Intro
The real number systems carries a set of properties which help us define how numbers should behave. These properties become even more important when we begin to work with algebraic expressions. Below you will find a table which lists the defining laws of real numbers.
Law Addition Multiplication
Commutative
For all real a, b : a + b = b + a
Example : 1 + 3 = 3 + 1
For all real a, b : ab = ba
Example : 1(3) = 3(1)
Associative
Identity
Inverse
For all real a, b, c : a + (b + c) = (a + b) + c
Example : 1 + (2 + 3) = (1 + 2) + 3
For all real a, b, c : (ab)c = a(bc)
Example : (1 · 2)3 = 1(2 · 3)
There exists a real number 0 such that for every real a : a + 0 = a
Example : 3 + 0 = 3
In other words, adding “0” to any number will always yield that number.
There exists a real number 1 such that for every real a : a · 1 = a
Example : 3 · 1 = 3
In other words, multiplying any number by “1” will always yield that number.
For every real number a , there exists a real number, -a , such that: a + -a = 0
Example : 3 + -3 = 0
In other words, any number will always have its opposite and when you add them together, they will equal zero.
For every real number a, except 0, there
1 exists a real number, a
, such that: a ⋅
1 a
= 1
Example :
⋅
1
3 =
3
1
In other words, any number will always have its reciprocal and when you multiply them, they will always equal 1.
Distributive
For all real a, b, c : a(b + c) = ab + bc and (a + b)c = ac + bc
Example : 2(3 + 4) = 2(3) + 2(4) or (2 + 3)4 = 2(4) + 3(4)
1 Knowledge of number sense, concepts and operations
Identify the use of the field properties of the real number system (e.g., associative, commutative and distributive) in real-world situations and apply operations of real numbers.
1

Notice the laws in the table only showed addition and multiplication, but not division nor subtraction. This is because the commutative and associative laws do not work for division or subtraction.
Subtraction and Division… the oddballs .
Commutative for subtraction : a – b ≠ b – a
Example: 4 – 3 = 3 – 4. Is this true? Well, if we work it out we get 1 = -1, which is not true.
Therefore, the commutative law does not work for subtraction.
Commutative for division : a ÷ b ≠ b
÷
a
1
Example: 20 ÷ 5 = 5 ÷ 20. Let’s work it out and see what happens, 4 =
4 true; therefore the commutative law does not work for division.
. The statement is not
Associative for subtraction : (a – b) – c ≠ a – (b – c)
Example: (5 – 4) – 3 = 5 – (4 – 3)
1 – 3 = 5 – 1
-2 = 4 Not true, therefore subtraction does not work for the associative law.
Associative for division : (a ÷ b) ÷ c ≠ a ÷ (b ÷ c)
Example: 20 ÷ (16 ÷ 4) = (20 ÷ 16) ÷ 4
20 ÷ 4 =
5
4
÷ 4
5
5 =
16
Not true, therefore division does not work for the associative law.
As you can see, the grouping and order are not as important for multiplication and addition, but matter a great deal when working with subtraction and addition. Each of these laws is important as you move through your understanding of algebra; however, I would spend a little extra time learning the distributive law as it will come up again and again when working with algebraic expressions.
1 Knowledge of number sense, concepts and operations
Identify the use of the field properties of the real number system (e.g., associative, commutative and distributive) in real-world situations and apply operations of real numbers.
2