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(3/25/2) = 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 = (1/23/2)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.
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