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Real Numbers
1
2
3
Definition
Properties
Examples
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Definition
2

Real Numbers include:
 Integers
 -3,-2,-1,0,1,2,3
 Rational
Numbers
 Decimals
that can be represented in fraction form that are
either terminating or non-terminating and repeating
 5/4 = 1.25
 177/55 = 3.2181818…
 1/3 = .33333…
 Irrational
Numbers
 Non-terminating
and non-repeating decimals
 Π = 3.14159…, √2 = 1.41421…
Properties
3

Addition is commutative
a
+b=b+a
 Order does not matter

Addition is associative
a
+ (b + c) = (a +b) + c
 Grouping does not matter

0 is the additive identity
a
+0=a
 Adding 0 yields the same number
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Properties (Cont.)
4

-a is the additive inverse (negative) of a
a
+ (-a) = 0, 12+(-12)=0
 Adding a number and it’s inverse gives 0

Multiplication is commutative
 ab
= ba, 3*4=4*3=12
 Order of multiplication does not change the result

1 is the multiplicative identity
a
*1=a
 Multiplying 1 yields the same number
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Properties (Cont.)
5

If a ≠ 0, 1/a is the multiplicative inverse (reciprocal)
of a
a(1/a) = 1, 3(1/3)=1
 Multiplying a non-zero number by its reciprocal yields 1


Multiplication is distributive over addition
a(b + c) = ab + ac
 (a + b)c = ac + bc
 Multiplying a number and a sum of two numbers is the same
as multiplying each of the two numbers by the multiplier and
then adding the products

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Properties (Cont.)
6

Trichotomy Law


Definition of Absolute Value



If a ≥ 0, then |a|=a
If a <0, then |a|=-(a)
Distance on a number line


If a and b are real numbers, then exactly one of the following is
true: a=b, a<b, a>b
d(A, B) = |B-A|
Law of the signs


If a and b both have the same sign, then ab and a/b are positive
If a and b have different signs, then ab and a/b are negative
Examples
7

If p, q, r, and s denote real numbers, show that
(p+q)(r+s)=pr+ps+qr+qs


If x>0, and y<0, determine the sign of x/y + y/x



(p+q)(r+s)
=p(r+s)+q(r+s)
=(pr+ps)+(qr+qs)
= pr+ps+qr+qs
Since only y is negative, both x\y and y/x will be negative numbers
A negative number increased by another negative number will yield a
“more” negative number
If x<1, rewrite |x-1| without using the absolute value symbol


If x<1, then x-1<0 (negative)
By part 2 of the definition of absolute value, |x-1|=-(x-1)=-x+1 or 1-x
Examples
8

Let A, B, C, and D have coordinates -5, -3, 1, and 6
respectively. Find d(B,D)/\.
 d(B,D)
= d(-3,6)
=|6-(-3)|
=|6+3|
=|9|
=9
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Guided Practice
9

Do Problems on page 16, 1-40
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