1. Real Numbers 1
Definition 2
Properties 3
Examples

2. Definition Real Numbers include: Integers Rational Numbers
-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 = …
1/3 = …
Irrational Numbers
Non-terminating and non-repeating decimals
Π = …, √2 = …

3. Properties Addition is commutative Addition is associative
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

4. Properties (Cont.) -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

5. Properties (Cont.)
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

6. Properties (Cont.) Trichotomy Law Definition of Absolute Value
If a and b are real numbers, then exactly one of the following is true: a=b, a<b, a>b
Definition of Absolute Value
If a ≥ 0, then |a|=a
If a <0, then |a|=-(a)
Distance on a number line
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

7. Examples
If p, q, r, and s denote real numbers, show that (p+q)(r+s)=pr+ps+qr+qs
(p+q)(r+s)
=p(r+s)+q(r+s)
=(pr+ps)+(qr+qs)
= pr+ps+qr+qs
If x>0, and y<0, determine the sign of x/y + y/x
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

8. Examples
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

9. Guided Practice
Do Problems on page 16, 1-40