Math Lesson 1.3.3 - Multiplication

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Welcome to our Math lesson on Multiplication, this is the third lesson of our suite of math lessons covering the topic of Operations with Numbers and Properties of Operations, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.

Multiplication

Multiplication is a shorter representation of the repeated addition of equal numbers. It is represented in expressions through the symbols ( × ) or ( · ). For example, if we see written somewhere 5 × 4 (or 5 · 4), we read "5 multiplied by 4" or "5 times 4" which means (5 + 5 + 5 + 5).

In general, we have

The numbers participating in a multiplication operation are known as factors and the result of a multiplication is called the product. For example, in the multiplication we saw earlier

5 and 4 are both factors and 20 is the product.

Properties of multiplication

As an addition of equal numbers, multiplication has all properties of addition, but it also has some extra properties. These are:

1. Closure property of multiplication

In simple words, this property states that if two (or more) factors belong to a given number set, the product also belongs to this set.

For example, in 4 × 7 = 28 both factors are natural numbers. As a results, the product is also a natural number.

Another example: (-4) × (-2) = (+8). Both factors are integers, hence the product is also an integer.

Symbolically, we write the closure property of multiplication as

a × b = c a ϵ X and b ϵ X then c ϵ X

where a and b are the factors, c is the product, X is the given set of numbers which a and b belong to.

2. Commutative property of multiplication

According to this property, when we switch the place of factors the product does not change. For example, since 4 × 9 = 36, then 9 × 4 = 36 as well. The commutative property is often used when there are more than two factors in a multiplication and we change the place of two of them to make the operations easier, for example, to obtain partial products which end with any zero. Let's consider a couple of examples to clarify this point.

Example 4

Find the product of the following multiplications:

Solution 4

1. Expressions involving multiplication are normally solved by completing the operations from left to right. However, here we switch the places of 7 and 6 to make the operations easier. Thus, we have
   5 × 7 × 6
   = 5 × 6 × 7
   = 30 × 7
   = 210
   If the commutative property was not used, we would have
   5 × 7 × 6
   = 35 × 6
   = 210
   The operation 35 × 6 is more challenging than 30 × 7, as in the second case, we can find the value of 3 × 7 = 21 and then, add a zero to the product.

In symbols, the commutative property of multiplication is written as

3. Associative property of multiplication

The associative property of multiplication is similar to that of addition in the sense that we can start doing the operations, not from the two leftmost factors but, from somewhere else for convenience as the result does not change. For example, in the mathematical expression 8 × 4 × 5, we can do 4 × 5 first, i.e.

In symbols, the associative property of multiplication is written as

4. Distributive property

This is a new property that integrates addition (or subtraction) and multiplication. In simple words:

When an expression inside brackets containing addition or subtraction is multiplied by a number, the expression can be written without brackets where the given number multiplies every element of the expression separately.

In symbols, we can write the distributive property as

Example 5

Calculate the value of expressions

Solution 5

1. Using the distributive property of multiplication, we obtain
   (18 + 4) × 5
   = 18 × 5 + 4 × 5
   = 90 + 20
   = 110
   If the distributive property was not used, we would have
   (18 + 4) × 5
   = 22 × 5
   = 110
   Hence, the result is the same in both cases.
2. Again, using the distributive property of multiplication, we obtain
   (31 - 7) × 3
   = 31 × 3 - 7 × 3
   = 93 - 21
   = 72
   If the distributive property was not used, we would have
   (31 - 7) × 3
   = 24 × 3
   = 72
   Again, the result is the same; this fact confirms the correctness of the distributive property of multiplication.

5. Multiplicative identity property

Recall that the number 0 was the identity element of addition, i.e. an element that didn't change the value of expression. In multiplication, this identity element is the number 1. This means that if we multiply a number by 1, the product is the same as the number itself.

We can write the multiplicative identity property in symbols as

Multiplication in column

We can multiply two numbers in column in a similar way to addition in column. Thus, we multiply each number of the upper factor to each number of the lower factor and the products obtained are written in separate rows below each other by starting (due right) from the position of the digit of the lower number involved in the process. When an individual product is more than 10, we carry numbers in the same way as we did in addition. Then these separate products are added in column. Look at the examples below.

Example 6

Calculate the following products in columns.

Solution 6

1. We write the factors in columns where the factor with the largest number of digits is written in the upper part for convenience. Hence, in column we write 56 × 7 instead of 7 × 56. We have
2. We multiply 37 by 2 and 37 by 5 separately. However, we write the second product (37 × 5) one position more in left. We have We may write a zero in the empty position at the end of the second product derived from the fact that we begin writing this product one position more in left. However, this is not obligatory.
3. We use the same procedure as in (b). Despite the fact that the second product gives zero, we write it in a separate column anyway. We have

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