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Encapsulate
math
RAtIoNAl NuMBERS

A number which can be expressed in the form of p , where p and q are integers and q ≠ 0 is
q
called a rational number.

Facts for rational numbers operations

• There are infinite rational numbers between any two rational For any two rational numbers, a and c :
numbers. b d
+
a c 1  a  c Addition: a + c = ad bc
• A rational number between and = ×  +  . b d bd
b d 2  b  d
−
• All natural numbers are rational numbers but all rational a c a  c −  ad bc
numbers need not be natural numbers. Subtraction: b − d = b +   d   = bd
• All integers are rational numbers but all rational numbers
need not be integers. Multiplication: a c ac
0 0 b × d = bd
• 0 is a rational number as it can be written as , , etc.
1 2
• All fractions are rational numbers but all rational numbers Division: a ÷ c = a × d = ad , where c ≠ 0
need not be fractions. b d b c bc d

Properties

For all four operations: For addition and multiplication:

Property Addition Subtraction Multiplication Division • Zero → Additive Identity
Closure     • 1 → Multiplicative identity
Property • −a and a are additive inverse of
b
b
Commutative     each other.
Property a b
Associative • b and a are multiplicative inverse
Property     or reciprocal of each other.

Distributive Property of Multiplication over addition and subtraction


a c e   a c   a e  a c e   a c   a e 
•  +  =  ×  +  ×  •  −  =  ×  −  × 
b d f   b d  b f  b d f   b d  b f 


Brain Sizzlers

Find the value of:
1 1
1. 5 + 2. 12÷
2 1 × 2
3 3
3+ 5 1 2 +− 5
3 7
1+
2

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