Distributive property


Definition of Distributive property:

The distributive property is one of the most frequently used math properties.  Some of the other well-known fundamental math properties are associative property, commutative property, closure property, etc.

Essentially, this property explains the distribution of multiplication over other fundamental arithmetic operations such as addition and subtraction.

The property states that a number or an algebraic variable such as X, Y or Z, can be independently multiplied to either each of the numbers involved in the sum or difference Or to their sum or difference to give a same final result – sum or difference.

The following examples involving numbers and variables will illustrate the application of distributive property of multiplication over addition and subtraction.

Example 1:

  • 2 x (3+4) = 2 x 3 + 2 x 4 = 6 + 8 = 14.
  • 2 x (3 - 4) = 2 x 3 – 2 x (4) = 6 – 8 = -2.
  • 5 x (Y + 2) = 5 x Y + 5 x 2 = 5Y + 10.

Types of Distributive Property:

  • Left Distributive.

When a factor, which may be either a number or a variable is to be multiplied to the sum or difference of other terms, then the operation of multiplication can be distributed by multiplying the factor to each of the individual terms in the sum or difference.

Next, the two sums or differences so obtained can be added or subtracted as the case may be to give the final result.

  • The Right Distributive.

When a sum or difference of terms is to be multiplied to a factor, then from the right distributive property, the operation can be performed by multiplying each of the individual terms with the factor and again as above the two sums can be added or subtracted to give a final sum or difference.

NOTE:
There is a difference between the Left-Distributive and Right Distributive properties.

  • Multiplication follows both Left and Right Distributive rules.

 X * (Y + Z) = X * Y + X * Z.  (Left)
(Y + Z) * X = Y * X + Z * X.  (Right)

  • Division follows only the Right Distributive rule.

  (Y + Z) / X = Y/X + Z/X.                          (Right)
But, division does not obey the left distributive property.
X / (Y + Z) is not equal to X/Y + X/Z.   (Left)

 Therefore, Distributive property doesn’t apply to division in all the cases.


Example:

4 / (2 + 3)  is not equal to 4/2 + 4/3.

But (2 + 3) / 4 = 2/4 + 3/4.

Other Examples

Consider multiplying 9 with 73 using the distributive property (the left one).

1. 9 X 73
: 9 X (70 + 3)
: 9 X (7 X 10 + 3)
: 9 X 7 X 10 + 9 X 3
: 63 X 10 + 9 X 3
: 630 + 27
: 657.

2. (a + b) × (a – b)

: a . (a - b) + b . (a – b)
: a.a – a.b + b.a – b.b
: a2 - ab + ba – b2
: a2- b2

3. The distributive law can be used in taking out common factors.

Eg: 12ab + 18b +30bc
The common factorof 12ab, 18b and 30bc, i.e. 6b can be taken out in the following way:
6.b.(2a + 3 + 5c)

Application of Distributive Property in Sets and Matrices.

  • Distributive Property over the set operations such as Union And Intersection:
  • Union over intersection

 Let A, B and C be three sets. Then the distributive property of union over intersection will be as

 A (B ∩ C) = (A B) (A C)

Let A = {1, 2, 3, 4, 5}
B = {8, 9, 10}
C = {9, 10}

The Left Hand Side, i.e. LHS is:
B ∩ C is the set of elements common to the two sets B and C.
Therefore,    (B C) = {9,10}
And, A (B ∩ C) is the set of all elements that are in the sets A, B and C.
i.e. A (B ∩ C) = = {1 , 2 , 3 , 4, 5, 9, 10}
The Right Hand Side, i.e RHS is:
(A B) = {1, 2, 3 , 4, 5, 8, 9, 10}
(AC) = {1 , 2 , 3 , 4, 5, 9, 10}

And finally,

(AB) ∩ (AC) = {1 , 2 , 3 , 4, 5, 9, 10}

Thus LHS = RHS.

Hence, it can be said that distributive property applies to Union over Intersection.

A∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

  • Intersection over Union.

Eg: A(BC) = (AB)  (B C)

Let A = {1 , 2 , 3 , 4, 5}
B = {5, 6, 10}
C = {5, 10}

LHS =>
BC = {5,10}
A (B C) = {5}

RHS =>
(AB)  = {5}
(A C) = {5}
(A B) (B C) = {5}

Thus LHS = RHS.

Therefore, it can be said that distributive property applies to Intersection over Union also.

A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

  • Distributive property on Matrix multiplication

Note: Matrix is a rectangular array of numbers arranged in rows and columns- that is treated in a certain prescribed way.

Note:
An m by n matrix means a matrix having m rows and n columns.
We say, the order of the matrix is m by n, written as m × n

Example:

If P and Q are m×n matrices and R is a nxq matrix, then
(P + Q) x R = P x R + Q x R.

  • Distributive property applies to Maximum and Minimum functions also.

Note : Maximum function is denoted by max(a, b, c…) which is equal to the maximum of the numbers denoted by a, b, c, …..

Eg: Maximum function
x + max(y , z) = max(x +y , x + z)

Let x = 2, y=4, z =6.

The LHS is:
2 + max(4,6)
2 + 6
8

The RHS is:
: max(2 + 4, 2 +6)
:max( 6, 8)
:8

Thus, LHS = RHS.
Therefore, Maximum and Minimum functions also follow the distributive property.

Commutative property

Math properties.



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