Chapter 2 Real Number System for Student New
Transcript of Chapter 2 Real Number System for Student New
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Chapter 2
The Real Number System
By
A.Suthin Khankhua(AOF)
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2.1 Real Number
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Exercise 2.1
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2.2 Properties of real numbers
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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:
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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.
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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 -
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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.
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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.
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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 less thanb. 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 -
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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 and b = c; then a > co Ifa < b and b = c; then a < c
The properties that deal with addition and subtraction state:
For any real numbers,a, b, c:o Ifa < b, then a + c < b + c and ac < bco
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< b
The properties for the multiplicative inverse state:
For any real numbers a and b that are both positive or both negativeo Ifa < b then 1/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/bo Ifa > b then 1/a > 1/b
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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 -
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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
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2. Determine set ofreal numbersforSolution
Answer
Exercise 2.5
1. Write sets ofreal numbers and draw its interval.
2. Write sets of interval.
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3. Determine sets of answer of these inequality of real numbers
4. Determine sets of answer of these inequality of real numbers
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2.6 Absolute Value
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