THE multiplication it is one of the four basic mathematical operations and has properties that can contribute to mental calculation and to expedite math.
THE multiplication is also known as “product”. Thus, when we talk about the product of two numbers, we are referring to the result of the multiplication between them. Each number that is multiplied is called a factor. Therefore, in the 9·3·7 multiplication, the factors are: 9, 3 and 7.
We will discuss each of the properties of multiplication. Come on?
→ First property: Commutativity
That property is so famous that it is used by many as the saying: “The order of factors does not change the product”. This means that, in a multiplication, the order in which the numbers are multiplied does not change the result. Mathematically:
Data The and B belonging to the real, we will have:
a·b = b·a
For example, 9·7 = 7·9 = 63.
This property is useful for mental calculation coupled with the next one.
→Second property: Associativity
That property involves the multiplication of three or more numbers. This type of multiplication is always done two by two and the property states that you can first multiply any pairs of numbers that are side by side. Mathematically, it is written as follows:
Given the real numbers The, B and ç, we will have:
(a·b)·c = a·(b·c)
For example:
(3·4)·5 = 12·5 = 60
3·(4·5) = 3·20 = 60
Joining these two properties (commutativity and associativity), we can say that a chain of multiplications can be done in any order. So, multiply the factors you already know the result first and leave the other factors last. Often the digits that appear in the results change and make multiplication easier.
→ Third property: Powers of base 10
When the multiplication involves a power of base 10, which is the numbers 1, 10, 100, 1000, etc., it is not necessary to do any multiplication. Just count how many zeros the power of 10 has and put them at the end of the other factor. Look at the example:
326·10000 = 3260000
The result will always follow this logic.
→ Fourth property: Multiples of 10
When one of the factors is a multiple of 10, the result will follow a logic similar to the previous one, however, only for the zeros that appear after the last non-zero digit (different from zero). Note the example below:
200·304000
Note that there will be two zeros of factor 200 and three zeros of factor 304000 that will be placed at the end of the result. So just multiply 2 times 304 and put the five zeros (2 caught in 200 and 3 caught in 304000) at the end.
2·304 = 608. Then:
200·304000 = 60800000
→ Fifth property: distributivity
this is the only property which involves addition and multiplication at the same time. Remember that you need to do multiplications first and then go on to additions and subtracts. Here's what the property says: “The product of the sum is equal to the sum of the products”.
In other words, when the factor of a multiplication is a real number The and there is a sum between the real numbers B and ç, we can choose to multiply The per B and The per ç and then add up the results. Mathematically:
Given the real numbers The, B and ç, we will have:
a·(b + c) = a·b + a·c
→ Multiplication by various factors
The previous properties joined together allow the following to be done: When it is necessary to perform a multiplication, decompose one of the factors into multiples of 10, multiply each by the other factor - using the knowledge of multiplication by multiples of 10 - and finally add the results. For example:
325·50
(300 + 20 + 5)·50
Knowing that 3·5 = 15, we conclude that 300·50 = 15000. Similarly, we find the other results:
15000 + 1000 + 250 = 16250
By Luiz Paulo Moreira
Graduated in Mathematics
