Closure: rationals closed under +,-,×. Division closed except by zero. |
| Commutativity: a+b=b+a, a×b=b×a. NOT for subtraction or division. |
| Associativity: (a+b)+c=a+(b+c), (a×b)×c=a×(b×c). NOT for subtraction or division. |
| Distributivity: a×(b+c)=a×b+a×c. |
| Additive identity: 0. Multiplicative identity: 1. |
| Additive inverse of p/q is -p/q. Multiplicative inverse (reciprocal) of p/q is q/p. |
| REPRESENTATION ON NUMBER LINE: |
| Between any two rational numbers there are infinite rational numbers (dense property). |
| Standard form: denominator positive and HCF of numerator and denominator=1. |
| FINDING RATIONAL NUMBERS BETWEEN TWO NUMBERS: |
| Method 1: find mean (average) of the two numbers repeatedly. |
| Method 2: convert to same denominator with larger LCM. |
| Between -1/2 and 1/3: mean=(-1/2+1/3)/2=(-1/6)/2=-1/12. Continue for more. |
| SOLVED EXAMPLES: |
| Example 1: Find 5 rational numbers between -1/2 and 1/3. |
| Convert: -1/2=-9/18 and 1/3=6/18. Between -9/18 and 6/18: -8/18,-7/18,-6/18,-5/18,-4/18. |
| Simplified: -4/9, -7/18, -1/3, -5/18, -2/9. |
| Example 2: Verify -2/3 + 4/5 = 4/5 + (-2/3) (commutativity) |
| LHS: (-10+12)/15=2/15. RHS: (12-10)/15=2/15. Equal, commutativity verified. |
| COMMON MISTAKES: |
| Between any two rationals there are INFINITE rationals, not finite. |
| Reciprocal of 0 does NOT exist. Reciprocal of 1 is 1. Reciprocal of -1 is -1. |
