1. Real Numbers and Algebra
Chapter 1
Real Numbers and Algebra

2. 1.1 Describing Data with Set of numbers
Natural Numbers are counting numbers
and can be expressed as
N = { 1, 2, 3, 4, 5, 6, …. } 
Set braces { }, are used to enclose the
elements of a set.
A whole numbers is a set of numbers, is
given by
W = { 0, 1, 2, 3, 4, 5, ……}

3. …Continued The set of integers include both natural
and the whole numbers and is given by  
I = { …, -3, -2, -1, 0, 1, 2, 3, ….}
A rational number is any number can be
written as the ratio of two integers
where q = 0. Rational numbers can be
written as fractions and include all
integers.
Some examples of rational numbers are 
, 1.2, and 0.

4. …continued
Rational numbers may be expressed in decimal form that either repeats or terminates. 
The fraction may be expressed as 0.3, a repeating decimal, and the fraction may be expressed as 0.25, a terminating decimal. The overbar indicates that 0.3 = ….
Some real numbers cannot be expressed by fractions. They are called irrational numbers.
2, 15, and  are examples of irrational numbers.

5. Identity Properties For any real number a, a + 0 = 0 + a = a,
0 is called the additive identity and
a . 1 = 1 . a = a,
The number 1 is called the multiplicative identity.
Commutative Properties
For any real numbers a and b,
a + b = b + a (Commutative Properties of addition)
a.b = b.a (Commutative Properties of multiplication)

6. (a.b) . c = a . (b . c) (Associative Properties for multiplication)
…Continued
Associative Properties 
For any real numbers a, b, c,
(a + b) + c = a + (b + c) (Associative Properties of addition)
(a.b) . c = a . (b . c) (Associative Properties for multiplication)
Distributive Properties
a(b + c) = ab + ac and
a(b- c) = ab - ac

7. 1.2 Operation on Real Numbers
The Real Number Line
Origin
-2 = Absolute value
cannot be
negative
2 = 2
Origin

8. …Continued If a real number a is located to the left of a
real number b on the number line, we say
that a is less than b and write a<b.
Similarly, if a real number a is located to the
right of a real number b, we say that a is
greater than b and write a>b.
Absolute value of a real number a, written a
, is equal to its distance from the origin on
the number line. Distance may be either
positive number or zero, but it cannot be a
negative number.

9. Arithmetic Operations
Addition of Real Numbers
To add two numbers that are either both positive or both
negative, add their absolute values. Their sum has the same
sign as the two numbers.
Subtraction of real numbers
For any real numbers a and b, a-b = a + (-b).
Multiplication of Real Numbers
The product of two numbers with like signs is positive.
The product of two numbers with unlike signs is negative.
Division of Real Numbers
For real numbers a and b, with b = 0, = a .
That is, to divide a by b, multiply a by the reciprocal of b.

10. 1.3 Bases and Positive Exponents
Squared Cubed 4 4 4
4 . 4 = = 43
Exponent
Base

11. Powers of Ten Power of 10 Value 103 1000 102 100 101 10 1 10-1 = 0.1
10-2
= 0.01
10-3
= 0.001

12. Integer Exponents
Let a be a nonzero real number and n be a positive integer. Then  
an = a. a. a. a……a (n factors of a ) 
a0 = 1, and  a –n =
a -n b m
b -m = a n
a -n b n
b = a

13. … cont The Product Rule am . an = a m+n
  For any non zero number a and integers m and n,
am . an = a m+n
The Quotient Rule
For any nonzero number a and integers m and n, am
= a m – n
a n

14. bn Raising Products To Powers
For any real numbers a and b and integer n,
(ab) n = a n b n
Raising Powers to Powers
For any real number a and integers m and n,
(am)n = a mn
Raising Quotients to Powers
For nonzero numbers a and b and any integer
a n = an b bn

15. …Continued
A positive number a is in scientific notation when a is written as b x 10n, where 1 < b < 10 and n is an integer. 
Scientific Notation
Example : 52,600 = 5.26 x 104 and = 6.8 x 10 -3

16. 1.4 Variables, Equations , and Formulas
A variable is a symbol, such as x, y, t, used to represent any unknown number or quantity.
An algebraic expression consists of numbers, variables, arithmetic symbols, parenthesis, brackets, square roots.
Example 6, x + 2, 4(t – 1)+ 1,
X + 1

17. …cont
An equation is a statement that says two mathematical expressions are equal. 
Examples of equation 
3 + 6 = 9, x + 1 = 4,  d = 30t, and x + y = 20
A formula is an equation that can be used to calculate one quantity by using a known value of another quantity.
The formula y = computes the no. of yards in x feet. If x= 15, then y= = 5.

18. Square roots The number b is a square root of a number a if b2 = a.
Example - One square root of 9 is 3 because 32 = 9.
The other square root of 9 is –3 because (-3)2 = 9. We use the
symbol to 9 denote the positive or principal square root of 9.
That is,
9 = +3. The following are examples of how to evaluate
the square root symbol. A calculator is sometimes needed to approximate square roots,
4 = + 2 -
The symbol ‘ + ‘ is read ‘plus or minus’. Note that 2 represents
the numbers 2 or –2.

19. Cube roots The number b is a cube root of a number a if b3 = a
The number b is a cube root of a number a if b3 = a
The cube root of 8 is 2 because 23 = 8, which may
be written as 3 8 = 2. Similarly 3 –27 = -3 because
(- 3)3 = - 27.
Each real number has exactly one cube root.

20. 1.5 Introduction to graphing
Relations is a set of Ordered pairs.
If we denote the ordered pairs in a relation
(x,y), then the set of all x-values is called the
Domain (D) of the relation and the set of all y
values is called the Range (R)
S = {(2, -2), (3, 4), (8, 9), (11, 13 )}
D= {2, 3, 8, 11}
R= { -2, 4, 9, 13 }

21. Example 1. Find the domain and range for the relation given by
Find the domain and range for the relation
given by
S = {( -3, -1), (0,3), (2, 4), (4,5), (6,5)} Solution
The domain D is determined by the first
element in each ordered pair, or
D ={-3, 0,2, 4,6}
The range R is determined by the second
R = {-1,3,4,5}

22. The Cartesian Coordinate System
Quadrant II y Quadrant I y
(1, 3) 3 2 1 -1 -2 2 1 -1 -2
Origin x x
Quadrant III Quadrant IV
The xy – plane Plotting a point

23. Scatterplots and Line Graphs
If distinct points are plotted in the xy- plane, the resulting graph is called a scatterplot. Y 7 6 5 4 3 2 1
(4, 6)
(3, 4)
(6, 3)
(1, 1)
(5, 0) X

24. Using Graphing Calculator

25. Using Graphing Calculator
Make a table for y = , starting at x = 10 and incrementing by 10 and compare
The table for example 4 ( pg 41)
Go to Y= and enter Go to 2nd then table set and enter Go to 2nd then table
Graph

26. Viewing Rectangle ( Page 57 )
Ymax
}Ysc1
Xmax
Xmin
Xsc1
Ymin
[ -2, 3, 0.5] by [-100, 200, 50]

27. Making a scatterplot with a graphing calculator
Plot the points (-2, -2), (-1, 3), (1, 2) and (2, -3) in [ -4, 4, 1] by [-4, 4, 1]
(Example 10, page 58)
Go to 2nd then stat plot
Go to Stat Edit then enter points
Scatter plot
[ -4, 4, 1] by [-4, 4, 1]

28. Example 11 Cordless Phone Sales
Year
1987
1990
1993
1996
2000
Phones (millions)
6.2
9.9
18.7
22.8
33.3
Go to Stat edit and enter data
Enter line graph
Hit graph
Enter datas in window
[1985, 2002, 5] by [0, 40, 10]