CBSE Class 8 Mathematics Rational Numbers Notes Set 19

Rational Numbers and Their Operations

A rational number is a number that can be written as p/q where p and q are integers and q is not zero. Every integer is rational (n = n/1). A rational number is in standard form when q is positive and p and q have no common factor.
To add or subtract, rewrite the fractions with the LCM of the denominators and then add or subtract the numerators. To multiply, multiply numerators and denominators. To divide, multiply by the reciprocal of the divisor.

Properties

Closure: sum, difference and product of rational numbers are rational; division is closed except by zero.
Commutative: a + b = b + a and a x b = b x a (not true for subtraction and division).
Associative: (a + b) + c = a + (b + c) and the same for multiplication (not for subtraction and division).
Distributive: a x (b + c) = a x b + a x c. Use it to simplify long expressions quickly.
Identities: 0 is the additive identity and 1 is the multiplicative identity.
Inverses: the additive inverse of a is -a; the multiplicative inverse (reciprocal) of a is 1/a, which exists for every non-zero a. Only 1 and -1 are their own reciprocals.

Rational Numbers Between Two Numbers

Between any two rational numbers there are infinitely many more. The mean (a + b)/2 is one of them. To list several, write both numbers with the same denominator and, if needed, multiply top and bottom by a larger number to create space.

Worked example: three numbers between 3/5 and 3/4. Write them as 12/20 and 15/20, or 48/80 and 60/80, and take 50/80, 52/80, 55/80.

Number Line

To show p/q on the number line, divide each unit length into q equal parts and count p parts from zero (to the left if p is negative). For -5/6 divide the unit left of zero into six parts and take the fifth mark.

Negative Exponents

a to the power -n equals 1/a^n, and (a/b)^-n = (b/a)^n. The laws a^m x a^n = a^(m+n), a^m / a^n = a^(m-n), (a^m)^n = a^(mn) and (ab)^n = a^n b^n hold for all integer powers.

Worked example: (2/3)^-2 x (3/4)^-1 = (9/4) x (4/3) = 3.