real numbers and their properties รายวิชา ค 40102...

Real Numbers and Their Properties Real Numbers and Their Properties

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Real numbers Real numbers are used in everyday life to desc ribe quantities such as age, miles per gallon, and population. Real numbers are

represented by symbols such as

Real Numbers Real Numbers

345,9,0, ,0.666...,28.21, 2, , and 32

3

subsets of the real numbers

Set of natural numbers{1,2,3,4,...}

Set of whole numbers{0,1,2,3,4,...}

S et of integers{ 3, 2, 1,0,1,2,3,4,...}

A real number is rationalrational if it can be written as the ratio of two integers, where . For instance, the numbers0q

/p q

1 1 1250.3333... 0.3, 0.125, and 1.126126... 1.126

3 8 111

A real number that cannot be written as the ratio of two integers is called irrationalirrational . Irrational numbers have infin ite nonrepeating

decimal representations. For instance, the numbers

2 1.4142315... 1.41 and 3.1415926... 3.14

Real numbers are represented graphically by a real number line.

Subsets of real numbers Subsets of real numbers

T here is a - - one to one correspondence between real numbers and points on t

he real number line.

Solving Equations Solving Equations Equations and Solutions of Equations

An equationequation in x is a statement that two algebraic expressions are equal. For example

3 5 7x

- -SolveSolve

- -SolutionSolution

2 6 0 x x

2 4x

An equation that is truetrue for every real number in the domain of the variable is called an identityidentity.The domaindomain is the set of all number for which the equation is defined .

For example2 9 ( 3)( 3)x x x Identity

2

1

3 3

x

x x Is an identity ?

An equation that is true for just some (or even none) of the real numbers in the domain of the variable is called a conditional equation. conditional equation.

For example, the equation2 9 0x

2 4 2 1x x

conditional equation

Is the conditional equation ?

Linear Equations in One Variable Linear Equations in One Variable

A linear equation has exactly one solution. To see this, consider the following steps. (Remember that .)0a

0ax b

ax bb

xa

Write original equation.

Subtract b from each side.

Divide each side by a.

To solve an equation involving fractional expressions, find the least common denominator (LCD) of all terms and multiply every term by

the LCD. This process will clear the original equation of fractions and produce a simpler equation to work with.

When multiplying or dividing an equation by a variable quantity, it is possible to introduce an extraneous solution. An extraneous

solution is one that does not satisfy the original equation. Therefore, it is essential that you check your solutions.

Quadratic Equations Quadratic Equations

A quadratic equation quadratic equation in x is an equation that can be written in the general form

2 0ax bx c

where a, b, and c are real numbers, with . A quadratic equation in x - is also known as a second degree polynomial equation

in x.

0a

Solving a Quadratic Equation Solving a Quadratic Equation

Solving a Quadratic Equation Solving a Quadratic Equation

Solving a Quadratic Equation

Solving a Quadratic Equation Solving a Quadratic Equation

Solving a Quadratic Equation Solving a Quadratic Equation

Solving a Quadratic Equation Solving a Quadratic Equation

Solving a Quadratic Equation Solving a Quadratic Equation

Solving a Quadratic Equation Solving a Quadratic Equation

Solving a Quadratic Equation Solving a Quadratic Equation

Polynomial Equations of Higher Degree Polynomial Equations of Higher Degree

Polynomial Equations of Higher Degree Polynomial Equations of Higher Degree

Equations Involving Radicals Equations Involving Radicals

Polynomial Equations of Higher Degree Polynomial Equations of Higher Degree

Polynomial Equations of Higher Degree

Ordering Real Numbers Ordering Real Numbers Definition of Order on the Real Number Line Definition of Order on the Real Number Line

If a and b are real numbers, a is less than b if b - a is positive. The orderorder of a and b is denoted by the inequalityinequality a < b. This relationship can also be described by saying that b is greater than a and writing b > a. The inequality a b means that a is less than or equal to b, and the inequality b a means that b is greater than or equal to a. The symbols <, >, , and are inequality

symbols.

Geometrically, this definition implies that a < b if and only if a lies to the left of b on the real number line, as shown in Figure

Inequalities can be used to describe subsets of real numbers called intervals. In the bounded intervals below, the real numbers a and b

are the endpoints of each interval. The endpoints of a closed interval are included in the interval, whereas the endpoints of an open

interval are not included in the interval.

The symbol , positive infinitypositive infinity, and , negative infinitynegative infinity, do notrepresent real numbers, They are simply convenient symbols used to describe the unboundedness of an interval such as or

(1, )( ,3]

The Law of TrichotomyLaw of Trichotomy states that for any two real numbers a and b, precisely one of three relationships is possible:

Law of Trichotomy, , and a b a b a b

Absolute Value Absolute Value and Distanceand Distance The absolute value of a real number is its magnitude, or the distance

between the origin and the point representing the real number on the real number line.

Notice in this definition that the absolute value of a real number is - never negative. For instance, if a = 5, then |- 5| - - = ( 5) = 5.

The absolute value of a real number is either positive or zero. Moreover, 0 is the only real number whose absolute value is 0 .

So, |0| 0= .

Absolute value can be used to define the distance between two points on the real number line. For instance, the distance between

- 3 and 4 is | 3 4 | | 7 | 7

Linear Inequalities in One Variable Linear Inequalities in One Variable

- - solve an inequality solve an inequality

- - solution set solution set

For instance, 1 4x

the solution set is all real numbers that are less than 3.

Properties of lnequalities Properties of lnequalities

Solving a Linear Inequality in One Variable Solving a Linear Inequality in One Variable

Sometimes it is possible to write two inequalities as a doubleiiiiiiiiiii iii iiiiiiiii iii iii i iiii iii ii i iiiiiiiiiiii

i iii iii iii ii4 5 2 and 5 2 7x x

4 5 2 7x

Inequalities Involving Absolute Values Inequalities Involving Absolute Values

Algebraic Algebraic ExpressionsExpressions One characteristic of algebra is the use of letters to represent

numbers. The letters are iiiiiiiiiiiiiiiiii , and combinations of letteri

and numbers are algebraic expressions algebraic expressions . Here are a fewexamples of algebraic expressions.

2

45 , 2 3, , 7 7

2x x x

x

The termsterms of an algebraic expression are those parts that are separated by addition. For example,2 25 8 ( 5 ) 8x x x x

has three terms: iii - 5 x are the variable terms variable terms i ii iii 8

constant term constant term . The numerical factor of a variable term is thecoefficientcoefficient of the variable term. For instance, the coefficient of- 5 ii - 5 and the coefficient of 1is .

2x

2x

Basic Rules of Basic Rules ofAlgebraAlgebra There are four arithmetic operations with real numbers: addition,

, , iii division, denoted by the symbolsi i i ii i -i iii ii /. Of these, addition and multiplication are

the two primary operations. Subtraction and division are the inverse operations of addition and multiplication, respectively.

If a, b, and c are integers such that ab = c, then a and b are factors factors or divisors divisors of c.

A prime number prime number is an integer that has exactly two positive- - factors itself and 1 such as 2, 3,5,7, and 11. The numbers 4, 6, 8,9,

10and are compositecomposite because each can be written as the product of two or more prime numbers.

The number 1 is neither prime nor composite. The Fundamental Fundamental Theorem of Arithmetic Theorem of Arithmetic states that every positive integer greater than

1 can be written as the product of prime numbers in precisely one way 24(disregarding order). For instance, the prime factorization of is

24 2 2 2 3