source: ncert
topic: numbers
class_level: class_8
chapter: ch1
difficulty: beginner
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.