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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 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.
As an addition of equal numbers, multiplication has all properties of addition, but it also has some extra properties. These are:
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.
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.
Find the product of the following multiplications:
In symbols, the commutative property of multiplication is written as
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
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
Calculate the value of expressions
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
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.
Calculate the following products in columns.
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