Basic Number Properties – Commutative, Associative and Distributive

Basic number properties like commutative, associative, and the distributive properties are explored here. However, we can extend them to include the properties of zero and one. We also called these properties rules of arithmetic

Commutative property

In commutative property, we see the word commute which means exchange from the Latin word ‘commutare’.

The word exchange in turn may mean switch. For examples, washing my face and combing my hair is a good example of this property.

Another good example is doing my math homework and then finishing my science reading.

The important thing to notice in the two examples above is that the order we do things can be switched, so does not matter or will never cause any problems or conflicts.

However, reading a math lesson and then answering the review questions is not commutative.

Here the order does matter because I have to read the lesson before knowing how to answer the review questions

In mathematics, we know that

2 + 5 = 5 + 2

12 + 4 = 4 + 12

-1 + 8 = 8 + -1

All the above illustrates the commutative property of addition. This means that when adding two numbers, the order in which the two numbers are added does not change the sum

All three examples given above will yield the same answer when the left and right side of the equation are added

For example, 2 + 5 = 7 and 5 + 2 is also equal to 7

The property is still valid if we are doing multiplication

Again, we know that

More examples:

Although addition is commutative, subtraction is not commutative.

Notice that 3-2 is not equal to 2-3

3- 2 = 1 , but 2- 3 = -1

Therefore, switching the order yield different results

Associative property

The word associate in associative property may mean to join or to combine.

For examples, suppose I go to the supermarket and buy ice cream for 12 dollars, bread for 8 dollars, and milk for 15 dollars.

How much money do I owe the cashier? The situation above is associative

When I do my total in my head, I can combine or add the price of the ice cream and the bread first and add the result to the price of milk.

Otherwise, I can combine or add the price of bread and milk first and add the result to the price of ice cream

Both ways of approaching the problem gives the same answer

Mathematically, you are trying to do the following:

12 + 8 + 15

You can add these three numbers in the order they appear

12 + 8 = 20 ( This is adding price of ice cream and bread first)

20 + 15 = 35

You can use parentheses to show the order in which you are adding

(12 + 8) + 15

Another way to add is to add not according the order in which they appear

You may decide you will add first 8 and 15

8 + 15 = 23 ( This is adding price of bread and milk first)

12 + 23 = 35

Again, using parentheses to show the order in which you are adding, you get:

12 + (8 + 15)

We conclude that (12 + 8) + 15 = 12 + ( 8 + 15)

The above example illustrates the associative property of addition

Terms added in different combinations or grouping yield the same answer

Associative property of multiplication

Again, we know that

All three examples given above will yield the same answer when the left and right side of the equation are multiplied.

For example,

Also,

Although multiplication is associative, division is not associative.

Notice that is not equal to

However,

Therefore, different combination may yield different results.

Notice that it may happen that a different grouping gives the same result.

However, we shall not make a rule out of this because it is not true for all cases

Finally, note that unlike the commutative property which plays around with two numbers, the associative property combines at least three numbers

Other examples:

( 1 + 5) + 2 = 1 + ( 5 + 2)

( 6 + 9) + 11 = 6 +( 9 + 11)

Distributive property

Named the ‘Distributive Property (sometimes referred to as the distributive law) because in essence, you are distributing something as you separate or break it into parts. The distributive property makes numbers easier to work with. In algebra when we use the distributive property, we’re expanding (distributing). The Distributive Property lets you multiply a sum by multiplying each addend separately and then add the products.

We can explain the distributive property with three good examples

Example 1:

Width= 6 Length =4 extended length= 10.

Since width = 6 and length = 4 + 10, area .

You can do the math two ways.

You can add 4 and 10 and multiply what you get by 6.

Otherwise, you can use the distributive property illustrated above by multiplying 6 by 4 and 6 by 10 and adding the results

Example 2: You go to the supermarket. 1 bag of apples costs 4 dollars. 1 gallon of olive oil costs 10 dollars. You get 6 bags of apples and 6 gallons of olive oil. How much money do you pay the cashier?

Total cost = number of items you get \times (cost for apples + cost for olive oil)

Total cost

Example 3: Robert has 8 notebooks and his brother has 6. If we double both amount, how many do they now have altogether?

We get

Notice that we get the same answer if we add 8 and 6 and multiply the result by 2

Properties of Zero

The two properties of zero are the addition property and the multiplication property.

Addition property:

The addition property says that a number does not change when adding or subtracting zero from that number

Examples:

2 + 0 = 2

12 + 0 = 12

5 − 0 = 5

48 − 0 = 48

0 + 1 = 1

0 − 9 = – 9

Additive inverse property

If you add two numbers and the sum is zero, we call the two numbers additive inverses or opposites of each other

For example, 2 is the additive inverse of -2 because 2 + -2 = 0

-2 is also the additive inverse of 2 because -2 + 2 = 0

Multiplication property

The multiplication property says that zero times any number is equal to zero

Examples

Exercise

1. Fill in the blanks:
   1. (2+3)+9=2+(3+9)=____ +9=2+ ____=______
   2. If (2 + 8) + 9 = 19, then what is 2 + (8 + 9)?
   3. If , then what is ?
   4. Rewrite the expression  using the associative property.
   5. If , then what is ?
2. Simplify . Justify your steps.
3. Simplify .   Justify your steps.
4. Rewrite the expression 4 + 1 using the commutative property.
5. Complete the equation using the commutative property.
   If 4 + 8 + 10 = 22, then 10 + 4 + 8 =____.
6. Rewrite the expression 5 + 9 + 10 using the commutative property.
7.  Multiply using Distributive Property:
   1. 
   2. 
   3. 
   4. 
8. I go to the market and buy 6 oranges, 6 apples and 6 pears. If the cost of each orange is rs 4, the cost of each apple is Rs 5 and the cost of each pear is Rs 3, how much money did I spend?
9. Simplify:
10. Find the additive inverse of:
    1. 7
    2. -11
    3. 19
    4. -25