Question

Show all the property of rational no. With example

Answer 1

Properties of Rational Numbers are-

• Closure Property

Addition

$\frac{3}{8} +\frac{-5}{7} =\frac{-19}{56}$

∴ Rational numbers are closed under addition. That is, for any two rational numbers 'a' and 'b', a+b is also a rational number.

Subtraction

$\frac{-8}{5} -\frac{3}{2} =\frac{-31}{10}$

∴ Rational numbers are closed under subtraction. That is, for any two rational numbers 'a' and 'b', a-b is also a rational number.

Multiplication

$\frac{3}{7}*\frac{-4}{5} =\frac{-12}{35}$

∴ Rational numbers are closed under subtraction. That is, for any two rational numbers 'a' and 'b', a×b is also a rational number.

Division

$\frac{7}{2} /\frac{0}{6} = \frac{7}{2} *\frac{6}{0}$

∴ Rational numbers are not closed under division. However, if we exclude zero then the collection of, all other rational numbers are closed under division.

∴Rational numbers are closed under addition, subtraction and multiplication.

• Commutative Property

Addition

$\frac{-2}{3} +\frac{5}{7} = \frac{1}{21} ,\frac{5}{7} +\frac{-2}{3} = \frac{1}{21}$

∴ Rational numbers can be added in any order. We say that addition is commutative for rational numbers. That is for any two rational numbers 'a' and 'b', a+b=b+a.

Multiplication

$\frac{-7}{3} *\frac{6}{5} =\frac{-42}{15} , \frac{6}{5} *\frac{-7}{3} =\frac{-42}{15}$

∴ Rational numbers can be multiplied in any order. We say that multiplication is commutative for rational numbers. That is for any two rational numbers 'a' and 'b', ab=ba.

Subtraction

$\frac{2}{3} -\frac{5}{4} = \frac{-7}{12} , \frac{5}{4} -\frac{2}{3} = \frac{7}{12}$

∴ Rational numbers cannot be subtracted in any order.

Division

$\frac{-5}{4} /\frac{3}{7} = \frac{-35}{12},\frac{3}{7} /\frac{-5}{4} = \frac{12}{-35}$

∴ Rational numbers cannot be divided in any order.

∴ Rational numbers are commutative under addition and multiplication.

• Associative Property

Addition

$\frac{-2}{3} + (\frac{3}{5} +\frac{-5}{6}) = \frac{-9}{10}$ , $(\frac{-2}{3} +\frac{3}{5})+\frac{-5}{6} = \frac{-9}{10}$

∴ Addition is associative for rational numbers. That is, three rational numbers 'a', 'b' and 'c' , a+(b+c) = (a+b)+c.

Multiplication

$\frac{2}{3}*(\frac{3}{4}*\frac{5}{7}) =\frac{5}{7} ,\ (\frac{2}{3}*\frac{3}{4}) *\frac{5}{7} =\frac{5}{7}$

∴ Multiplication is associative for rational numbers. That is, three rational numbers 'a', 'b' and 'c', a×(b×c) = (a×b)×c

Subtraction

$(\frac{2}{3} -\frac{1}{4}) -\frac{3}{8} = \frac{1}{24} ,\ \frac{2}{3} -(\frac{1}{4} -\frac{3}{8})= \frac{19}{24}$

Clearly, $(\frac{2}{3} -\frac{1}{4}) -\frac{3}{8} \neq \frac{2}{3} -(\frac{1}{4} -\frac{3}{8})$

∴ Subtraction is not associative for rational numbers.

Division

$\frac{1}{3} /(\frac{4}{5}/\frac{6}{9}) = \frac{5}{18} , \ (\frac{1}{3} /\frac{4}{5})/\frac{6}{9} =\frac{5}{13}$

Clearly, $\frac{1}{3} /(\frac{4}{5}/\frac{6}{9}) \neq (\frac{1}{3} /\frac{4}{5})/\frac{6}{9}$

∴ Division is not associative for rational numbers.

∴ Rational numbers are associative under addition and multiplication.

• Additive Identity

-5 + 0 = 0 + -5 = -5

Zero is called the identity of addition of rational numbers.

So, for every rational number $\frac{p}{q}$ ,

$\frac{p}{q} +0=0+\frac{p}{q} = \frac{p}{q}$

• Multiplicative Identity

9 × 1 = 1 × 9 = 9

One is the multiplicative identity for rational numbers.

So, for every rational number $\frac{x}{y}$ ,

$\frac{x}{y} *1=1*\frac{x}{y} =\frac{x}{y}$