Properties of real numbers
In this concept, you will be working with 10 properties of real numbers:
1. Commutative Property of Addition
2. Associative Property of Addition
3. Additive Identity Property
4. Additive Inverse Property
5. Distributive Property
6. Commutative Property of Multiplication
7. Associative Property of Multiplication
8. Multiplicative Identity Property
9. Multiplication Property of Zero
Keep the following chart handy:
Commutative (order)
Associative (grouping)
Identity (returns itself)
PROPERTIES OF ADDITION
x + y = y + x
(x + y) + z = x + (y +
z)
x + 0 = x
or 0 + x = x
PROPERTIES OF MULTIPLICATION
x • y = y • x
or
xy = yx
(x • y) • z = x • (y • z)
or (xy)z = x(yz)
x • 1 = x
or 1 • x = x
x + (-x) = 0
x•
Inverse (opposite  0 or 1)
or
Distributive ( • through + )
Multiplicative Property of
Zero (multiply by 0 = 0)
-x + x = 0
1
 1
x
1
or
•x  1
x
x(y + z) = xy + xz
or
(x + y)z = xz + yz
0 • x = 0
or
x • 0 = 0
Now, let’s look at the chart and try to understand what it is telling us:
Commutative: In both cases (addition and multiplication), you have two numbers. The
order of the numbers switches. This property tells us that it doesn’t matter if we add 3
to 4 or add 4 to 3, the answer is the same. This property tells us that it doesn’t matter if
we multiply 3 times 4 or multiply 4 times 3, the answer is the same.
Associative: In both cases (addition and multiplication), you have three numbers. The
order of the numbers remains the same. The parenthesis move to form different
groups. This property tells us that if we have three numbers to add (or to multiply), we
can do the first two numbers first then the third number, or we can do the last two
numbers first then the first one.
Identity: This property is talking about the two special numbers 1 and 0. If you add 0 to
anything, it doesn’t change that anything. Thus, zero is the additive identity. Also, if
you multiply 1 to anything, it doesn’t change that anything. Thus, one is the
multiplicative identity. The multiplicative identity is extremely important when adding and
subtracting fractions.
Inverse: This property talks about numbers that are opposites of other numbers.
When you add a number and its opposite (inverse), you get 0. When you multiply a
number by its opposite (inverse), you get 1.
Distributive Property: This one is easy to recognize because it is the only property that
deals with BOTH addition and multiplication. This property tells you that if two
numbers are added inside grouping symbols (parenthesis) and a number is multiplied
outside those grouping symbols, you can distribute (multiply) to each number inside the
parenthesis first then add. This appears to be contrary to the order of operations, but
there are times when one of the two numbers inside the parenthesis is an unknown
variable. Thus, you can’t add inside the parenthesis first. So, you can multiply the
number outside the parenthesis to EACH number/variable inside the parenthesis.
IMPORTANT: When all you see outside the parenthesis is a minus sign, remember
that this represents a negative one so multiply (distribute) -1 to EACH number/variable
inside the parenthesis.
Multiplication Property of Zero: This property is so intuitive and simple that it is
difficult to define, yet it is an extremely important property when solving equations. The
multiplication property of zero states that if you multiply any number or any variable
by zero, the answer will be zero.
Let’s try some examples – for each equation below, indicate the property that justifies
the equation:
1) 3 = 1 • 3 Notice that you are multiplying by one? Identity Property of multiplication
2) a + d = d + a Notice that the same two numbers are used but they are switched?
Commutative Property of addition
3) 4(5 + a) = 4 • 5 + 4a Notice how this has both addition and multiplication?
Distributive Property
4) ½ = ½ + 0 Notice how we are adding zero? Identity Property of addition
5) (6 + x) + 5 = 6 + (x + 5) Notice how we have three numbers/variables in the
6)
same order? The parenthesis are grouping different pairs of these numbers. Associative Property
of addition
-7 + 7 = 0 Notice how a number and its opposite are added to get 0? Inverse Property of
addition
7) 3 •
1
 1
3
Notice that you are multiplying by a number’s opposite and you get 1? Inverse
Property of multiplication