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Operations On Rational Numbers

May 27, 2020
Category: Operations On Rational Numbers

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Properties of Addition on Q :

▪ Q is closed for addition.

▪ Addition on Q is commutative as well as associative.

▪ 0 $\in$ Q is the additive identity.

▪ For each $\frac{a}{b}\in$ Q there exists $-\frac{a}{b}\in$ Q such that

$\frac{a}{b}$ + $\left (\frac{-a}{b} \right )$ = $\left (\frac{-a}{b} \right )$ + $\frac{a}{b}$ = 0. We call $\left (\frac{-a}{b} \right )$ the additive inverse of $\frac{a}{b}$ .

▪ Q is closed for subtraction.

Properties of Multiplication on Q :

▪ Q is closed for multiplication.

▪ Multiplication on Q is commutative as well as associative.

▪ 1 is the multiplicative identity in Q.

▪ Every non-zero rational number (a / b) has its multiplicative inverse or reciprocal (b / a). Note that the rational number ‘0’ has no reciprocal.

An Important Property : Between any two rational numbers, there are an infinite number of rational numbers.

Note : If ‘x’ and ‘y’ are any two rational numbers such that x < y,

then $\frac{1}{2}$ (x + y) is a rational number lying between ‘x’ and ‘y’.

e.g. Write three rational numbers between $\frac{1}{2}$ and $\frac{3}{4}$ .

Soln. Let a = $\frac{1}{2}\left (\frac{1}{2}+\frac{5}{8} \right )=\frac{9}{16}$ and c = $\frac{1}{2}\left (\frac{5}{8}+\frac{3}{4} \right )=\frac{11}{16}$