Download Properties of Real Numbers

Properties of
Real Numbers
The properties of real numbers help us simplify math
expressions and help us better understand the
concepts of algebra.
Commutative Property of
Addition

a+b=b+a

Example: 7 + 3 = 3 + 7

Two real numbers can be added in either
order to achieve the same sum.
Commutative Property of
Multiplication

axb=bxa

Example:3 x 7 = 7 x 3

Two real numbers can be multiplied
in either order to achieve the same
product.
Associative Property of
Multiplication

(a x b) x c = a x (b x c)

Example: (6 x 4) x 5 = 6 x (4 x 5)

When three real numbers are multiplied, it
makes no difference which are multiplied
first.

Notice how multiplying the 4 and 5 first
makes completing the problem easier.
Associative Property of
Addition

(a + b) + c = a + (b + c)

Example: (29 + 13) + 7 = 29 + (13 + 7)

When three real numbers are added, it
makes no difference which are added
first.

Notice how adding the 13 + 7 first
makes completing the problem easier
mentally.
Additive Identity Property

a+0=a

Example:

The sum of zero and a real number
equals the number itself.

Memory note: When you add zero to a
number, that number will always keep
its identity.
9+0=9
Multiplicative Identity
Property

ax1=a

Example:

The product of one and a number equals
the number itself.

Memory note: When you multiply any
number by one, that number will keep
its identity.
8x1=8
Additive Inverse Property

a + (-a) = 0

Example:
3 + (-3) = 0

The sum of a real number and its
opposite is zero.
Multiplicative Inverse

𝐚·
𝟏
𝐚
=𝟏
however,
𝟒
𝟏
·
𝟏
𝟒
a≠0

Example:
=𝟏

The product of a nonzero real
number and its reciprocal is one.
Property of Zero
(Multiplication)
 When
any number is
multiplied with zero, the
answer is zero.
 98,756,432
X0=0
Property of Opposites
a
 If
+ (-a) = 0
you added opposite #’s and
ended with 0
Distributive Property

a(b + c) = ab + ac
ab – ac

Example: 2(3 + 4) = (2 x 3) + (2 x
4)



or a(b – c) =
or
2(3 - 4) = (2 x 3) - (2 x 4)
Distributive Property is the sum or
difference of two expanded
products.
Properties of Equality
Addition property
of equality
Subtraction
property of equality
Multiplication
property of equality
Division property
of equality
If a = b, then a + c = b + c.
Adding the same number to
both sides of an equation
does not change the
equality of the equation.
If a = b, then
a – c = b – c.
Subtracting the same
number from both sides of
an equation does not
change the equality of the
equation.
If a = b and
c ≠ 0, then
a • c = b • c.
Multiplying both sides of the
equation by the same
number, other than 0, does
not change the equality of
the equation.
If a = b and
c ≠ 0, then
a ÷ c = b ÷ c.
Dividing both sides of the
equation by
the same number, other
than 0, does not change the
equality of the equation.
Transitive Property
If a = b and b = c, then a = c
If one quantity equals a second
quantity and the second quantity
equals a third quantity, then the first
equals the third.
If 1000 mm = 100 cm and 100 cm = 1 m,
Then 1000 mm = 1m
Symmetric Property
If a + b = c then c = a + b
If one quantity equals a second
quantity, then the second quantity
equals the first.
If 10 = 4 + 6, then 4 + 6 = 10
Reflexive Property
a=a
a+b=a+b
Any quantity is equal to itself.
7=7
2+3=2+3
Justify Steps in Solving
Equations

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