Check sibling questions

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.

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Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths and Science at Teachoo.