Category :
8th Class
Rational Numbers
1, 2, 3, 4,.... etc., are called natural numbers, denoted by N.
All natural numbers together with zero are called whole numbers, denoted by W.
W = {0, 1, 2, 3, 4,......}
All whole numbers together with negatives of natural numbers are called integers, denoted by Z.
Z = {.....-4,-3,-2,-1, 0.1.2, 3, 4,...}
(i) -1, -2, -3, - 4,….. are called negative integers.
(ii) 1,2,3,4 ... are called positive integers.
Note: Zero is neither positive nor negative.
- The numbers of the form -\[\frac{a}{b}\], where 'a' and 'b' are natural numbers are called fractions.
e.g., \[\frac{3}{5},\frac{7}{11},\frac{13}{213}\],….etc.
- The numbers of the form \[\frac{p}{q}\], where 'p' and 'q' are integers and 'q'\[\ne \]0 are called rational numbers, denoted by Q.
\[\frac{-3}{5},\frac{7-}{-11},\frac{-13}{-213}\],….etc.
Properties of rational numbers
- Closure property of addition: The sum of two rational numbers is always a rational number.
- Commutative law of addition: For any two rational numbers \['a'\] and 'b', a + b = b + a.
- Associative law of addition: For any three rational numbers 'a'. 'b' and 'c', (a + b) + c = a + (b + c).
- Existence of additive identity: Zero is the additive identity.
For any rational number 'a', a + 0 = 0 + a = a
- Existence of additive inverse: For each rational number \['a'\], there exists a rational number \['-a'\] such that +(-a) =(-a) +a is the additive inverse of \['a'\]
- Closure property for multiplications: The product of two rational numbers.
- Commutative law of multiplication: For any three rational numbers \['a'\],\['b'\]and \['c'\](ab)c For any rational number \['a'\],1.a=a.1=a.
- Existence of multiplication identity: 1 is called the multiplication identity.
- Existence of multiplicative inverse: Every non – Zero rational number \['a'\] has its multiplicative inverse\[\frac{1}{a}\].
Note: Zero is a rational number which has no multiplicative inverse.
- of multiplication over addition:
For rational numbers \['a'\,and\,'b'\]and \['c'\]a (b + c) = ab+ ac
- Rational numbers can be represented on a number line.
- Between any two rational numbers, there exist infinitely many rational numbers.
- To find rational numbers between any two given rational numbers, we find average or mean.
