Properties of Rational Number

Chapter 1 Class 8 Rational Numbers
Concept wise
 Operation Associativity True / False Addition a  + (b + c) = (a + b) + c True Subtraction a  − (b − c) = (a − b) − c False Multiplication (a × b) × c = a × (b × c) True Division (a ÷ b) ÷ c = a ÷ (b ÷ c) False

For Rational Numbers

Let us take three rational numbers 1/2, 3/2  &5/2

 Operation Number Remark Addition a  + (b + c) = (a + b) + c Take a = 1/2, b = 3/2 & c = 5/2   L.H.S a + (b + c) = 1/2+(3/2+5/2) = 1/2+((3  +  5)/2) = 1/2+(8/2) = (1  +  8)/2 = 9/2 ∴ (a + b) + c = 9/2   R .H.S (a + b) + c = (1/2+3/2)+5/2 = ((1  +  3)/2)+5/2 = 4/2+5/2 = (4  +  5)/2 = 9/2 ∴ a + (b + c) = 9/2 Since a  + (b + c) = (a + b) + c ∴ Addition is associative. Subtraction a − (b − c) = (a − b) − c Take a = 1/2, b = 3/2 & c = 5/2   L.H.S a − (b − c) = 1/2-(3/2-5/2) = 1/2-((3  -  5)/2) = 1/2-((-2)/2) = (1  -  (-2))/2 = (1 +  2)/2=3/2 ∴ (a − b) − c = 3/2   R.H.S (a − b) − c = (1/2-3/2)-5/2 = ((1  -  3)/2)-5/2 = ((-2)/2)-5/2 = (-2  -  5)/2 = (-7)/2 ∴ a − (b − c) = (-7)/2 Since  a − (b − c) ≠ (a − b) − c ∴ Subtraction is not associative. Multiplication a × (b × c) = (a × b) × c Take a = 1/2, b = 3/2 & c = 5/2   L.H.S a × (b × c) = 1/2×(3/2×5/2) = 1/2×((3 ×  5)/(2 ×  2)) = 1/2×15/4 = (1  ×  15)/(2  ×  4) = 15/8 ∴ (a × b) × c = 15/8   R.H.S (a × b) × c = (1/2×3/2) ×5/2 = ((1 ×  3)/(2 ×  2))×5/2 = 3/4×5/2 = (3  ×  5)/(4  ×  2) = 15/8 ∴ a × (b × c) = 15/8 Since  a × (b × c) = (a × b) × c ∴ Multiplication is associative. Division (a ÷ b) ÷ c = a ÷ (b ÷ c) Take a = 1/2, b = 3/2 & c = 5/2   L.H.S (a ÷ b) ÷ c = (1/2÷3/2)÷5/2 = (1/2  ×2/3)÷5/2 = (1/3)÷5/2 = 1/3×2/5 = 2/( 15)   R.H.S a ÷ (b ÷ c) = 1/2÷(3/2÷5/2) = 1/2÷(3/2  ×2/5) = 1/2÷(3/5) = 1/2×5/3 = 5/6   (a ÷ b) ÷ c ≠ a ÷ (b ÷ c) Since (a ÷ b) ÷ c ≠ a ÷ (b ÷ c) ∴ Division is  not associative.

To summarize

 Numbers Associative for Addition Subtraction Multiplication Division Natural numbers Yes No Yes No Whole numbers Yes No Yes No Integers Yes No Yes No Rational Numbers Yes No Yes No

So associativity is always possible for addition & multiplication,

but not for subtraction & division.