This document provides an overview of different types of numbers and their relationships. It discusses:
1) Real numbers which include rational numbers like fractions and irrational numbers like square roots. Rational numbers have repeating decimals while irrational numbers do not.
2) Complex numbers which are numbers of the form a + bi, where a and b are real numbers. They were invented to allow solutions to equations like x^2 = -1.
3) How René Descartes linked algebra and geometry by establishing a correspondence between real numbers and points on a coordinate line, allowing geometric shapes to be described with algebraic equations.
This document provides an overview of solving polynomial equations. It defines polynomials and their key properties like degree, coefficients, and roots. It introduces several theorems for finding roots, including the Remainder Theorem, Factor Theorem, and the idea that a polynomial of degree n has n roots when counting multiplicities. Methods discussed include factoring, long division, and the quadratic formula. The document explains it is not possible to express solutions of polynomials of degree 5 or higher using radicals.
This document provides proofs of several basic limit theorems and properties from calculus. It includes:
1) Proofs of three parts of a limit theorem about combining constant multiples, sums, and products of functions with limits.
2) A proof of a basic continuity property regarding limits of composite functions.
3) Proofs of the chain rule of differentiation and that relative extrema of functions occur at critical points.
4) Proofs of two summation formulas involving sums of integers and sums of squared integers.
The proofs illustrate fundamental limit concepts and techniques like choosing appropriate δ values, using preceding results about limits, and algebraic manipulations of expressions involving limits.
This chapter introduces exponential and logarithmic functions. Exponential functions take the form f(x) = bx and model continuous exponential growth or decay. Their inverse functions are logarithmic functions of the form x = logb y. Key points covered include:
- Exponential growth occurs when b > 1 and decay when 0 < b < 1.
- Logarithmic functions have a domain of positive real numbers and range of all real numbers. Their graphs are reflections of exponential graphs in the line y = x.
- Important properties are established, such as logb(bn) = n, logb1 = 0, and the relationship between exponential and logarithmic forms of an equation.
This document provides an introduction to surds and indices. It discusses different types of numbers including rational and irrational numbers. It explains that surds like the square root of integers are either integers or irrational. The key properties of surds including simplifying expressions with surds are described. Index notation is also introduced as a shorthand for exponents. The basic rules for multiplying and dividing terms with indices are outlined.
The document is a presentation on polynomials. It defines a polynomial as an expression that can contain constants, variables, and exponents, but cannot contain division by a variable. It discusses the key characteristics of polynomials including their degree, standard form, zeros, factoring, and algebraic identities. Examples are provided to illustrate different types of polynomials like monomials, binomials, trinomials, and how to add, subtract, multiply and divide polynomials.
This document discusses types of polynomials including constant, linear, quadratic, cubic, and bi-quadratic polynomials. It defines a zero of a polynomial as a real number where the polynomial equals 0. Geometrically, linear polynomials intersect the x-axis at one point, quadratic polynomials form parabolas that can open up or down, and cubic polynomials can have up to three zeros where they intersect the x-axis. Polynomials are used in engineering, economics, physics, and industry to model and describe real-world phenomena like roller coaster curves, price variations over time, energy and voltage differences.
The document discusses polynomial division algorithms. It introduces long division and synthetic division as methods for dividing polynomials. Long division is analogous to dividing numbers, while synthetic division is simpler but only applies when dividing a polynomial by a monomial. The key points are:
- Long division allows dividing any polynomial P(x) by any polynomial D(x) to obtain a quotient Q(x) and remainder R(x) such that P(x) = Q(x)D(x) + R(x) and the degree of R(x) is less than the degree of D(x).
- Synthetic division is more efficient than long division when dividing a polynomial by a monomial of the form (
The document discusses solving numerical equations involving logarithmic and exponential functions in base 10 or base e. It provides examples of solving log and exponential equations by isolating the part containing the unknown, then rewriting the equation in the opposite form (log to exponential or exponential to log). The key steps outlined are: 1) isolate the exponential/log part containing the unknown, 2) rewrite the equation by "bringing down" exponents as logarithms or vice versa. Several examples are worked through demonstrating these steps.
This document provides an overview of solving polynomial equations. It defines polynomials and their key properties like degree, coefficients, and roots. It introduces several theorems for finding roots, including the Remainder Theorem, Factor Theorem, and the idea that a polynomial of degree n has n roots when counting multiplicities. Methods discussed include factoring, long division, and the quadratic formula. The document explains it is not possible to express solutions of polynomials of degree 5 or higher using radicals.
This document provides proofs of several basic limit theorems and properties from calculus. It includes:
1) Proofs of three parts of a limit theorem about combining constant multiples, sums, and products of functions with limits.
2) A proof of a basic continuity property regarding limits of composite functions.
3) Proofs of the chain rule of differentiation and that relative extrema of functions occur at critical points.
4) Proofs of two summation formulas involving sums of integers and sums of squared integers.
The proofs illustrate fundamental limit concepts and techniques like choosing appropriate δ values, using preceding results about limits, and algebraic manipulations of expressions involving limits.
This chapter introduces exponential and logarithmic functions. Exponential functions take the form f(x) = bx and model continuous exponential growth or decay. Their inverse functions are logarithmic functions of the form x = logb y. Key points covered include:
- Exponential growth occurs when b > 1 and decay when 0 < b < 1.
- Logarithmic functions have a domain of positive real numbers and range of all real numbers. Their graphs are reflections of exponential graphs in the line y = x.
- Important properties are established, such as logb(bn) = n, logb1 = 0, and the relationship between exponential and logarithmic forms of an equation.
This document provides an introduction to surds and indices. It discusses different types of numbers including rational and irrational numbers. It explains that surds like the square root of integers are either integers or irrational. The key properties of surds including simplifying expressions with surds are described. Index notation is also introduced as a shorthand for exponents. The basic rules for multiplying and dividing terms with indices are outlined.
The document is a presentation on polynomials. It defines a polynomial as an expression that can contain constants, variables, and exponents, but cannot contain division by a variable. It discusses the key characteristics of polynomials including their degree, standard form, zeros, factoring, and algebraic identities. Examples are provided to illustrate different types of polynomials like monomials, binomials, trinomials, and how to add, subtract, multiply and divide polynomials.
This document discusses types of polynomials including constant, linear, quadratic, cubic, and bi-quadratic polynomials. It defines a zero of a polynomial as a real number where the polynomial equals 0. Geometrically, linear polynomials intersect the x-axis at one point, quadratic polynomials form parabolas that can open up or down, and cubic polynomials can have up to three zeros where they intersect the x-axis. Polynomials are used in engineering, economics, physics, and industry to model and describe real-world phenomena like roller coaster curves, price variations over time, energy and voltage differences.
The document discusses polynomial division algorithms. It introduces long division and synthetic division as methods for dividing polynomials. Long division is analogous to dividing numbers, while synthetic division is simpler but only applies when dividing a polynomial by a monomial. The key points are:
- Long division allows dividing any polynomial P(x) by any polynomial D(x) to obtain a quotient Q(x) and remainder R(x) such that P(x) = Q(x)D(x) + R(x) and the degree of R(x) is less than the degree of D(x).
- Synthetic division is more efficient than long division when dividing a polynomial by a monomial of the form (
The document discusses solving numerical equations involving logarithmic and exponential functions in base 10 or base e. It provides examples of solving log and exponential equations by isolating the part containing the unknown, then rewriting the equation in the opposite form (log to exponential or exponential to log). The key steps outlined are: 1) isolate the exponential/log part containing the unknown, 2) rewrite the equation by "bringing down" exponents as logarithms or vice versa. Several examples are worked through demonstrating these steps.
english mathematics dictionary
kamus bahassa inggris untuk matematika
oleh neneng
Nurwaningsih
(06081281520066)
Nurwaningsih30@gmail.com
PROGRAM STUDI PENDIDIKAN MATEMATIKA
FAKULTAS KEGURUAN DAN ILMU PENDIDIKAN
UNIVERSITAS SRIWIJAYA
INDRALAYA
2017
semoga bermanfaat
The document discusses solving literal equations by isolating the variable of interest on one side of the equation. It provides examples of solving equations for various variables by adding, subtracting, multiplying, or dividing both sides of the equation by the same quantity. The goal is to isolate the variable being solved for so it stands alone on one side of the equal sign. Steps include clearing fractions, moving all other terms to the other side of the equation, and then dividing both sides by the coefficient of the variable being solved for.
The document discusses the Remainder Theorem, which states that the remainder obtained when dividing a polynomial f(x) by a linear polynomial x-c is equal to the value of f(x) when x is substituted with c. It provides examples to show how to use the Remainder Theorem to determine whether a given polynomial is divisible by x-c. The document also contains practice exercises for readers to apply the Remainder Theorem.
The document discusses the Remainder Theorem, which provides a way to factorize polynomials by dividing them by factors and obtaining a remainder. There are two methods for finding the remainder: long division/evaluation and synthetic division. Evaluation involves substituting the factor value into the polynomial, while synthetic division arranges the coefficients and repeatedly multiplies and adds down the line. The document provides examples of using both methods and notes that synthetic division allows determining the full quotient polynomial.
The document discusses the remainder theorem for polynomials. It defines the division algorithm for polynomials which divides a polynomial P(x) by (x-c) to get a unique quotient polynomial Q(x) and remainder R. The remainder theorem then states that the remainder R is equal to the value of P(c). The document proves the theorem and provides examples of using it to find the remainder when one polynomial is divided by another. It also provides exercises for students to find remainders using the theorem.
The document discusses evaluating the formula log[(2x+1)/(sin1/3(x)+1)] at x=0 and x=10 degrees using a scientific calculator. It explains that the answer is 0 at x=0 and approximately 1.13 at x=10 degrees. It then describes the keyboard of a typical scientific calculator, noting the number, operation, yx, sin, log and formula keys. The rest of the document provides examples and definitions of algebraic, trigonometric and exponential-log formulas.
The document discusses functions and their graphical representations. It defines key terms like domain, range, and one-to-one and many-to-one mappings. It then focuses on quadratic functions, showing that their graphs take characteristic U-shaped or inverted U-shaped forms. The document also examines inequalities involving quadratic expressions and how to determine the range of values satisfying such inequalities by analyzing the graph of the quadratic function.
The document discusses arithmetic sequences and provides examples to illustrate how to determine if a sequence is arithmetic, derive the specific formula for an arithmetic sequence from the general formula, and use the specific formula to calculate future terms. It defines an arithmetic sequence as one where the terms follow a linear formula of an = d*n + c. Examples show how to identify the common difference d between terms and plug into the general formula along with the first term a1 to derive the specific formula for different sequences.
The document discusses linear equations and how to graph them. It explains that linear equations relate the x-coordinate and y-coordinate of points in a straight line. To graph a linear equation, one finds ordered pairs that satisfy the equation by choosing values for x and solving for y, then plots the points. An example demonstrates graphing the linear equation y = 2x - 5 by making a table of x and y values and plotting the line.
The document discusses the Remainder Theorem and Factor Theorem. The Remainder Theorem states that if a polynomial p(x) is divided by a factor x - a, the remainder will be zero if x - a is a factor of p(x). The Factor Theorem is the reverse - if dividing a polynomial by x = a gives a zero remainder, then x - a is a factor of the polynomial. Both theorems relate the remainder of polynomial division to the factors of the polynomial.
55 inequalities and comparative statementsalg1testreview
The document discusses inequalities and the real number line. It explains that real numbers are associated with positions on a line, with positive numbers to the right of zero and negative numbers to the left. This line is called the real number line. Two numbers are related by an inequality if their corresponding positions on the real number line have one number being further to the right than the other. Inequalities can be used to represent intervals of numbers on the real number line.
The document discusses factorable polynomials and how to graph them. It defines a factorable polynomial as one that can be written as the product of linear factors using real numbers. For large values of x, the leading term of a polynomial dominates so the graph resembles that of the leading term. To graph a factorable polynomial, one first graphs the individual factors like x^n and then combines them, which gives smooth curves tending to the graphs of the leading terms for large x.
Factor Theorem and Remainder Theorem. Mathematics10 Project under Mrs. Marissa De Ocampo. Prepared by Danielle Diva, Ronalie Mejos, Rafael Vallejos and Mark Lenon Dacir of 10- Einstein. CNSTHS.
The document provides an overview of topics covered in Grade 9 math term 1, including:
1) Different types of numbers and their properties such as natural numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers.
2) Representation of real numbers on the number line and their decimal expansions.
3) Polynomials, including their classification based on terms and degrees. Properties such as zeros of polynomials and dividing one polynomial by another are discussed.
4) Factorization of polynomials using algebraic identities.
This document discusses factorials and the binomial theorem. It begins by defining factorials and providing examples of simplifying expressions with factorials. It then explains the binomial theorem, which gives a formula for expanding binomial expressions as binomial series. Specifically, it shows that the coefficients of terms in the binomial expansion can be determined using Pascal's triangle and factorials. It provides examples of using the binomial theorem to expand binomial expressions and find specific terms. In the examples, it demonstrates expanding binomials, finding coefficients, and determining terms with given exponents.
This document discusses algebraic expressions and polynomial expressions. It provides examples of algebraic expressions and defines them as formulas constructed with variables and numbers using basic arithmetic operations. Polynomials are defined as expressions of the form anxn + an-1xn-1 + ... + a1x + a0, where the ai's are numbers. The document gives examples of factoring polynomials and evaluating polynomial expressions at given values. It also discusses using factoring to find the roots of polynomial equations.
The document discusses methods for graphing quadratic equations. It explains that the graphs of quadratic equations are called parabolas, which are symmetric around a center line with a highest or lowest point called the vertex. It provides an example of graphing the equation y = x^2 - 4x - 12 by first finding the vertex, then making a table of symmetric x and y values around the vertex and plotting the points. The document also describes how to find the x-intercepts and y-intercepts of a parabola and provides an alternate two-step method for graphing a parabola.
The document outlines various formulae that students are expected to know, understand, or be able to use for the GCSE Mathematics exam. It presents formulae for the quadratic formula, circumference and area of a circle, Pythagoras' theorem, and trigonometry. It also lists formulae for perimeter, area, surface area, volume, compound interest, and probability that students should understand but won't be provided. Finally, it mentions kinematics formulae and calculators that may be provided or useful for questions involving various required formulae.
The document discusses the factor theorem and how to determine if a polynomial is a factor of another polynomial. It provides examples of using the factor theorem to show that (x + 1) is a factor of 2x^3 + 5x^2 - 3 and that (x - 2) is a factor of x^4 + x^3 - x^2 - x - 18. It also gives an example of finding a polynomial function given its zeros as -2, 1, -1. The document provides exercises for using the factor theorem to determine unknown values in polynomials.
This document discusses inverse functions, including exponential, logarithmic, and inverse trigonometric functions. It begins by defining an inverse function as two functions f and g where g(f(x)) = x and f(g(y)) = y. It then discusses how to find the inverse of a function by solving an equation like y = f(x) for x in terms of y. For a function to have an inverse, it must assign distinct outputs to distinct inputs. The document provides examples of finding inverses and discusses domains, ranges, and interpretations of inverse functions.
This chapter discusses various forms of asynchronous communication including electronic mailing lists, newsgroups, web-based forums, weblogs (blogs), and wikis. It defines each technology and explains how they work, how to participate in them, and basic rules for their use.
english mathematics dictionary
kamus bahassa inggris untuk matematika
oleh neneng
Nurwaningsih
(06081281520066)
Nurwaningsih30@gmail.com
PROGRAM STUDI PENDIDIKAN MATEMATIKA
FAKULTAS KEGURUAN DAN ILMU PENDIDIKAN
UNIVERSITAS SRIWIJAYA
INDRALAYA
2017
semoga bermanfaat
The document discusses solving literal equations by isolating the variable of interest on one side of the equation. It provides examples of solving equations for various variables by adding, subtracting, multiplying, or dividing both sides of the equation by the same quantity. The goal is to isolate the variable being solved for so it stands alone on one side of the equal sign. Steps include clearing fractions, moving all other terms to the other side of the equation, and then dividing both sides by the coefficient of the variable being solved for.
The document discusses the Remainder Theorem, which states that the remainder obtained when dividing a polynomial f(x) by a linear polynomial x-c is equal to the value of f(x) when x is substituted with c. It provides examples to show how to use the Remainder Theorem to determine whether a given polynomial is divisible by x-c. The document also contains practice exercises for readers to apply the Remainder Theorem.
The document discusses the Remainder Theorem, which provides a way to factorize polynomials by dividing them by factors and obtaining a remainder. There are two methods for finding the remainder: long division/evaluation and synthetic division. Evaluation involves substituting the factor value into the polynomial, while synthetic division arranges the coefficients and repeatedly multiplies and adds down the line. The document provides examples of using both methods and notes that synthetic division allows determining the full quotient polynomial.
The document discusses the remainder theorem for polynomials. It defines the division algorithm for polynomials which divides a polynomial P(x) by (x-c) to get a unique quotient polynomial Q(x) and remainder R. The remainder theorem then states that the remainder R is equal to the value of P(c). The document proves the theorem and provides examples of using it to find the remainder when one polynomial is divided by another. It also provides exercises for students to find remainders using the theorem.
The document discusses evaluating the formula log[(2x+1)/(sin1/3(x)+1)] at x=0 and x=10 degrees using a scientific calculator. It explains that the answer is 0 at x=0 and approximately 1.13 at x=10 degrees. It then describes the keyboard of a typical scientific calculator, noting the number, operation, yx, sin, log and formula keys. The rest of the document provides examples and definitions of algebraic, trigonometric and exponential-log formulas.
The document discusses functions and their graphical representations. It defines key terms like domain, range, and one-to-one and many-to-one mappings. It then focuses on quadratic functions, showing that their graphs take characteristic U-shaped or inverted U-shaped forms. The document also examines inequalities involving quadratic expressions and how to determine the range of values satisfying such inequalities by analyzing the graph of the quadratic function.
The document discusses arithmetic sequences and provides examples to illustrate how to determine if a sequence is arithmetic, derive the specific formula for an arithmetic sequence from the general formula, and use the specific formula to calculate future terms. It defines an arithmetic sequence as one where the terms follow a linear formula of an = d*n + c. Examples show how to identify the common difference d between terms and plug into the general formula along with the first term a1 to derive the specific formula for different sequences.
The document discusses linear equations and how to graph them. It explains that linear equations relate the x-coordinate and y-coordinate of points in a straight line. To graph a linear equation, one finds ordered pairs that satisfy the equation by choosing values for x and solving for y, then plots the points. An example demonstrates graphing the linear equation y = 2x - 5 by making a table of x and y values and plotting the line.
The document discusses the Remainder Theorem and Factor Theorem. The Remainder Theorem states that if a polynomial p(x) is divided by a factor x - a, the remainder will be zero if x - a is a factor of p(x). The Factor Theorem is the reverse - if dividing a polynomial by x = a gives a zero remainder, then x - a is a factor of the polynomial. Both theorems relate the remainder of polynomial division to the factors of the polynomial.
55 inequalities and comparative statementsalg1testreview
The document discusses inequalities and the real number line. It explains that real numbers are associated with positions on a line, with positive numbers to the right of zero and negative numbers to the left. This line is called the real number line. Two numbers are related by an inequality if their corresponding positions on the real number line have one number being further to the right than the other. Inequalities can be used to represent intervals of numbers on the real number line.
The document discusses factorable polynomials and how to graph them. It defines a factorable polynomial as one that can be written as the product of linear factors using real numbers. For large values of x, the leading term of a polynomial dominates so the graph resembles that of the leading term. To graph a factorable polynomial, one first graphs the individual factors like x^n and then combines them, which gives smooth curves tending to the graphs of the leading terms for large x.
Factor Theorem and Remainder Theorem. Mathematics10 Project under Mrs. Marissa De Ocampo. Prepared by Danielle Diva, Ronalie Mejos, Rafael Vallejos and Mark Lenon Dacir of 10- Einstein. CNSTHS.
The document provides an overview of topics covered in Grade 9 math term 1, including:
1) Different types of numbers and their properties such as natural numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers.
2) Representation of real numbers on the number line and their decimal expansions.
3) Polynomials, including their classification based on terms and degrees. Properties such as zeros of polynomials and dividing one polynomial by another are discussed.
4) Factorization of polynomials using algebraic identities.
This document discusses factorials and the binomial theorem. It begins by defining factorials and providing examples of simplifying expressions with factorials. It then explains the binomial theorem, which gives a formula for expanding binomial expressions as binomial series. Specifically, it shows that the coefficients of terms in the binomial expansion can be determined using Pascal's triangle and factorials. It provides examples of using the binomial theorem to expand binomial expressions and find specific terms. In the examples, it demonstrates expanding binomials, finding coefficients, and determining terms with given exponents.
This document discusses algebraic expressions and polynomial expressions. It provides examples of algebraic expressions and defines them as formulas constructed with variables and numbers using basic arithmetic operations. Polynomials are defined as expressions of the form anxn + an-1xn-1 + ... + a1x + a0, where the ai's are numbers. The document gives examples of factoring polynomials and evaluating polynomial expressions at given values. It also discusses using factoring to find the roots of polynomial equations.
The document discusses methods for graphing quadratic equations. It explains that the graphs of quadratic equations are called parabolas, which are symmetric around a center line with a highest or lowest point called the vertex. It provides an example of graphing the equation y = x^2 - 4x - 12 by first finding the vertex, then making a table of symmetric x and y values around the vertex and plotting the points. The document also describes how to find the x-intercepts and y-intercepts of a parabola and provides an alternate two-step method for graphing a parabola.
The document outlines various formulae that students are expected to know, understand, or be able to use for the GCSE Mathematics exam. It presents formulae for the quadratic formula, circumference and area of a circle, Pythagoras' theorem, and trigonometry. It also lists formulae for perimeter, area, surface area, volume, compound interest, and probability that students should understand but won't be provided. Finally, it mentions kinematics formulae and calculators that may be provided or useful for questions involving various required formulae.
The document discusses the factor theorem and how to determine if a polynomial is a factor of another polynomial. It provides examples of using the factor theorem to show that (x + 1) is a factor of 2x^3 + 5x^2 - 3 and that (x - 2) is a factor of x^4 + x^3 - x^2 - x - 18. It also gives an example of finding a polynomial function given its zeros as -2, 1, -1. The document provides exercises for using the factor theorem to determine unknown values in polynomials.
This document discusses inverse functions, including exponential, logarithmic, and inverse trigonometric functions. It begins by defining an inverse function as two functions f and g where g(f(x)) = x and f(g(y)) = y. It then discusses how to find the inverse of a function by solving an equation like y = f(x) for x in terms of y. For a function to have an inverse, it must assign distinct outputs to distinct inputs. The document provides examples of finding inverses and discusses domains, ranges, and interpretations of inverse functions.
This chapter discusses various forms of asynchronous communication including electronic mailing lists, newsgroups, web-based forums, weblogs (blogs), and wikis. It defines each technology and explains how they work, how to participate in them, and basic rules for their use.
1) Speaker recognition uses characteristics extracted from voices to validate a user's claimed identity. It recognizes who is speaking, whereas speech recognition recognizes what is being said.
2) Speaker recognition technologies have evolved alongside speech recognition and synthesis since the 1960s as researchers have studied vocal tract physiology and developed systems to analyze speech acoustics and match samples to templates.
3) Speaker recognition systems extract features from speech like duration, pitch, and intensity to generate likelihood ratios comparing a sample to the claimed identity versus other speakers. Updates help models cope with voice changes over time.
This document provides an overview of trigonometric functions and identities. It defines angles and their measurement in degrees and radians. It discusses trigonometric functions for right triangles, extending the definitions to angles in rectangular coordinate systems. Examples are provided to illustrate evaluating trigonometric functions of various angles. Key relationships between arc length, angle, radius, and area are also summarized.
This document provides an introduction to PHP, including:
- What scripting languages and PHP are, and how PHP works as a server-side scripting language
- The history and origins of PHP
- How to set up a PHP development environment using XAMPP
- PHP programming fundamentals like syntax, operators, and control structures
- How to handle forms and files in PHP
- How to connect to and manipulate databases like MySQL from PHP
- Several tasks as examples of working with forms, files, and databases in PHP
Discussion of the science, collection and availability of lidar, specifically topobathymetric lidar. Use of NOAA/USGS Interagency Elevation Inventory leveraged
The document provides an introduction to PHP including:
- PHP is an open source scripting language especially suited for web development and can be embedded into HTML.
- PHP code is executed on the server, generating HTML which is then sent to the client.
- PHP supports variables, operators, conditional statements, arrays, loops, functions, and forms. Key functions like $_GET and $_POST are used to collect form data submitted via GET and POST methods respectively.
The document discusses new features in PHP 5.3 including new syntax, functions, extensions, and changes from previous versions. Some key changes include new array, date, and PHP core functions; the addition of namespaces, lambda functions, and closures; and removal of support for certain Windows versions and extensions. Backwards compatibility issues are addressed for magic methods, reserved words, and deprecations.
The document provides an introduction to PHP, covering topics such as what PHP is, PHP files, why use PHP, PHP requests, the PHP development environment, PHP syntax, comments, mixing PHP with HTML, variables, operators, control structures like if/else statements, loops, functions, arrays, cookies, and sessions. PHP is introduced as a server-side scripting language used to build dynamic web applications. Key aspects explained include PHP files containing HTML, text, and scripts which are executed on the server and returned to the browser as HTML.
Groundwater data management techniques such as permitting wells, monitoring wells and DFC, and developing groundwater databases and GIS systems were discussed. Specific examples included the Northern Trinity GCD's online well registration database and mapping tool, and an analysis of groundwater banking in Texas using GIS. Historical and DFC availability of the Ogallala Aquifer was also examined for the High Plains UWCD. Contact information was provided for follow up.
Guidelines for Modelling Groundwater Surface Water Interaction in eWater SourceeWater
One of the key challenges in modelling GW-SW interactions is the significant time-scale
differences between surface water and groundwater processes. Because groundwater
movement can be orders of magnitude slower than surface water movement, the
responses of groundwater systems to hydrological and management drivers such as
climate variability, land use change, and groundwater extraction can be very damped and
lagged. Hence, a key requirement in modelling GW-SW interactions in river system
models is to account for these time lags.
The modelling of GW-SW interactions in river system models is still very much in its
infancy, not just in Australia, but also throughout the world. As such, there is no consensus
on implementation of this functionality in river system models, and hence the little
discussion in the literature so far on what constitutes Best Practice Modelling in this
domain.
Offshore pipelines transport oil and gas from offshore oil platforms and drilling rigs to onshore facilities. They are made of steel and can extend for hundreds of miles underwater, buried below the seabed. Maintaining and repairing offshore pipelines presents unique challenges due to their subsea environment and remote locations.
This document provides an introduction to PHP by summarizing its history and key features. PHP was created in 1994 by Rasmus Lerdorf as a set of Common Gateway Interface scripts for tracking visits to his online resume. It has since evolved into a full-featured programming language used widely by major companies like Google, Facebook, and Bank of America. The document outlines PHP's core syntax like variables, constants, includes, and flow control structures. It also discusses databases, MVC patterns, classes, and tools that employers seek like contributions to open source projects.
The document provides an overview of how to connect to and use the Internet. It discusses the history and development of the Internet from its origins in ARPANET in the 1960s to the creation of the World Wide Web in the early 1990s. Key events included the development of packet switching, TCP/IP, email, web browsers, and commercialization of the Internet. The document describes how individuals and businesses connect to the Internet using options like dial-up, DSL, cable, or wireless. Common activities on the Internet are discussed like browsing websites, emailing, downloading files, and e-commerce.
This chapter discusses accessing information resources on the web, including the difference between the surface web and deep web. It covers various search tools like search engines, subject directories, and meta search engines. Boolean logic and search syntax are explained to refine queries. Advanced search features and evaluating results are also summarized. Methods to define search questions and formulate strategies are provided to efficiently find relevant information online.
1) The document introduces computers and their components, including input/output devices, the system unit, storage, and communications devices.
2) It discusses the advantages and disadvantages of using computers and defines key terms like digital literacy and the information processing cycle.
3) Networks and the internet are introduced, including how they connect computers and allow sharing of resources. The functions of servers and how the world wide web works are also summarized.
The document discusses different computer components related to power and electricity, including form factors, power supplies, cases, and how electricity is measured. It covers topics like ATX, MicroATX, and BTX form factors; desktop and tower cases; voltages, amps, ohms, and watts; AC and DC power; surge protection; UPS systems; and how computers meet Energy Star standards through power management features. The document provides information to help understand and troubleshoot electrical issues in personal computers.
The document provides an overview of topics in number theory including:
- Number systems such as natural numbers, integers, and real numbers
- Properties of real numbers like closure, commutativity, associativity, identity, and inverse properties
- Rational and irrational numbers
- Order of operations
- Absolute value
- Intervals on the number line
- Finite and repeating decimals
- Converting between fractions and decimals
4 ESO Academics - Unit 01 - Real Numbers and PercentagesGogely The Great
Rational numbers can be expressed as fractions with integer numerators and denominators, while irrational numbers cannot. Some important irrational numbers include π, the square root of 2, the golden ratio φ, and e. Together, the sets of rational and irrational numbers form the set of real numbers, which can be represented on the real number line. Approximating numbers involves rounding them to a specified place value or number of significant figures.
The document discusses operations on real numbers, including:
1) Addition and subtraction of real numbers follows rules based on sign, where numbers with the same sign are added and different signs are subtracted.
2) Multiplication and division of real numbers results in a positive number if the signs are the same, and negative if different.
3) Properties like commutativity, associativity, identity, inverse and distribution apply to real numbers as they do to other types of numbers.
The document defines different types of numbers including rational numbers, integers, fractions, algebraic numbers, and real numbers. It also discusses sets and set operations including union, intersection, difference, symmetric difference, and complement. Finally, it covers absolute value, absolute value inequalities, and mathematical inequalities.
This document discusses number systems and different types of numbers. It begins by introducing rational numbers as numbers that can be written as fractions with integer numerators and denominators. Irrational numbers are then defined as numbers that cannot be expressed as rational numbers. Some examples of irrational numbers like square roots and pi are provided. Finally, it is explained that the set of all rational and irrational numbers together make up the real numbers, and every real number can be represented by a unique point on the number line.
1) The document discusses the real number system, including identifying integers, rational numbers, and irrational numbers. Real numbers have important properties like closure under addition and multiplication.
2) Intervals of real numbers can be described using inequality notation like x ≥ 1 or -4 < x < 1, or interval notation like [1, +∞) or (-4, 1).
3) Absolute value of a real number a, written |a|, represents the distance of the number from the origin on the number line and is always non-negative.
Real numbers include rational numbers like integers and fractions as well as irrational numbers like square roots and pi. Real numbers can be represented on a continuous number line and include both countable and uncountable infinite numbers. Real numbers have the properties of a field where they can be added, multiplied, and ordered on the number line in a way compatible with these operations. Rational numbers are numbers that can be represented as fractions of integers, and they include integers, whole numbers, and natural numbers. Irrational numbers cannot be represented as fractions.
This chapter reviews real numbers including:
[1] Classifying numbers as natural numbers, integers, rational numbers, irrational numbers, and real numbers. Rational numbers can be written as fractions while irrational numbers cannot.
[2] Approximating irrational numbers like π as decimals to a given number of decimal places by rounding or truncating.
[3] How calculators handle decimals by either truncating or rounding values based on their display capabilities. A scientific or graphing calculator is recommended for this course.
This document discusses different number systems including real numbers, rational numbers, and irrational numbers. Real numbers include all rational and irrational numbers and can be represented on a continuous number line. Rational numbers are numbers that can be expressed as fractions of integers, and their decimal representations will either terminate or repeat. Irrational numbers cannot be expressed as fractions and have non-repeating, non-terminating decimal representations. The document further explains rational numbers are divided into integers, whole numbers, and natural numbers, and provides definitions and properties for each.
This document provides a summary of key concepts from Grade 9 Math Term I, including:
- Different types of numbers (natural, whole, integers, rational, irrational, real) and their properties
- Polynomials, including definitions of terms, degrees, zeros, and factoring polynomials
- The Cartesian coordinate system for identifying points on a plane using perpendicular x and y axes, with the origin at their intersection
The document defines and describes various types of number systems. It discusses natural numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers. It also describes their properties and relationships. Different types of polynomials are defined based on their degree, number of terms, zeros, and factors. Methods for factorizing polynomials including taking common factors, grouping, and splitting the middle term are explained. Algebraic identities are also introduced.
The document discusses different number systems used in mathematics. It begins by explaining that a number system is defined by its base, or the number of unique symbols used to represent numbers. The most common system is decimal, which uses base-10. Other discussed systems include binary, octal, hexadecimal, and those used historically by cultures like the Babylonians. Rational and irrational numbers are also defined. Rational numbers can be written as fractions of integers, while irrational numbers cannot.
The document discusses real numbers and their subsets. It defines natural numbers, whole numbers, integers, rational numbers, and irrational numbers. It explains that rational numbers can be expressed as terminating or repeating decimals, while irrational numbers are non-terminating and non-repeating. Examples are provided of different types of numbers. Classification of numbers using Venn diagrams is demonstrated. Rounding and truncating decimals is also covered.
The document discusses the history and development of the Arabic numeral system. It explains that the numerals originated from the Phoenicians but were popularized by Arabs. It then provides a theory about how the shapes of the numerals may have been derived from representing different numbers of angles, showing examples. The document also briefly outlines some key developments in the history of algebra.
God gave us natural numbers, while other types of numbers were created by humans. The German mathematician Kronecker expressed that natural numbers play a significant role in human thought. The document then defines various types of numbers - natural numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers. It provides properties and examples for each number type. Rational numbers can be expressed as terminating or repeating decimals, while irrational numbers cannot be expressed as fractions. Together, rational and irrational numbers form the set of real numbers.
The document discusses Catalan numbers and provides several problems that are equivalent in that they all generate the same sequence of numbers called the Catalan numbers. It introduces problems involving balanced parentheses, mountain ranges, diagonal-avoiding paths, polygon triangulations, handshakes around a table, binary trees, and more. It then derives a recursive definition for the Catalan numbers by analyzing the balanced parentheses problem and showing the number of configurations for n pairs is equal to the sum of configurations of i pairs times configurations of n-i-1 pairs, for i from 0 to n-1. Finally, it shows this same recursive relationship holds for the polygon triangulations and non-crossing handshake problems.
This document discusses various mathematical symbols and constants, including their origins and meanings. It explains that symbols like π, e, i, Σ, Π originated from Greek words and letters related to their mathematical definitions. Symbols like N, Z, Q, R, C are used to represent sets of numbers like natural numbers, integers, rational numbers, real numbers, and complex numbers. The document also notes that mathematical notation can help relieve the brain and allow it to focus on more advanced problems.
This document defines and provides examples of different types of real numbers including rational and irrational numbers. It discusses how rational numbers can be represented as fractions or decimals, and how to convert between fraction and decimal representations. Irrational numbers are defined as having non-terminating, non-repeating decimal representations. The key types of real numbers - natural numbers, integers, rational numbers, and irrational numbers - are related in a Venn diagram with their union being the set of all real numbers.
sets of numbers and interval notation, operation on real numbers, simplifying expression, linear equation in one variable, aplication of linear equation in one variable, linear equation and aplication to geometry, linear inequaloties in one variable, properties of integers exponents and scientific notation
This document introduces the concept of a limit, which is the foundation of calculus. It provides an intuitive approach to understanding limits by examining how the instantaneous velocity of an object can be approximated by calculating average velocities over increasingly small time intervals. The key idea is that as the time intervals approach zero, the average velocities approach a limiting value that represents the instantaneous velocity. More generally, a limit describes how a function's values approach a specific number L as the input values approach a particular value a. The process of determining a limit involves first conjecturing the limit based on sampling function values, and then verifying it with a more rigorous analysis.
This document introduces the concept of functions. It defines a function as a rule that associates a unique output value with each input value. Functions are used to describe relationships between variables in mathematics and science. The document discusses using tables, graphs, and equations to represent functional relationships and extract information from graphical representations of data. It provides examples of how various quantities can be expressed as functions of other variables.
The document discusses analyzing functions using calculus concepts like derivatives. It introduces analyzing functions to determine if they are increasing, decreasing, or constant on intervals based on the sign of the derivative. The sign of the derivative indicates whether the graph of the function has positive, negative, or zero slope at points, relating to whether the function is increasing, decreasing, or constant. It also introduces the concept of concavity, where the derivative indicates whether the curvature of the graph is upward (concave up) or downward (concave down) based on whether tangent lines have increasing or decreasing slopes. Examples are provided to demonstrate these concepts.
This document discusses limits and continuity in calculus. It begins by explaining how limits were used to define instantaneous rates of change in velocity and acceleration, which were fundamental to the development of calculus. The chapter then aims to develop the concept of the limit intuitively before providing precise mathematical definitions. Limits are introduced as the value a function approaches as the input gets arbitrarily close to a given value, without actually reaching it. Several examples are provided to illustrate how to determine limits through sampling inputs and making conjectures.
This document introduces the concept of functions in calculus. It defines a function as a rule that associates a unique output with each input. Functions can be represented and analyzed through tables, graphs, and equations. The document uses several examples, like qualifying speeds in auto racing and cigarette consumption over time, to illustrate how graphs convey information about relationships between variables and can be interpreted to extract insights. It also discusses how equations define functions by determining a unique output value for each allowable input.
This document contains an exercise set with 46 problems involving real numbers, intervals, and inequalities. The problems cover topics such as determining whether numbers are rational or irrational, solving equations, graphing inequalities on number lines, factoring polynomials, and solving compound inequalities.
The document discusses using the discriminant of a quadratic equation to determine the type of conic section represented by the graph of the equation. It defines the discriminant as B^2 - 4AC and explains that:
(a) If the discriminant is negative, the graph is an ellipse, circle, point, or has no graph.
(b) If the discriminant is positive, the graph is a hyperbola or intersecting lines.
(c) If the discriminant is 0, the graph is a parabola, line, parallel lines, or has no graph.
This document discusses distance, circles, and quadratic equations in three parts:
1) It derives the formula for finding the distance between two points in a plane as the square root of the sum of the squares of the differences of their x- and y-coordinates.
2) It derives the midpoint formula for finding the midpoint between two points as the average of their x-coordinates and the average of their y-coordinates.
3) It discusses the standard equation of a circle, gives methods for finding the center and radius from different forms of the circle equation, and notes degenerate cases where the equation does not represent a circle.
This document describes rectangular coordinate systems and how to plot points and graphs in them. It contains the following key points:
- A rectangular coordinate system uses two perpendicular axes (typically x and y) that intersect at the origin to locate points in a plane.
- The coordinates of a point P are ordered pairs (x,y) where x is the point's distance from the y-axis and y is its distance from the x-axis.
- An equation in x and y defines a graph - the set of all points whose coordinates satisfy the equation. Graphs can be approximated by plotting sample points but this has limitations.
- Intercepts are points where a graph crosses an axis,
This document defines and provides examples of absolute value, including:
- Absolute value strips away the minus sign if a number is negative and leaves it unchanged if nonnegative.
- The absolute value of a product is the product of the absolute values.
- Absolute value has a geometric interpretation as distance on a number line.
- Important inequalities involving absolute value, like |x-a|<k, have solution sets that consist of values within k units of a.
- The triangle inequality states the absolute value of a sum is less than or equal to the sum of the absolute values.
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PAGE PROOFS
A31
a p p e n d i x d
REAL NUMBERS, INTERVALS,
AND INEQUALITIES
REAL NUMBERS
Figure D.1 describes the various categories of numbers that we will encounter in this text.
The simplest numbers are the natural numbers
1, 2, 3, 4, 5, . . .
These are a subset of the integers
. . . , −4, −3, −2, −1, 0, 1, 2, 3, 4, . . .
and these in turn are a subset of the rational numbers, which are the numbers formed by
taking ratios of integers (avoiding division by 0). Some examples are
2
3
, 7
5
, 23 = 23
1
, 0.19 = 19
100
, −5
2
= −5
2
= 5
−2
Complex numbers:
a + bi, where i = √–1
Real numbers:
Rational and irrational numbers
Integers:
..., –4, –3, –2, –1, 0, 1, 2, 3, 4, ...
Natural numbers:
1, 2, 3, 4, 5, ...
Rational
numbers:
2
3
7
5
23
1
5
2
–, , , 0.19,
Figure D.1
The early Greeks believed that every measurable quantity had to be a rational number.
However, this idea was overturned in the fifth century B.C. by Hippasus of Metapontum
who demonstrated the existence of irrational numbers, that is, numbers that cannot be
expressed as the ratio of two integers. Using geometric methods, he showed that the length
of the hypotenuse of the triangle in Figure D.2 could not be expressed as a ratio of integers,
thereby proving that
√
2 is an irrational number. Some other examples of irrational numbers
are √
3,
√
5, 1 +
√
2,
3√
7, π, cos 19◦
Therationalandirrationalnumberstogethercomprisewhatiscalledtherealnumbersystem,
and both the rational and irrational numbers are called real numbers.
Hippasus of Metapon-
tum (circa 500 B.C.)
A Greek Pythagorean
philosopher. Accord-
ing to legend, Hippasus
made his discovery
at sea and was thrown overboard
by fanatic Pythagoreans because
his result contradicted their doc-
trine. The discovery of Hippasus
is one of the most fundamental in
the entire history of science.
1
1
√2
Figure D.2
COMPLEX NUMBERS
Because the square of a real number cannot be negative, the equation
x2
= −1
has no solutions in the real number system. In the eighteenth century mathematicians
remedied this problem by inventing a new number, which they denoted by
i =
√
−1
and which they defined to have the property i2
= −1. This, in turn, led to the development
of the complex numbers, which are numbers of the form
a + bi
where a and b are real numbers. Some examples are
2 + 3i 3 − 4i 6i 2
3
|a = 2, b = 3| |a = 3, b = −4| |a = 0, b = 6| |a = 2
3 , b = 0|
Observe that every real number a is also a complex number because it can be written as
a = a + 0i
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PAGE PROOFS
A32 Appendix D: Real Numbers, Intervals, and Inequalities
Thus, the real numbers are a subset of the complex numbers. Although we will be concerned
primarily with real numbers in this text, complex numbers will arise in the course of solving
equations. For example, the solutions of the quadratic equation
ax2
+ bx + c = 0
which are given by the quadratic formula
x =
−b ±
√
b2 − 4ac
2a
are not real if the quantity b2
− 4ac is negative.
DIVISION BY ZERO
Division by zero is not allowed in numerical computations because it leads to mathematical
inconsistencies. For example, if 1/0 were assigned some numerical value, say p, then it
would follow that 0 · p = 1, which is incorrect.
DECIMAL REPRESENTATION OF REAL NUMBERS
Rational and irrational numbers can be distinguished by their decimal representations. Ra-
tional numbers have decimals that are repeating, by which we mean that at some point in
the decimal some fixed block of numbers begins to repeat indefinitely. For example,
4
3
= 1.333 . . . ,
3 repeats
3
11
= .272727 . . . ,
27 repeats
1
2
= .50000 . . . ,
0 repeats
5
7
= .714285714285714285 . . .
714285 repeats
Decimals in which zero repeats from some point on are called terminating decimals. For
brevity, it is usual to omit the repetitive zeros in terminating decimals and for other repeating
decimals to write the repeating digits only once but with a bar over them to indicate the repe-
tition. For example,
1
2
= .5, 12
4
= 3, 8
25
= .32, 4
3
= 1.3, 3
11
= .27, 5
7
= .714285
Irrational numbers have nonrepeating decimals, so we can be certain that the decimals
√
2 = 1.414213562373095 . . . and π = 3.141592653589793 . . .
do not repeat from some point on. Moreover, if we stop the decimal expansion of an
irrational number at some point, we get only an approximation to the number, never an
exact value. For example, even if we compute π to 1000 decimal places, as in Figure D.3,
we still have only an approximation.
3.141592653589793238462643383279502884197169
39937510582097494459230781640628620899862803
48253421170679821480865132823066470938446095
50582231725359408128481117450284102701938521
10555964462294895493038196442881097566593344
61284756482337867831652712019091456485669234
60348610454326648213393607260249141273724587
00660631558817488152092096282925409171536436
78925903600113305305488204665213841469519415
11609433057270365759591953092186117381932611
79310511854807446237996274956735188575272489
12279381830119491298336733624406566430860213
94946395224737190702179860943702770539217176
29317675238467481846766940513200056812714526
35608277857713427577896091736371787214684409
01224953430146549585371050792279689258923542
01995611212902196086403441815981362977477130
99605187072113499999983729780499510597317328
16096318595024459455346908302642522308253344
68503526193118817101000313783875288658753320
83814206171776691473035982534904287554687311
59562863882353787593751957781857780532171226
8066130019278766111959092164201989
Figure D.3
Beginning mathematics students are sometimes taught to approximate π by 22
7
. Keep in mind,
however, that this is only an approximation, since
22
7
= 3.142857
is a rational number whose decimal representation begins to differ from π in the third decimal
place.
COORDINATE LINES
In 1637 René Descartes published a philosophical work called Discourse on the Method
of Rightly Conducting the Reason. In the back of that book was an appendix that the Brit-
ish philosopher John Stuart Mill described as “the greatest single step ever made in the
progress of the exact sciences.” In that appendix René Descartes linked together algebra
and geometry, thereby creating a new subject called analytic geometry; it gave a way of
describing algebraic formulas by geometric curves and, conversely, geometric curves by
algebraic formulas.
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PAGE PROOFS
Appendix D: Real Numbers, Intervals, and Inequalities A33
The key step in analytic geometry is to establish a correspondence between real numbers
and points on a line. To do this, choose any point on the line as a reference point, and call
it the origin; and then arbitrarily choose one of the two directions along the line to be the
positive direction, and let the other be the negative direction. It is usual to mark the positive
direction with an arrowhead, as in Figure D.4, and to take the positive direction to the right
when the line is horizontal. Next, choose a convenient unit of measure, and represent each
positive number r by the point that is r units from the origin in the positive direction, each
negative number −r by the point that is r units from the origin in the negative direction
from the origin, and 0 by the origin itself (Figure D.5). The number associated with a point
P is called the coordinate of P, and the line is called a coordinate line, a real number line,
or a real line.
Origin +–
Figure D.4
Figure D.5 -4 -3 -2 -1 0 1 2 3 4
-1.75 c2
1
-
√2
INEQUALITY NOTATION
The real numbers can be ordered by size as follows: If b − a is positive, then we write
either a < b (read “a is less than b”) or b > a (read “b is greater than a”). We write a ≤ b
to mean a < b or a = b, and we write a < b < c to mean that a < b and b < c. As one
traverses a coordinate line in the positive direction, the real numbers increase in size, so
on a horizontal coordinate line the inequality a < b implies that a is to the left of b, and
the inequalities a < b < c imply that a is to the left of c, and b lies between a and c. The
meanings of such symbols as
a ≤ b < c, a ≤ b ≤ c, and a < b < c < d
should be clear. For example, you should be able to confirm that all of the following are
true statements:
3 < 8, −7 < 1.5, −12 ≤ −π, 5 ≤ 5, 0 ≤ 2 ≤ 4,
8 ≥ 3, 1.5 > −7, −π > −12, 5 ≥ 5, 3 > 0 > −1 > −3
REVIEW OF SETS
In the following discussion we will be concerned with certain sets of real numbers, so it will
be helpful to review the basic ideas about sets. Recall that a set is a collection of objects,
called elements or members of the set. In this text we will be concerned primarily with sets
whose members are numbers or points that lie on a line, a plane, or in three-dimensional
space. We will denote sets by capital letters and elements by lowercase letters. To indicate
that a is a member of the set A we will write a ∈ A (read “a belongs to A”), and to indicate
that a is not a member of the set A we will write a /∈ A (read “a does not belong to A”).
For example, if A is the set of positive integers, then 5 ∈ A, but −5 /∈ A. Sometimes sets
René Descartes (1596–1650) Descartes, a French aristo-
crat, was the son of a government official. He graduated
from the University of Poitiers with a law degree at age 20.
After a brief probe into the pleasures of Paris he became
a military engineer, first for the Dutch Prince of Nassau
and then for the German Duke of Bavaria. It was dur-
ing his service as a soldier that Descartes began to pursue mathemat-
ics seriously and develop his analytic geometry. After the wars, he
returned to Paris where he stalked the city as an eccentric, wearing
a sword in his belt and a plumed hat. He lived in leisure, seldom
arose before 11 D.M., and dabbled in the study of human physiology,
philosophy, glaciers, meteors, and rainbows. He eventually moved
to Holland, where he published his Discourse on the Method, and
finally to Sweden where he died while serving as tutor to Queen
Christina. Descartes is regarded as a genius of the first magnitude.
In addition to major contributions in mathematics and philosophy,
he is considered, along with William Harvey, to be a founder of
modern physiology.
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PAGE PROOFS
A34 Appendix D: Real Numbers, Intervals, and Inequalities
arise that have no members (e.g., the set of odd integers that are divisible by 2). A set with
no members is called an empty set or a null set and is denoted by the symbol л.
Some sets can be described by listing their members between braces. The order in which
the members are listed does not matter, so, for example, the set A of positive integers that
are less than 6 can be expressed as
A = {1, 2, 3, 4, 5} or A = {2, 3, 1, 5, 4}
We can also write A in set-builder notation as
A = {x : x is an integer and 0 < x < 6}
which is read “A is the set of all x such that x is an integer and 0 < x < 6.” In general,
to express a set S in set-builder notation we write S = {x : } in which the line is
replaced by a property that identifies exactly those elements in the set S.
If every member of a set A is also a member of a set B, then we say that A is a subset
of B and write A ⊆ B. For example, if A is the set of positive integers and B is the set
of all integers, then A ⊆ B. If two sets A and B have the same members (i.e., A ⊆ B and
B ⊆ A), then we say that A and B are equal and write A = B.
INTERVALS
In calculus we will be concerned with sets of real numbers, called intervals, that correspond
to line segments on a coordinate line. For example, if a < b, then the open interval from a
to b, denoted by (a, b), is the line segment extending from a to b, excluding the endpoints;
and the closed interval from a to b, denoted by [a, b], is the line segment extending from
a to b, including the endpoints (Figure D.6). These sets can be expressed in set-builder
notation as
(a, b) = {x : a < x < b} The open interval from a to b
[a, b] = {x : a ≤ x ≤ b} The closed interval from a to b
a b
a b
The open interval (a, b)
The closed interval [a, b]
Figure D.6
Observe that in this notation and in the corresponding Figure D.6, parentheses and open dots mark
endpoints that are excluded from the interval, whereas brackets and closed dots mark endpoints
that are included in the interval. Observe also that in set-builder notation for the intervals, it is
understood that x is a real number, even though it is not stated explicitly.
As shown in Table 1, an interval can include one endpoint and not the other; such
intervals are called half-open (or sometimes half-closed). Moreover, the table also shows
that it is possible for an interval to extend indefinitely in one or both directions. To indicate
that an interval extends indefinitely in the positive direction we write +ϱ (read “positive
infinity”) in place of a right endpoint, and to indicate that an interval extends indefinitely
in the negative direction we write −ϱ (read “negative infinity”) in place of a left endpoint.
Intervals that extend between two real numbers are called finite intervals, whereas intervals
that extend indefinitely in one or both directions are called infinite intervals.
By convention, infinite intervals of the form [a, +ϱ) or (−ϱ, b] are considered to be closed because
they contain their endpoint, and intervals of the form (a, +ϱ) and (−ϱ, b) are considered to be
open because they do not include their endpoint. The interval (−ϱ, +ϱ), which is the set of all real
numbers, has no endpoints and can be regarded as both open and closed. This set is often denoted
by the special symbol R. To distinguish verbally between the open interval (0, +ϱ) = {x : x > 0}
and the closed interval [0, +ϱ) = {x : x ≥ 0}, we will call x positive if x > 0 and nonnegative if x ≥ 0.
Thus, a positive number must be nonnegative, but a nonnegative number need not be positive,
since it might possibly be 0.
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PAGE PROOFS
Appendix D: Real Numbers, Intervals, and Inequalities A35
Table 1
a b
a b
a b
a b
b
b
a
a
interval
notation
set
notation
geometric
picture
(a, b)
[a, b]
[a, b)
(a, b]
(–∞, b]
(–∞, b)
[a, +∞)
(a, +∞)
(–∞, +∞)
Finite; open
Finite; closed
Finite; half-open
Finite; half-open
Infinite; closed
Infinite; open
Infinite; closed
Infinite; open
Infinite; open and closed
{x : a < x < b}
{x : a ≤ x ≤ b}
{x : a ≤ x < b}
{x : a < x ≤ b}
{x : x ≤ b}
{x : x < b}
{x : x ≥ a}
{x : x > a}
classification
UNIONS AND INTERSECTIONS OF INTERVALS
If A and B are sets, then the union of A and B (denoted by A ∪ B) is the set whose members
belong to A or B (or both), and the intersection of A and B (denoted by A ∩ B) is the set
whose members belong to both A and B. For example,
{x : 0 < x < 5} ∪ {x : 1 < x < 7} = {x : 0 < x < 7}
{x : x < 1} ∩ {x : x ≥ 0} = {x : 0 ≤ x < 1}
{x : x < 0} ∩ {x : x > 0} = л
or in interval notation, (0, 5) ∪ (1, 7) = (0, 7)
(−ϱ, 1) ∩ [0, +ϱ) = [0, 1)
(−ϱ, 0) ∩ (0, +ϱ) = л
ALGEBRAIC PROPERTIES OF INEQUALITIES
The following algebraic properties of inequalities will be used frequently in this text. We
omit the proofs.
D.1 theorem (Properties of Inequalities). Let a, b, c, and d be real numbers.
(a) If a < b and b < c, then a < c.
(b) If a < b, then a + c < b + c and a − c < b − c.
(c) If a < b, then ac < bc when c is positive and ac > bc when c is negative.
(d) If a < b and c < d, then a + c < b + d.
(e) If a and b are both positive or both negative and a < b, then 1/a > 1/b.
If we call the direction of an inequality its sense, then these properties can be paraphrased
as follows:
(b) The sense of an inequality is unchanged if the same number is added to or subtracted
from both sides.
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PAGE PROOFS
A36 Appendix D: Real Numbers, Intervals, and Inequalities
(c) The sense of an inequality is unchanged if both sides are multiplied by the same positive
number, but the sense is reversed if both sides are multiplied by the same negative
number.
(d) Inequalities with the same sense can be added.
(e) If both sides of an inequality have the same sign, then the sense of the inequality is
reversed by taking the reciprocal of each side.
These properties remain true if the
symbols < and > are replaced by ≤ and
≥ in Theorem D.1.
Example 1
starting
inequality
resulting
inequality
Add 7 to both sides.
Subtract 8 from both sides.
Multiply both sides by 3.
Multiply both sides by –3.
Multiply both sides by 4.
Multiply both sides by –4.
Take reciprocals of both sides.
Take reciprocals of both sides.
Add corresponding sides.
operation
–2 < 6
–2 < 6
–2 < 6
–2 < 6
3 < 7
3 < 7
3 < 7
–8 < –6
4 < 5, –7 < 8
5 < 13
–10 < –2
–6 < 18
12 < 28
–12 > –28
1
3
1
7
>
1
8
1
6
>– –
6 > –18
–3 < 13
SOLVING INEQUALITIES
A solution of an inequality in an unknown x is a value for x that makes the inequality a true
statement. For example, x = 1 is a solution of the inequality x < 5, but x = 7 is not. The
set of all solutions of an inequality is called its solution set. It can be shown that if one does
not multiply both sides of an inequality by zero or an expression involving an unknown,
then the operations in Theorem D.1 will not change the solution set of the inequality. The
process of finding the solution set of an inequality is called solving the inequality.
Example 2 Solve 3 + 7x ≤ 2x − 9.
Solution. We will use the operations of Theorem D.1 to isolate x on one side of the
inequality.
3 + 7x ≤ 2x − 9 Given.
7x ≤ 2x − 12 We subtracted 3 from both sides.
5x ≤ −12 We subtracted 2x from both sides.
x ≤ −12
5
We multiplied both sides by 1
5 .
Because we have not multiplied by any expressions involving the unknown x, the last
inequality has the same solution set as the first. Thus, the solution set is the interval
−ϱ, −12
5
shown in Figure D.7.
12
5
–
Figure D.7
Example 3 Solve 7 ≤ 2 − 5x < 9.
Solution. The given inequality is actually a combination of the two inequalities
7 ≤ 2 − 5x and 2 − 5x < 9
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PAGE PROOFS
Appendix D: Real Numbers, Intervals, and Inequalities A37
We could solve the two inequalities separately, then determine the values of x that satisfy
both by taking the intersection of the two solution sets. However, it is possible to work
with the combined inequalities in this problem:
7 ≤ 2 − 5x < 9 Given.
5 ≤ −5x < 7 We subtracted 2 from each member.
−1 ≥ x > −
7
5
We multiplied by − 1
5 and reversed
the sense of the inequalities.
−
7
5
< x ≤ −1 For clarity, we rewrote the inequalities
with the smaller number on the left.
Thus, the solution set is the interval −7
5
, −1 shown in Figure D.8.
7
5
– –1
Figure D.8
Example 4 Solve x2
− 3x > 10.
Solution. By subtracting 10 from both sides, the inequality can be rewritten as
x2
− 3x − 10 > 0
Factoring the left side yields
(x + 2)(x − 5) > 0
The values of x for which x + 2 = 0 or x − 5 = 0 are x = −2 and x = 5. These points
divide the coordinate line into three open intervals,
(−ϱ, −2), (−2, 5), (5, +ϱ)
on each of which the product (x + 2)(x − 5) has constant sign. To determine those signs
we will choose an arbitrary point in each interval at which we will determine the sign; these
are called test points. As shown in Figure D.9, we will use −3, 0, and 6 as our test points.
The results can be organized as follows:
sign of
(x + 2)(x – 5)
at the test point
–3
0
6
test pointinterval
(–∞, –2)
(–2, 5)
(5, +∞)
(–)(–) = +
(+)(–) = –
(+)(+) = +
The pattern of signs in the intervals is shown on the number line in the middle of Figure D.9.
We deduce that the solution set is (−ϱ, −2) ∪ (5, +ϱ), which is shown at the bottom of
Figure D.9.
Figure D.9 –2 5
–2
–2
5
5
–3 0 6
0 0– – – – – – + + + ++ + +
Test points
Sign of (x + 2)(x – 5)
Solution set for
(x + 2)(x – 5) > 0
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PAGE PROOFS
A38 Appendix D: Real Numbers, Intervals, and Inequalities
Example 5 Solve
2x − 5
x − 2
< 1.
Solution. We could start by multiplying both sides by x − 2 to eliminate the fraction.
However, this would require us to consider the cases x − 2 > 0 and x − 2 < 0 separately
because the sense of the inequality would be reversed in the second case, but not the first.
The following approach is simpler:2x − 5
x − 2
< 1 Given.
2x − 5
x − 2
− 1 < 0 We subtracted 1 from both sides
to obtain a 0 on the right.
(2x − 5) − (x − 2)
x − 2
< 0 We combined terms.
x − 3
x − 2
< 0 We simplified.
The quantity x − 3 is zero if x = 3, and the quantity x − 2 is zero if x = 2. These points
divide the coordinate line into three open intervals,
(−ϱ, 2), (2, 3), (3, +ϱ)
on each of which the quotient (x − 3)/(x − 2) has constant sign. Using 0, 2.5, and 4 as
test points (Figure D.10), we obtain the following results:
sign of
(x – 3)/(x – 2)
at the test point
0
2.5
4
test pointinterval
(–∞, 2)
(2, 3)
(3, +∞)
(–)/(–) = +
(–)/(+) = –
(+)/(+) = +
The signs of the quotient are shown in the middle of Figure D.10. From the figure we see
that the solution set consists of all real values of x such that 2 < x < 3. This is the interval
(2, 3) shown at the bottom of Figure D.10.
Figure D.10
2 3
2 3
2 3
0 2.5 4
0– + + + ++ + +
Solution set for
+++ + + –
Sign of
x – 2
x – 3
x – 2
x – 3
Test points
< 0
EXERCISE SET D
1. Among the terms integer, rational, and irrational, which
ones apply to the given number?
(a) −3
4
(b) 0 (c) 24
8
(d) 0.25
(e) −
√
16 (f ) 21/2
(g) 0.020202 . . . (h) 7.000 . . .
2. Which of the terms integer, rational, and irrational apply
to the given number?
(a) 0.31311311131111 . . . (b) 0.729999 . . .
(c) 0.376237623762 . . . (d) 174
5
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PAGE PROOFS
Appendix D: Real Numbers, Intervals, and Inequalities A39
3. The repeating decimal 0.137137137 . . . can be expressed as
a ratio of integers by writing
x = 0.137137137 . . .
1000x = 137.137137137 . . .
and subtracting to obtain 999x = 137 or x = 137
999
. Use this
idea, where needed, to express the following decimals as
ratios of integers.
(a) 0.123123123 . . . (b) 12.7777 . . .
(c) 38.07818181 . . . (d) 0.4296000 . . .
4. Show that the repeating decimal 0.99999 . . . represents the
number 1. Since 1.000 . . . is also a decimal representation
of 1, this problem shows that a real number can have two
different decimal representations. [Hint: Use the technique
of Exercise 3.]
5. The Rhind Papyrus, which is a fragment of Egyptian math-
ematical writing from about 1650 B.C., is one of the oldest
known examples of written mathematics. It is stated in the
papyrus that the area A of a circle is related to its diameter
D by
A = 8
9
D
2
(a) What approximation to π were the Egyptians using?
(b) Use a calculating utility to determine if this approxi-
mation is better or worse than the approximation 22
7
.
6. The following are all famous approximations to π:
333
106
Adrian Athoniszoon, c. 1583
355
113
Tsu Chung-Chi and others
63
25
17 + 15
√
5
7 + 15
√
5
Ramanujan
22
7
Archimedes
223
71
Archimedes
(a) Use a calculating utility to order these approximations
according to size.
(b) Which of these approximations is closest to but larger
than π?
(c) Which of these approximations is closest to but smaller
than π?
(d) Which of these approximations is most accurate?
7. In each line of the accompanying table, check the blocks,
if any, that describe a valid relationship between the real
numbers a and b. The first line is already completed as an
illustration.
a
1
6
–3
5
–4
0.25
6
1
5
–3
–4
b a < b a ≤ b a > b a ≥ b a = b
4
1
4
3
3
1
––
Table Ex-7
8. In each line of the accompanying table, check the blocks,
if any, that describe a valid relationship between the real
numbers a, b, and c.
a
–1
2
–5
0.75
0
4
–5
1.25
2
–3
–5
1.25
b c a < b < c a ≤ b ≤ c a < b ≤ c a ≤ b < c
2
1
2
1
4
3
Table Ex-8
9. Which of the following are always correct if a ≤ b?
(a) a − 3 ≤ b − 3 (b) −a ≤ −b (c) 3 − a ≤ 3 − b
(d) 6a ≤ 6b (e) a2
≤ ab (f ) a3
≤ a2
b
10. Which of the following are always correct if a ≤ b and
c ≤ d?
(a) a + 2c ≤ b + 2d (b) a − 2c ≤ b − 2d
(c) a − 2c ≥ b − 2d
11. For what values of a are the following inequalities valid?
(a) a ≤ a (b) a < a
12. If a ≤ b and b ≤ a, what can you say about a and b?
13. (a) If a < b is true, does it follow that a ≤ b must also be
true?
(b) If a ≤ b is true, does it follow that a < b must also be
true?
14. In each part, list the elements in the set.
(a) {x : x2
− 5x = 0}
(b) {x : x is an integer satisfying −2 < x < 3}
15. In each part, express the set in the notation {x : }.
(a) {1, 3, 5, 7, 9, . . .}
(b) the set of even integers
(c) the set of irrational numbers
(d) {7, 8, 9, 10}
16. Let A = {1, 2, 3}. Which of the following sets are equal
to A?
(a) {0, 1, 2, 3} (b) {3, 2, 1}
(c) {x : (x − 3)(x2
− 3x + 2) = 0}
10. October 22, 2004 12:43 k34-appd Sheet number 10 Page number 40 cyan magenta yellow black
PAGE PROOFS
A40 Appendix D: Real Numbers, Intervals, and Inequalities
17. In the accompanying figure, let
S = the set of points inside the square
T = the set of points inside the triangle
C = the set of points inside the circle
and let a, b, and c be the points shown. Answer the follow-
ing as true or false.
(a) T ⊆ C (b) T ⊆ S
(c) a /∈ T (d) a /∈ S
(e) b ∈ T and b ∈ C (f ) a ∈ C or a ∈ T
(g) c ∈ T and c /∈ C
c
b
a
Figure Ex-17
18. List all subsets of
(a) {a1, a2, a3} (b) л.
19. In each part, sketch on a coordinate line all values of x that
satisfy the stated condition.
(a) x ≤ 4 (b) x ≥ −3 (c) −1 ≤ x ≤ 7
(d) x2
= 9 (e) x2
≤ 9 (f ) x2
≥ 9
20. In parts (a)–(d), sketch on a coordinate line all values of x,
if any, that satisfy the stated conditions.
(a) x > 4 and x ≤ 8
(b) x ≤ 2 or x ≥ 5
(c) x > −2 and x ≥ 3
(d) x ≤ 5 and x > 7
21. Express in interval notation.
(a) {x : x2
≤ 4} (b) {x : x2
> 4}
22. In each part, sketch the set on a coordinate line.
(a) [−3, 2] ∪ [1, 4] (b) [4, 6] ∪ [8, 11]
(c) (−4, 0) ∪ (−5, 1) (d) [2, 4) ∪ (4, 7)
(e) (−2, 4) ∩ (0, 5] (f ) [1, 2.3) ∪ (1.4,
√
2)
(g) (−ϱ, −1) ∪ (−3, +ϱ) (h) (−ϱ, 5) ∩ [0, +ϱ)
23–44 Solve the inequality and sketch the solution on a co-
ordinate line.
23. 3x − 2 < 8 24. 1
5
x + 6 ≥ 14
25. 4 + 5x ≤ 3x − 7 26. 2x − 1 > 11x + 9
27. 3 ≤ 4 − 2x < 7 28. −2 ≥ 3 − 8x ≥ −11
29.
x
x − 3
< 4 30.
x
8 − x
≥ −2
31.
3x + 1
x − 2
< 1 32.
1
2
x − 3
4 + x
> 1
33.
4
2 − x
≤ 1 34.
3
x − 5
≤ 2
35. x2
> 9 36. x2
≤ 5
37. (x − 4)(x + 2) > 0 38. (x − 3)(x + 4) < 0
39. x2
− 9x + 20 ≤ 0 40. 2 − 3x + x2
≥ 0
41.
2
x
<
3
x − 4
42.
1
x + 1
≥
3
x − 2
43. x3
− x2
− x − 2 > 0 44. x3
− 3x + 2 ≤ 0
45–46 Find all values of x for which the given expression
yields a real number.
45.
√
x2 + x − 6 46.
x + 2
x − 1
47. Fahrenheit and Celsius temperatures are related by the for-
mula C = 5
9
(F − 32). If the temperature in degrees Celsius
ranges over the interval 25 ≤ C ≤ 40 on a certain day, what
is the temperature range in degrees Fahrenheit that day?
48. Every integer is either even or odd. The even integers are
those that are divisible by 2, so n is even if and only if
n = 2k for some integer k. Each odd integer is one unit
larger than an even integer, so n is odd if and only if
n = 2k + 1 for some integer k. Show:
(a) If n is even, then so is n2
(b) If n is odd, then so is n2
.
49. Prove the following results about sums of rational and
irrational numbers:
(a) rational + rational = rational
(b) rational + irrational = irrational.
50. Prove the following results about products of rational and
irrational numbers:
(a) rational · rational = rational
(b) rational · irrational = irrational (provided the rational
factor is nonzero).
51. Show that the sum or product of two irrational numbers can
be rational or irrational.
52. Classify the following as rational or irrational and justify
your conclusion.
(a) 3 + π (b) 3
4
√
2 (c)
√
8
√
2 (d)
√
π
(See Exercises 49 and 50.)
53. Prove: The average of two rational numbers is a rational
number, but the average of two irrational numbers can be
rational or irrational.
54. Can a rational number satisfy 10x
= 3?
55. Solve: 8x3
− 4x2
− 2x + 1 < 0.
56. Solve: 12x3
− 20x2
≥ −11x + 2.
57. Prove: If a, b, c, and d are positive numbers such that a < b
and c < d, then ac < bd. (This result gives conditions un-
der which inequalities can be “multiplied together.”)
58. Is the number represented by the decimal
0.101001000100001000001 . . .
rational or irrational? Explain your reasoning.