Using Properties in Multiplication
Multiplication has several properties and these have been explained below:
Identity Property of Multiplication
For each real number, there is a unique real number 1.
\( 2\times 1=2 \) and \( 1\times 2=2 \)
In the above example, one (1) is the identity element of multiplication.
Example: Find the value of S if S \( \times \) 1 = 6 using the multiplicative identity property.
According to the identity property of multiplication, when we multiply any number by one, we get the same number as the result.
Therefore, S = 6
6 \( \times \) 1 = 6
Zero Property of Multiplication
The zero property of multiplication can also be termed the multiplicative property of zero. This property states that if any term or value is multiplied by zero (0) then the result will be zero.
For each real number R,
R \( \times \) 0 = 0 and 0 \( \times \) R = 0
Example: Find the missing number by using the zero property of multiplication: 21 \( \times \) 0 = _.
When we multiply a number by zero, then the result will be zero.
21 \( \times \) 0 = 0
Therefore, the missing number is 0.
Commutative Property
According to the commutative property, the order in which two real numbers are multiplied does not affect the result.
For the real numbers 2 and 3,
2 \( \times \) 3 = 3 \( \times \) 2
6 = 6
Commutative property is also referred to as the order property of multiplication.
Example 1: If we take two balloons and multiply them by three, the result will be six apples. Even if we change the order of multiplication, the result will be the same. Hence, the commutative property holds true.
Example 2: If there are three fruit baskets, each with four fruits, write two multiplication sentences for the total number of fruits.
We have four fruits in one basket, and the total number of baskets is four. So, the total number of fruits can be found in two ways.
Case 1: Multiply three baskets with four fruits.
Case 2: Multiply four fruits with three baskets.
Distributive Property
The distributive property of binary operations is a generalization of the distributive law, which states that in elementary algebra, equality is always true.
Distributive Property (for Addition)
The distributive property for addition states that when we multiply the sum of two or more addends by a number, it produces the same result as multiplying each addend individually by the number and then adding up the products together.
Suppose the given numbers are 1, 2, and 3. So, the distributive property for addition will be:
1 ( 2 + 3 ) = 1 \( \times \) 2 + 1 \( \times \) 3
1 \( \times \) 5 = 2 + 3
5 = 5
Distributive Property (for Subtraction)
The distributive property for multiplication over subtraction is similar to multiplication over addition. You can first multiply, and then subtract, or you may first subtract and then multiply. This is referred to as “multiplying the multiplier.”
Suppose the given numbers are 1, 3, and 2. So, the distributive property for subtraction will be:
1 ( 3 – 2 ) = 1 \( \times \) 3 – 1 \( \times \) 2
1 \( \times \) 1 = 3 – 2
1 = 1
Example: What are the two ways to break apart 4 x 7 using the distributive property.
The two ways to break apart 4 x 7 using the distributive property are:
Case 1: Let’s break 7 as 9 – 2
So, 4 ( 9 – 2 )
( 4 x 9 ) – ( 4 x 2 ) [Distributive Property]
36 – 8 [Simplify]
28
Case 2: Let us break apart 4 as 2 + 2
So, 7 ( 2 + 2 )
( 7 x 2 ) + ( 7 + 2 ) [Distributive Property]
14 + 14
28
Associative Property
When we multiply any three real numbers, the order in which they are grouped (or associated) does not affect the outcome.
For the real numbers 1, 2, and 3
1 \( \times \) ( 2 \( \times \) 3 ) = ( 2 \( \times \) 3 ) \( \times \) 1
1 \( \times \) 6 = 6 \( \times \) 1
6 = 6
The associative property is also referred to as the grouping property of multiplication.
Example: Multiply three factors and use the associative property to write the same equation in two different ways.
Let us take the three factors as 2, 4, and 6. Using the associative property, we can write the multiplication of the given factors in different ways.
Case 1: Let us assume the order as given above.
So, ( 2 \( \times \) 4 ) \( \times \) 6 = 2 \( \times \) ( 4 \( \times \) 6 ) [Associative Property]
8 \( \times \) 6 = 2 \( \times \) 24 [Simplify]
48 = 48
Thus, the above factors satisfy the condition of associative property.
Case 2: Let us change the position of factors to their successive positions.
Since, the factors are 6, 2, and 4
Hence, ( 6 \( \times \) 2 ) \( \times \) 4 = 6 \( \times \) ( 2 \( \times \) 4 ) [Associative Property]
12 \( \times \) 4 = 6 \( \times \) 8 [Simplify]
48 = 48
Thus, the above factors satisfy the condition of associative property.
They are equal because they follow the commutative property of multiplication.
A 0 in a multiplication equation means 0 equal groups or 0 objects in a group, and so the product is also 0.