Associative Property for numbers

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

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.

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