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As we can see, neither subtraction nor division is associative.

Distributive property

The distributive property    states that the product of a factor times a sum is the sum of the factor times each term in the sum.

a ( b + c ) = a b + a c

This property combines both addition and multiplication (and is the only property to do so). Let us consider an example.

The number four is separated by a multiplication symbol from a bracketed expression reading: twelve plus negative seven. Arrows extend from the four pointing to the twelve and negative seven separately. This expression equals four times twelve plus four times negative seven. Under this line the expression reads forty eight plus negative twenty eight. Under this line the expression reads twenty as the answer.

Note that 4 is outside the grouping symbols, so we distribute the 4 by multiplying it by 12, multiplying it by –7, and adding the products.

To be more precise when describing this property, we say that multiplication distributes over addition. The reverse is not true, as we can see in this example.

6 + ( 3 5 ) = ? ( 6 + 3 ) ( 6 + 5 ) 6 + ( 15 ) = ? ( 9 ) ( 11 ) 21   99

Multiplication does not distribute over subtraction, and division distributes over neither addition nor subtraction.

A special case of the distributive property occurs when a sum of terms is subtracted.

a b = a + ( b )

For example, consider the difference 12 ( 5 + 3 ) . We can rewrite the difference of the two terms 12 and ( 5 + 3 ) by turning the subtraction expression into addition of the opposite. So instead of subtracting ( 5 + 3 ) , we add the opposite.

12 + ( −1 ) ( 5 + 3 )

Now, distribute −1 and simplify the result.

12 ( 5 + 3 ) = 12 + ( −1 ) ( 5 + 3 ) = 12 + [ ( −1 ) 5 + ( −1 ) 3 ] = 12 + ( −8 ) = 4

This seems like a lot of trouble for a simple sum, but it illustrates a powerful result that will be useful once we introduce algebraic terms. To subtract a sum of terms, change the sign of each term and add the results. With this in mind, we can rewrite the last example.

12 ( 5 + 3 ) = 12 + ( −5 3 ) = 12 + ( −8 ) = 4

Identity properties

The identity property of addition    states that there is a unique number, called the additive identity (0) that, when added to a number, results in the original number.

a + 0 = a

The identity property of multiplication    states that there is a unique number, called the multiplicative identity (1) that, when multiplied by a number, results in the original number.

a 1 = a

For example, we have ( −6 ) + 0 = −6 and 23 1 = 23. There are no exceptions for these properties; they work for every real number, including 0 and 1.

Inverse properties

The inverse property of addition    states that, for every real number a , there is a unique number, called the additive inverse (or opposite), denoted− a , that, when added to the original number, results in the additive identity, 0.

a + ( a ) = 0

For example, if a = −8 , the additive inverse is 8, since ( −8 ) + 8 = 0.

The inverse property of multiplication    holds for all real numbers except 0 because the reciprocal of 0 is not defined. The property states that, for every real number a , there is a unique number, called the multiplicative inverse (or reciprocal), denoted 1 a , that, when multiplied by the original number, results in the multiplicative identity, 1.

a 1 a = 1

For example, if a = 2 3 , the reciprocal, denoted 1 a , is 3 2 because

a 1 a = ( 2 3 ) ( 3 2 ) = 1

Properties of real numbers

The following properties hold for real numbers a , b , and c .

Addition Multiplication
Commutative Property a + b = b + a a b = b a
Associative Property a + ( b + c ) = ( a + b ) + c a ( b c ) = ( a b ) c
Distributive Property a ( b + c ) = a b + a c
Identity Property There exists a unique real number called the additive identity, 0, such that, for any real number a
a + 0 = a
There exists a unique real number called the multiplicative identity, 1, such that, for any real number a
a 1 = a
Inverse Property Every real number a has an additive inverse, or opposite, denoted –a , such that
a + ( a ) = 0
Every nonzero real number a has a multiplicative inverse, or reciprocal, denoted 1 a , such that
a ( 1 a ) = 1

Questions & Answers

root under 3-root under 2 by 5 y square
Himanshu Reply
The sum of the first n terms of a certain series is 2^n-1, Show that , this series is Geometric and Find the formula of the n^th
amani Reply
cosA\1+sinA=secA-tanA
Aasik Reply
why two x + seven is equal to nineteen.
Kingsley Reply
The numbers cannot be combined with the x
Othman
2x + 7 =19
humberto
2x +7=19. 2x=19 - 7 2x=12 x=6
Yvonne
because x is 6
SAIDI
what is the best practice that will address the issue on this topic? anyone who can help me. i'm working on my action research.
Melanie Reply
simplify each radical by removing as many factors as possible (a) √75
Jason Reply
how is infinity bidder from undefined?
Karl Reply
what is the value of x in 4x-2+3
Vishal Reply
give the complete question
Shanky
4x=3-2 4x=1 x=1+4 x=5 5x
Olaiya
hi can you give another equation I'd like to solve it
Daniel
what is the value of x in 4x-2+3
Olaiya
if 4x-2+3 = 0 then 4x = 2-3 4x = -1 x = -(1÷4) is the answer.
Jacob
4x-2+3 4x=-3+2 4×=-1 4×/4=-1/4
LUTHO
then x=-1/4
LUTHO
4x-2+3 4x=-3+2 4x=-1 4x÷4=-1÷4 x=-1÷4
LUTHO
A research student is working with a culture of bacteria that doubles in size every twenty minutes. The initial population count was  1350  bacteria. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest whole number, what is the population size after  3  hours?
David Reply
v=lbh calculate the volume if i.l=5cm, b=2cm ,h=3cm
Haidar Reply
Need help with math
Peya
can you help me on this topic of Geometry if l help you
litshani
( cosec Q _ cot Q ) whole spuare = 1_cosQ / 1+cosQ
Aarav Reply
A guy wire for a suspension bridge runs from the ground diagonally to the top of the closest pylon to make a triangle. We can use the Pythagorean Theorem to find the length of guy wire needed. The square of the distance between the wire on the ground and the pylon on the ground is 90,000 feet. The square of the height of the pylon is 160,000 feet. So, the length of the guy wire can be found by evaluating √(90000+160000). What is the length of the guy wire?
Maxwell Reply
the indicated sum of a sequence is known as
Arku Reply
how do I attempted a trig number as a starter
Tumwe Reply
cos 18 ____ sin 72 evaluate
Het Reply

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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