Understanding Algebra

James W. Brennan
 

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Properties of Real Numbers

The following table lists the defining properties of the real numbers (technically called the field axioms). These laws define how the things we call numbers should behave.

 

Addition

Multiplication

Commutative

For all real a, b

a + b = b + a

Commutative

For all real a, b

ab = ba

Associative

For all real a, b, c

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

Associative

For all real a, b, c

(ab)c = a(bc)

Identity

There exists a real number 0 such that for every real a

a + 0 = a

Identity

There exists a real number 1 such that for every real a

a × 1 = a

Additive Inverse
(Opposite)

For every real number a there exist a real number, denoted (-a), such that

a + (–a) = 0

Multiplicative Inverse (Reciprocal)

For every real number a except 0 there exist a real number, denoted , such that

a ×  = 1

Distributive Law

For all real a, b, c

a(b + c) = ab + ac, and (a + b)c = ac + bc

The commutative and associative laws do not hold for subtraction or division:

a – b is not equal to b – a

a ÷ b is not equal to b ÷ a

a – (b – c) is not equal to (a – b) – c

a ÷ (b ÷ c) is not equal to (a ÷ b) ÷ c

Try some examples with numbers and you will see that they do not work.

What these laws mean is that order and grouping don't matter for addition and multiplication, but they certainly do matter for subtraction and division. In this way, addition and multiplication are “cleaner” than subtraction and division. This will become important when we start talking about algebraic expressions. Often what we will want to do with an algebraic expression will involve rearranging it somehow. If the operations are all addition and multiplication, we don't have to worry so much that we might be changing the value of an expression by rearranging its terms or factors. Fortunately, we can always think of subtraction as an addition problem (adding the opposite), and we can always think of division as a multiplication (multiplying by the reciprocal).

You may have noticed that the commutative and associative laws read exactly the same way for addition and multiplication, as if there was no difference between them other than notation. The law that makes them behave differently is the distributive law, because multiplication distributes over addition, not vice-versa.. The distributive law is extremely important, and it is impossible to understand algebra without being thoroughly familiar with this law.

Example:  2(3 + 4)

According to the order of operations rules, we should evaluate this expression by first doing the addition inside the parentheses, giving us

2(3 + 4) = 2(7) = 14

But we can also look at this problem with the distributive law, and of course still get the same answer. The distributive law says that

 
      

 


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James W. Brennan