Real Numbers and Algebra Chapter 1 Real Numbers and Algebra

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, ……}  

…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.  

…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 = 0.3333333…. Some real numbers cannot be expressed by fractions. They are called irrational numbers. 2, 15, and  are examples of irrational numbers.

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)  

(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

1.2 Operation on Real Numbers The Real Number Line -3 -2 -1 0 1 2 3 Origin -2 2 -2 = 2 Absolute value cannot be negative 2 = 2 -3 -2 -1 0 1 2 3 Origin

…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.

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.

1.3 Bases and Positive Exponents Squared 4 Cubed 4 4 4 4 4 4 . 4 = 42 4 . 4. 4 = 43 Exponent Base

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

1.3 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  

… 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

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

…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 0.0068 = 6.8 x 10 -3  

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

…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.

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.  

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.

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 }

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}

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

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 1 2 3 4 5 6 7

Using Graphing Calculator

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

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

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]

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]