chapter 2 real number system for student new

8/9/2019 Chapter 2 Real Number System for Student New

1/39Real Number System By A. AOF Page1

Chapter 2

The Real Number System

By

A.Suthin Khankhua(AOF)



2.1 Real Number



Exercise 2.1



2.2 Properties of real numbers
http://www.google.co.th/url?sa=t&source=web&cd=1&ved=0CBUQFjAA&url=http%3A%2F%2Fstaff.pausd.org%2F%7Epjorgens%2Fpapers%2Fproperties.pdf&ei=9gEqTJWQA5S0rAfoof1z&usg=AFQjCNEDAXmP-LbLscmLNaUi5yACJwo6wQ&sig2=u3w9_3sstWa4JJ9UncE5awhttp://www.google.co.th/url?sa=t&source=web&cd=1&ved=0CBUQFjAA&url=http%3A%2F%2Fstaff.pausd.org%2F%7Epjorgens%2Fpapers%2Fproperties.pdf&ei=9gEqTJWQA5S0rAfoof1z&usg=AFQjCNEDAXmP-LbLscmLNaUi5yACJwo6wQ&sig2=u3w9_3sstWa4JJ9UncE5aw



2.3 Polynomials in One Variable

The study of systems of polynomial equations in many variables requires a good understanding

of what can be said about one polynomial equation in one variable. The purpose of this chapter is

to provide some basic tools for this problem. We shall consider the problem of how to compute

and how to represent the zeros of a general polynomial of degree d in one variable x:



We call the largest exponent of x appearing in a non-zero term of a polynomial the degree of that

polynomial.

Examples

1. 3x + 1 = 0 has degree 1, since the largest power of x that occurs is x = x1. Degree 1 equations

are called linear equations.

2. x2

- x - 1 = 0 has degree 2, since the largest power of x that occurs is x2. Degree 2 equations

are also called quadratic equations, or just quadratics.

3. x3 = 2x2 + 1 is a degree 3 polynomial (or cubic) in disguise. It can be rewritten as x

3 - 2x2 - 1 =

0, which is in the standard form for a degree 3 equation.

4. x4 - x = 0 has degree 4. It is called a quartic.

Solution of Linear Equations

By definition, a linear equation can be written in the formax + b = 0 a and b are fixed numbers with a 0

Solving this is a nice mental exercise: subtract b from both sides and then divide by a, getting x =

-b/a. Don't bother memorizing this formula, just go ahead and solve linear equations as they

arise.

Q1 2x + 3 = 0 has the unique solution x =

Q2 3x - 3 = 1 has the unique solution x =

Q3 ax + b = c (a 0) has the unique solution x =

Solution of Quadratic Equations

By definition, a quadratic equation has the form

ax2

+ bx + c = 0 a, b, and c are fixed numbers and a 0.

The solutions of this equation are also called the roots of ax2

+ bx + c. We''re assuming that you

saw quadratic equations somewhere in high school but may be a little hazy as to the details of

their solution. There are two ways of solving these equations -- one works sometimes, and theother works every time.



Solving Quadratic Equations by Factoring (works sometimes)

If we can factor a quadratic equation ax2 + bx + c = 0, we can solve the equation by setting each

factor equal to 0.

Examples

1.x2

+ 7x + 10 = 0

(x + 5)(x + 2) = 0 Factor the left-hand side

x + 5 = 0 or x + 2 = 0 If a product is zero, one or both factors is zero

Solutions: x = -5 and x =--2

2.2x2

- 5x - 12 = 0

(2x + 3)(x - 4) = 0 Factor the left-hand side

Solutions: x = -3/2 and x = 4

Test for FactoringThe quadratic ax

2+ bx + c, with a, b, and c being integers (whole numbers), factors into an

expression of the form (rx + s)(tx + u) with r, s, t and u being integers precisely when thequantity b2- 4ac is a perfect square (that is, it is the square of an integer). If this happens, we say

that the quadratic factors over the integers.

Examples

x2

+ x + 1 has a = 1, b = 1, and c = 1, so b2

- 4ac = -3, which is not a perfect square.Therefore, this quadratic does not factor over the integers.

2x2- 5x -12 has a = 2, b = -5 and c = -12, so b 2 - 4ac = 121. Since 121 = 112, thisquadratic does factor over the integers (we factored it above).
http://people.hofstra.edu/Stefan_waner/realworld/tut_alg_review/framesA_3B.htmlhttp://people.hofstra.edu/Stefan_waner/realworld/tut_alg_review/framesA_3B.html



Solving Quadratic Equations with the Quadratic Formula (works every time)

The solutions of the general quadratic equation ax2

+ bx + c = 0 (a 0) are given by

a2

ac4bbx

2

We call the quantity = b2

- 4ac the discriminant of the quadratic ( is the Greek letter delta)and we have the following general principle:

If is positive, there are two distinct real solutions.

If is zero, there is only one real solution: x = -b/2a (why?). If is negative, there are no real solutions.



Solution of Cubic Equations

By definition, a cubic equation can be written in the form

ax3

+ bx2

+ cx + d = 0 a, b, c, and d are fixed numbers and a 0

Now we get into something of a bind. While there is a perfectly respectable formula for thesolutions, it is very complicated and involves the use of complex numbers rather heavily 1. So we

discuss instead a much simpler method that sometimes works nicely. Here is the method in a

nutshell.

2.4 Properties of Inequality of Real Numbers

In mathematics, an inequality is a statement about the relative size or order of two objects, or

about whether they are the same or not

The notation a b means that a is greater thanb.

The notation a b means that a is not equal tob, but does not say that one is greater thanthe other or even that they can be compared in size.

In each statement above, a is not equal to b. These relations are known as strict inequalities.

The notation a < b may also be read as "a is strictly less than b".

In contrast to strict inequalities, there are two types of inequality statements that are not strict:

The notation a b means that a is less than or equal tob (or, equivalently, not greaterthanb)

The notation a b means that a is greater than or equal tob (or, equivalently, not lessthan

b)

An additional use of the notation is to show that one quantity is much greater than another,

normally by several orders of magnitude.

The notation ab means that a is much less thanb. The notation ab means that a is much greater thanb.

Properties

Inequalities are governed by the following properties. Note that, for the transitivity, reversal,

addition and subtraction, and multiplication and division properties, the property also holds ifstrict inequality signs (< and >) are replaced with their corresponding non-strict inequality sign

( and ).
http://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Orders_of_magnitudehttp://en.wikipedia.org/wiki/Propertieshttp://en.wikipedia.org/wiki/Propertieshttp://en.wikipedia.org/wiki/Orders_of_magnitudehttp://en.wikipedia.org/wiki/Mathematics

Transitivity

The transitivity of inequalities states:

For any real numbers,a, b, c:o Ifa > b and b > c; then a > co Ifa b and b = c; then a > co Ifa b, then a + c > b + c and ac > bc

i.e., the real numbers are an ordered group.

The properties that deal with multiplication and division state:

For any real numbers, a, b, co Ifc is positive and a < b, then ac < bco Ifc is negative and a < b, then ac > bc

More generally this applies for an ordered field, see below.

The properties for the additive inverse state:

For any real numbers a and bo Ifa < bthen a> bo Ifa > bthen a 1/bo

Ifa > b then 1/a < 1/b

If either a or b is negative then

o Ifa b then 1/a > 1/b
http://en.wikipedia.org/wiki/Transitive_relationhttp://en.wikipedia.org/wiki/Real_numberhttp://en.wikipedia.org/wiki/Additionhttp://en.wikipedia.org/wiki/Subtractionhttp://en.wikipedia.org/wiki/Real_numberhttp://en.wikipedia.org/wiki/Ordered_grouphttp://en.wikipedia.org/wiki/Multiplicationhttp://en.wikipedia.org/wiki/Division_%28mathematics%29http://en.wikipedia.org/wiki/Positive_numberhttp://en.wikipedia.org/wiki/Negative_numberhttp://en.wikipedia.org/wiki/Ordered_fieldhttp://en.wikipedia.org/wiki/Additive_inversehttp://en.wikipedia.org/wiki/Multiplicative_inversehttp://en.wikipedia.org/wiki/Positive_numberhttp://en.wikipedia.org/wiki/Negative_and_non-negative_numbershttp://en.wikipedia.org/wiki/Negative_and_non-negative_numbershttp://en.wikipedia.org/wiki/Positive_numberhttp://en.wikipedia.org/wiki/Multiplicative_inversehttp://en.wikipedia.org/wiki/Additive_inversehttp://en.wikipedia.org/wiki/Ordered_fieldhttp://en.wikipedia.org/wiki/Negative_numberhttp://en.wikipedia.org/wiki/Positive_numberhttp://en.wikipedia.org/wiki/Division_%28mathematics%29http://en.wikipedia.org/wiki/Multiplicationhttp://en.wikipedia.org/wiki/Ordered_grouphttp://en.wikipedia.org/wiki/Real_numberhttp://en.wikipedia.org/wiki/Subtractionhttp://en.wikipedia.org/wiki/Additionhttp://en.wikipedia.org/wiki/Real_numberhttp://en.wikipedia.org/wiki/Transitive_relation



2.5 Interval

In mathematics, a (real) interval is a set ofreal numbers with the property that any number that

lies between two numbers in the set is also included in the set. For example, the set of all

numbers x satisfying is an interval which contains 0 and 1, as well as all numbers

between them. Other examples of intervals are the set of all real numbers , the set of allnegative real numbers, and the empty set.

In fact, intervals are meaningful in any (totally or partially) ordered set, not just in the reals; so

one can have intervals ofrational numbers, integers, computer-representable floating point

numbers, or subsets of a set (ordered by inclusion), for example.

Real intervals play an important role in the theory ofintegration, because they are the simplest

sets whose "size" or "measure" or "length" is easy to define. The concept of measure can then beextended to more complicated sets of real numbers, leading to the Borel measure and eventually

to the Lebesgue measure.

Intervals are central to interval arithmetic, a general numerical computing technique that

automatically provides guaranteed enclosures for arbitrary formulas, even in the presence of

uncertainties, mathematical approximations, and arithmetic roundoff.

The interval of numbers between a and b, including a and b, is often denoted [a,b]. The two

numbers are called the endpoints of the interval.

Open interval

An open interval does not include its endpoints, and is indicated with parentheses. For

example (0,1) means greater than 0 and less than 1. bxa|xb,a

Closed interval

Conversely, a closed interval includes its endpoints, and is denoted with square brackets. Forexample [0,1] means greater than or equal to 0 and less than or equal to 1.

bxa|xb,a

Halfopen intervalAn interval is said to be left-bounded or right-bounded if there is some real number that is,

respectively, smaller than or larger than all its elements. An interval is said to be bounded if it isboth left- and right-bounded; and is said to be unbounded otherwise. Intervals that are bounded

at only one end are said to be half-bounded.

bxa|xb,a bxa|xb,a
http://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Set_%28mathematics%29http://en.wikipedia.org/wiki/Real_numberhttp://en.wikipedia.org/wiki/Empty_sethttp://en.wikipedia.org/wiki/Partially_ordered_set#Intervalhttp://en.wikipedia.org/wiki/Rational_numbershttp://en.wikipedia.org/wiki/Integershttp://en.wikipedia.org/wiki/Computerhttp://en.wikipedia.org/wiki/Floating_point_numbershttp://en.wikipedia.org/wiki/Floating_point_numbershttp://en.wikipedia.org/wiki/Integralhttp://en.wikipedia.org/wiki/Borel_measurehttp://en.wikipedia.org/wiki/Lebesgue_measurehttp://en.wikipedia.org/wiki/Interval_arithmetichttp://en.wikipedia.org/wiki/Numerical_methodhttp://en.wikipedia.org/wiki/Rounding_errorhttp://en.wikipedia.org/wiki/Rounding_errorhttp://en.wikipedia.org/wiki/Numerical_methodhttp://en.wikipedia.org/wiki/Interval_arithmetichttp://en.wikipedia.org/wiki/Lebesgue_measurehttp://en.wikipedia.org/wiki/Borel_measurehttp://en.wikipedia.org/wiki/Integralhttp://en.wikipedia.org/wiki/Floating_point_numbershttp://en.wikipedia.org/wiki/Floating_point_numbershttp://en.wikipedia.org/wiki/Computerhttp://en.wikipedia.org/wiki/Integershttp://en.wikipedia.org/wiki/Rational_numbershttp://en.wikipedia.org/wiki/Partially_ordered_set#Intervalhttp://en.wikipedia.org/wiki/Empty_sethttp://en.wikipedia.org/wiki/Real_numberhttp://en.wikipedia.org/wiki/Set_%28mathematics%29http://en.wikipedia.org/wiki/Mathematics



Infinite interval

The empty set is bounded, and the set of all reals is the only interval that is unbounded at both

ends. Bounded intervals are also commonly known as finite intervals.

1. ax|x,a 2. ax|x,a 3. ax|xa, 4. ax|xa,

Example

1. Determine set ofreal numbers forSolution

Answer
http://en.wikipedia.org/wiki/Set_%28mathematics%29http://en.wikipedia.org/wiki/Real_numberhttp://en.wikipedia.org/wiki/Real_numberhttp://en.wikipedia.org/wiki/Set_%28mathematics%29



2. Determine set ofreal numbersforSolution

Answer

Exercise 2.5

1. Write sets ofreal numbers and draw its interval.

2. Write sets of interval.
http://en.wikipedia.org/wiki/Set_%28mathematics%29http://en.wikipedia.org/wiki/Real_numberhttp://en.wikipedia.org/wiki/Real_numberhttp://en.wikipedia.org/wiki/Real_numberhttp://en.wikipedia.org/wiki/Real_numberhttp://en.wikipedia.org/wiki/Real_numberhttp://en.wikipedia.org/wiki/Set_%28mathematics%29



3. Determine sets of answer of these inequality of real numbers

4. Determine sets of answer of these inequality of real numbers



2.6 Absolute Value


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chapter 2 real number system for student new

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