Description: In theory, one's algebraic ability should be an indicator of one's calculus ability. That is, a student's algebra skills should be a direct indicator of his or.
Table of contents
In the future we may expand on this by providing introductions to other calculators or computer algebra systems. However, in order to keep this text as self contained as possible we always recall all results that are necessary to follow the core of the course even if we assume that the student has familiarity with the formula or topic at hand. After a first introduction to the abstract notion of a function, we study polynomials, rational functions, exponential functions, logarithmic functions, and trigonometric functions with the function viewpoint.
Throughout, we will always place particular importance to the corresponding graph of the discussed function which will be analyzed with the help of the TI calculator as mentioned above. These are in fact the topics of the first four of the five parts of this precalculus course. In the fifth and last part of the book, we deviate from the above theme and collect more algebraically oriented topics that will be needed in calculus or other advanced mathematics courses or even other science courses. This part includes a discussion of the algebra of complex numbers in particular complex numbers in polar form , the 2-dimensional real vector space R 2 sequences and series with focus on the arithmetic and geometric series which are again examples of functions, though this is not emphasized , and finally the generalized binomial theorem.
Precalculus is intended for college-level precalculus students. Since precalculus courses vary from one institution to the next, we have attempted to meet the needs of as broad an audience as possible, including all of the content that might be covered in any particular course. The result is a comprehensive book that covers more ground than an instructor could likely cover in a typical one- or two-semester course; but instructors should find, almost without fail, that the topics they wish to include in their syllabus are covered in the text.
Many chapters of Openstax College Precalculus are suitable for other freshman and sophomore math courses such as College Algebra and Trigonometry; however, instructors of those courses might need to supplement or adjust the material. Openstax will also be releasing College Algebra and Algebra and Trigonometry titles tailored to the particular scope, sequence, and pedagogy of those courses. Prior to , the performance of a student in precalculus at the University of Washington was not a predictor of success in calculus.
For this reason, the mathematics department set out to create a new course with a specific set of goals in mind:. A casual glance through the Table of Contents of most of the major publishers' College Algebra books reveals nearly isomorphic content in both order and depth. We then introduce a class of functions and discuss the equations, inequalities with a heavy emphasis on sign diagrams and applications which involve functions in that class. This book is for people who have never programmed before. As a result, the order of presentation is unusual.
The book starts with scalar values and works up to vectors and matrices very gradually. This approach is good for beginning programmers, because it is hard to understand composite objects until you understand basic programming semantics. But there are problems:. The goal of this text, as its name implies, is to allow the reader to become proficient in the analysis and design of circuits utilizing modern linear ICs. The text is intended for use in a second year Operational Amplifiers course at the Associate level, or for a junior level course at the Baccalaureate level.
In order to make effective use of this text, students should have already taken a course in basic discrete transistor circuits, and have a solid background in algebra and trigonometry, along with exposure to phasors. Calculus is used in certain sections of the text, but for the most part, its use is kept to a minimum. For students without a calculus background, these sections may be skipped without a loss of continuity. The sole exception to this being Chapter Ten, Integrators and Differentiators, which hinges upon knowledge of calculus.
The Open Logic Text is an open-source, collaborative textbook of formal meta-logic and formal methods, starting at an intermediate level i. Though aimed at a non-mathematical audience in particular, students of philosophy and computer science , it is rigorous. A one semester first course on differential equations, aimed at engineering students. Prerequisite for the course is the basic calculus sequence. This free online book e-book in webspeak should be usable as a stand-alone textbook or as a companion to a course using another book such as Edwards and Penney, Differential Equations and Boundary Value Problems: I have also taught Math 20D at University of California, San Diego with these notes a 3-day-a-week quarter-long course.
There is enough material to run a 2-quarter course, and even perhaps a two semester course depending on lecturer speed. This is a book on linear algebra and matrix theory. While it is self contained, it will work best for those who have already had some exposure to linear algebra. It is also assumed that the reader has had calculus. Some optional topics require more analysis than this, however. We believe the entire book can be taught in twenty five minute lectures to a sophomore audience that has been exposed to a one year calculus course.
Vector calculus is useful, but not necessary preparation for this book, which attempts to be self-contained. Key concepts are presented multiple times, throughout the book, often first in a more intuitive setting, and then again in a definition, theorem, proof style later on. We do not aim for students to become agile mathematical proof writers, but we do expect them to be able to show and explain why key results hold.
We also often use the review exercises to let students discover key results for themselves; before they are presented again in detail later in the book. This is an introductory text intended for a one-year introductory course of the type typically taken by biology majors, or for AP Physics B. Algebra and trig are used, and there are optional calculus-based sections. This is a text for a two-term course in introductory real analysis for junior or senior mathematics majors and science students with a serious interest in mathematics.
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Prospective educators or mathematically gifted high school students can also benefit from the mathematical maturity that can be gained from an introductory real analysis course. Probability theory began in seventeenth century France when the two great French mathematicians, Blaise Pascal and Pierre de Fermat, corresponded over two problems from games of chance. Problems like those Pascal and Fermat solved continuedto influence such early researchers as Huygens, Bernoulli, and DeMoivre in establishing a mathematical theory of probability. Today, probability theory is a wellestablished branch of mathematics that finds applications in every area of scholarlyactivity from music to physics, and in daily experience from weather prediction topredicting the risks of new medical treatments.
Our goal with this textbook is to provide students with a strong foundation in mathematical analysis. Such a foundation is crucial for future study of deeper topics of analysis. Students should be familiar with most of the concepts presented here after completing the calculus sequence. However, these concepts will be reinforced through rigorous proofs. This guide is heavy on linear algebra and makes a good supplement to a linear algebra textbook. But, it is assumed that any college student studying linear algebra will also be studying calculus and differential equations, maybe statistics.
Therefore it makes sense to apply the Octave skills learned for linear algebra to these subjects as well. Chapters 3 and 5 have several applications to calculus, differential equations, and statistics. The overarching objective is to enhance our understanding of calculus and linear algebra using Octave as a tool for computations. For the most part, we will not address issues of accuracy and round-off error in machine arithmetic. For more details about numerical issues, refer to [1], which also contains many useful Octave examples. This book presents standard intermediate microeconomics material and some material that, in the authors' view, ought to be standard but is not.
Introductory economics material is integrated. Standard mathematical tools, including calculus , are used throughout. The book easily serves as an intermediate microeconomics text, and can be used for a relatively sophisticated undergraduate who has not taken a basic university course in economics. The typical introductory real analysis text starts with an analysis of the real number system and uses this to develop the definition of a limit, which is then used as a foundation for the definitions encountered thereafter.
While this is certainly a reasonable approach from a logical point of view, it is not how the subject evolved, nor is it necessarily the best way to introduce students to the rigorous but highly non-intuitive definitions and proofs found in analysis. It covers a variety of topics at an introductory level. Chapter One introduces basic notions, such as arguments and explanations, validity and soundness, deductive and inductive reasoning; it also covers basic analytical techniques, such as distinguishing premises from conclusions and diagramming arguments.
Chapter Two discusses informal logical fallacies. Chapters Three and Four concern deductive logic, introducing the basics of Aristotelian and Sentential Logic, respectively.
Chapter Five deals with analogical and causal reasoning, including a discussion of Mill's Methods. The textbook meets high quality standards and has been used at Princeton, Vanderbilt, UMass Amherst, and many other schools. We look forward to expanding the reach of the project and working with teachers from all colleges and schools.
The chapters of this book are as follows:. This book consists of ten weeks of material given as a course on ordinary differential equations ODEs for second year mathematics majors at the University of Bristol. It is the first course devoted solely to differential equations that these students will take. Prealgebra is a textbook for a one-semester course that serves as a bridge between arithmetic and algebra.
The organization makes it easy to adapt the book to suit a variety of course syllabi. A casual glance through the Table of Contents of most of the major publishers' College Algebra books reveals nearly isomorphic content in both order and depth. We then introduce a class of functions and discuss the equations, inequalities with a heavy emphasis on sign diagrams and applications which involve functions in that class. Precalculus is intended for college-level precalculus students. Since precalculus courses vary from one institution to the next, we have attempted to meet the needs of as broad an audience as possible, including all of the content that might be covered in any particular course.
The result is a comprehensive book that covers more ground than an instructor could likely cover in a typical one- or two-semester course; but instructors should find, almost without fail, that the topics they wish to include in their syllabus are covered in the text. Many chapters of Openstax College Precalculus are suitable for other freshman and sophomore math courses such as College Algebra and Trigonometry; however, instructors of those courses might need to supplement or adjust the material. Openstax will also be releasing College Algebra and Algebra and Trigonometry titles tailored to the particular scope, sequence, and pedagogy of those courses.
Our approach is calculator based. For this, we will use the currently standard TI calculator, and in particular, many of the examples will be explained and solved with it. However, we want to point out that there are also many other calculators that are suitable for the purpose of this course and many of these alternatives have similar functionalities as the calculator that we have chosen to use. An introduction to the TI calculator together with the most common applications needed for this course is provided in appendix A.
In the future we may expand on this by providing introductions to other calculators or computer algebra systems. However, in order to keep this text as self contained as possible we always recall all results that are necessary to follow the core of the course even if we assume that the student has familiarity with the formula or topic at hand. After a first introduction to the abstract notion of a function, we study polynomials, rational functions, exponential functions, logarithmic functions, and trigonometric functions with the function viewpoint.
Throughout, we will always place particular importance to the corresponding graph of the discussed function which will be analyzed with the help of the TI calculator as mentioned above. These are in fact the topics of the first four of the five parts of this precalculus course. In the fifth and last part of the book, we deviate from the above theme and collect more algebraically oriented topics that will be needed in calculus or other advanced mathematics courses or even other science courses. This part includes a discussion of the algebra of complex numbers in particular complex numbers in polar form , the 2-dimensional real vector space R 2 sequences and series with focus on the arithmetic and geometric series which are again examples of functions, though this is not emphasized , and finally the generalized binomial theorem.
An Investigation of Functions is a free, open textbook covering a two-quarter pre-calculus sequence including trigonometry. The first portion of the book is an investigation of functions, exploring the graphical behavior of, interpretation of, and solutions to problems involving linear, polynomial, rational, exponential, and logarithmic functions. An emphasis is placed on modeling and interpretation, as well as the important characteristics needed in calculus. This free undergraduate textbook provides an introduction to proofs, logic, sets, functions, and other fundamental topics of abstract mathematics.
It is designed to be the textbook for a bridge course that introduces undergraduates to abstract mathematics, but it is also suitable for independent study by undergraduates or mathematically mature high-school students , or for use as a very inexpensive supplement to undergraduate courses in any field of abstract mathematics. The emphasis in this course is on problems—doing calculations and story problems.
To master problem solving one needs a tremendous amount of practice doing problems. The more problems you do the better you will be at doing them, as patterns will start to emerge in both the problems and in successful approaches to them. You will learn quickly and effectively if you devote some time to doing problems every day.
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Typically the most difficult problems are story problems, since they require some effort before you can begin calculating. Here are some pointers for doing story problems:. This is a text that covers the standard topics in a sophomore-level course in discrete mathematics: It explains and clarifies the unwritten conventions in mathematics, and guides the students through a detailed discussion on how a proof is revised from its draft to a final polished form. Hands-on exercises help students understand a concept soon after learning it. The text adopts a spiral approach: The goal is to slowly develop students' problem-solving and writing skills.
This initiative publishes high-quality, cost-effective course resources by engaging faculty as authors and peer-reviewers, and libraries as publishing service and infrastructure. The pilot launched in , providing an editorial framework and service to authors, students and faculty, and establishing a community of practice among libraries. More information can be found at http: This is a new approach to an introductory statistical inference textbook, motivated by probability theory as logic.
It is targeted to the typical Statistics college student, and covers the topics typically covered in the first semester of such a course. It is freely available under the Creative Commons License, and includes a software library in Python for making some of the calculations and visualizations easier. This trigonometry textbook is different than other trigonometry books in that it is free to download, and the reader is expected to do more than read the book and is expected to study the material in the book by working out examples rather than just reading about them.
So this book is not just about mathematical content but is also about the process of learning and doing mathematics. That is, this book is designed not to be just casually read but rather to be engaged.
Math, art, and ideas
This is a text on elementary multivariable calculus, designed for students who have completed courses in single-variable calculus. The traditional topics are covered: An introductory level single variable calculus book, covering standard topics in differential and integral calculus, and infinite series. Late transcendentals and multivariable versions are also available.
I intend this book to be, firstly, a introduction to calculus based on the hyperrealnumber system. In other words, I will use infinitesimal and infinite numbers freely. Just as most beginning calculus books provide no logical justification for the real number system, I will provide none for the hyperreals. The reader interested in questions of foundations should consult books such asAbraham Robinson's Non-standard Analysis or Robert Goldblatt's Lectures onthe Hyperreals.
A Gentle Introduction to the Art of Mathematics Joseph Fields, Southern Connecticut State University This book is designed for the transition course between calculus and differential equations and the upper division mathematics courses with an emphasis on proof and abstraction. Austin State University This text is intended for a one- or two-semester undergraduate course in abstract algebra. Advanced Problems in Mathematics: Algebra and Trigonometry Algebra and Trigonometry provides a comprehensive and multi-layered exploration of algebraic principles.
Applied Discrete Structures Alan Doerr, University of Massachusetts Lowell Kenneth Levasseur, University of Massachusetts Lowell In writing this book, care was taken to use language and examples that gradually wean students from a simpleminded mechanical approach andmove them toward mathematical maturity. Applied Finite Mathematics Rupinder Sekhon, De Anza College Cupertino Applied Finite Mathematics covers topics including linear equations, matrices, linear programming, the mathematics of finance, sets and counting, probability, Markov chains, and game theory.
Applied Probability Paul Pfeiffer, Rice University This is a "first course" in the sense that it presumes no previous course in probability. Calculus for the Life Sciences: Calculus One Multiple Authors, Mooculus Calculus is about the very large, the very small, and how things change—the surprise is that something seemingly so abstract ends up explaining the real world.
Calculus Volume 1 Gilbert Strang, MIT Edwin Herman, University of Wisconsin-Stevens Point Calculus is designed for the typical two- or three-semester general calculus course, incorporating innovative features to enhance student learning.
Calculus Volume 2 Gilbert Strang, Massachusetts Institute of Technology Calculus is designed for the typical two- or three-semester general calculus course, incorporating innovative features to enhance student learning. Calculus Volume 3 Gilbert Strang, Massachusetts Institute of Technology Calculus is designed for the typical two- or three-semester general calculus course, incorporating innovative features to enhance student learning.
College Algebra College Algebra provides a comprehensive and multi-layered exploration of algebraic principles. Combinatorics Through Guided Discovery Kenneth Bogart, Dartmouth College This book is an introduction to combinatorial mathematics, also known as combinatorics. Elementary Algebra John Redden, College of the Sequoias It is essential to lay a solid foundation in mathematics if a student is to be competitive in today's global market. Elementary Differential Equations with Boundary Value Problems William Trench, Trinity University Elementary Differential Equations with Boundary Value Problems is written for students in science, engineering, and mathematics who have completed calculus through partial differentiation.
Euclidean plane and its relatives Anton Petrunin, Penn State This book is designed for a semester-long course in Foundations of Geometry and meant to be rigorous, conservative, elementary and minimalistic. Fundamentals of Mathematics Denny Burzynski, College of Southern Nevada Wade Ellis, West Valley Community College Fundamentals of Mathematics is a work text that covers the traditional study in a modern prealgebra course, as well as the topics of estimation, elementary analytic geometry, and introductory algebra.
It is intended for students who: How We Got from There to Here: A Story of Real Analysis Robert Rogers, State University of New York Eugene Boman, The Pennsylvania State University The typical introductory real analysis text starts with an analysis of the real number system and uses this to develop the definition of a limit, which is then used as a foundation for the definitions encountered thereafter. Intermediate Algebra John Redden, College of the Sequoias It is essential to lay a solid foundation in mathematics if a student is to be competitive in today's global market.
Intermediate Algebra Lynn Marecek, Santa Ana College Intermediate Algebra is designed to meet the scope and sequence requirements of a one-semester Intermediate algebra course. Introduction to GNU Octave: A brief tutorial for linear algebra and calculus students Jason Lachniet, Wytheville Community College This guide is heavy on linear algebra and makes a good supplement to a linear algebra textbook. Laurie Snell, Dartmouth College Probability theory began in seventeenth century France when the two great French mathematicians, Blaise Pascal and Pierre de Fermat, corresponded over two problems from games of chance.
Introduction to Real Analysis William Trench, Trinity University This is a text for a two-term course in introductory real analysis for junior or senior mathematics majors and science students with a serious interest in mathematics. Introduction to Statistics David Lane, Rice University Introduction to Statistics is a resource for learning and teaching introductory statistics. Introductory Statistics Douglas Shafer, University of North Carolina Zhiyi Zhang, University of North Carolina In many introductory level courses today, teachers are challenged with the task of fitting in all of the core concepts of the course in a limited period of time.
Introductory Statistics Multiple Authors, Openstax College Introductory Statistics follows the scope and sequence of a one-semester, introduction to statistics course and is geared toward students majoring in fields other than math or engineering. Learning Statistics with R: A tutorial for psychology students and other beginners Danielle Navarro, University of New South Wales Learning Statistics with R covers the contents of an introductory statistics class, as typically taught to undergraduate psychology students, focusing on the use of the R statistical software.
Linear Algebra with Applications W. Keith Nicholson, University of Calgary After being traditionally published for many years, this formidable text by W. Writing and Proof, Version 2. The primary goals of the text are to help students: Notes on Diffy Qs: The chapters of this book are as follows: Ordinary Differential Equations Stephen Wiggins, University of Bristol This book consists of ten weeks of material given as a course on ordinary differential equations ODEs for second year mathematics majors at the University of Bristol.
Prealgebra Multiple Authors, Openstax College Prealgebra is a textbook for a one-semester course that serves as a bridge between arithmetic and algebra. Precalculus Multiple Authors, Openstax College Precalculus is intended for college-level precalculus students. The Fundamentals of Abstract Mathematics Dave Morris, University of Lethbridge Joy Morris, University of Lethbridge This free undergraduate textbook provides an introduction to proofs, logic, sets, functions, and other fundamental topics of abstract mathematics.
Single Variable Calculus I: If we jumped in steps of. Hey I have a question. Is it possible to integrate the volume of the sphere using the same method only with a pyramid? Wikipedia went about deriving the volume of a sphere on a completely different manner and when I differentiate the formula for the volume I get the formula for the surface area of a sphere.
What is the relation between these results? Because that would make calculating shapes above the third dimension very easy. I have never really bothered to read anything into calculus. After reading this, I actually feel that I would actually like to learn a lot more into this as this gave me a really good view of what Calculus is. I like what you said. Two Weeks ago I read an article on quantum entanglment and since then have been trying to figure out how to learn more.
This is exciting yet daunting especially since I am doing this on my own and not in a class. I do this so that I can really understand what people are saying about entanglement, and well learning is always a good thing. Thank you for describing it in a manner my liberal arts mind not only understood but enjoyed. I look forward to my journey in the math world. This single article has taught me more in terms of real understanding than my last 24 years of schooling.
These are some wise words. The education system does try and crush my love for maths but this has assisted to allow me to look past that and focus on the beauty of the subject. Calculus is a very lovely subject…. You will see how rigid and awesome it is later…. Ask questions and show curiosity till you understand everything…. How did you pass College Algebra? If nature is programmed, how does your statement [Like evolution, calculus expands your understanding of how Nature works. And, Who is th programmer? Who wrote or created the CODE?
Had to take Calc 1 three times to pass, three times to pass Calc 2, never used any of it in the next 40 years. But I feel to this day presentations were awful, full of theorems, not a bit of common sense real world problems solved or real world applications shown. Probably half the students flunked out of Parks College because of Calc. Reading your stuff tonight was a great refresher, and you have an excellent knack of simple explanation.
I wish I had you as an instructor 40 years ago! Hi Pete, thanks for the note. Really happy things resonated with you: Congratulations and keep up the good work! As the area of a triangle is half the one of a parallelogram. But I have to add, because I read your article, the formula for the area of the parallelogram made total sense, as being a bunch a lines stacked one next to the other, resulting in the formula: D I had to figure it out by myself.
With the few words of your text, you already unlocked a better math logic in me, and more admiration for Newton, as I begin to understand more of his genius: Again, thank you a lot for sharing your knowledge in a simple and understandable way! Hi Yashvardhan, thanks for the note and kind words — happy the articles are helping grow your math interest: Yep, I write all the articles!
Been pouring over your site last 3 to 4 hours. Worth every second of it. Just bought the book too. Keep up good work. Hi Louis, it can actually be really fun once you get into it. Physics and chemistry, especially, are presented as surrogate math courses when they could be presented as the wonder of discovery. Max Planck suggested that light travels in packets of energy because it was the only way he could solve an equation.
He expected someone to come up with a better answer in time. It explains so much. And there are so many others that could be told as well. I had a hard time accepting the reality of elliptic geometry until one author made the comparison with latitudes and longitudes. I am 49 years old, been a housewife most my later adult life, office manager in younger years, and am planning to go back to college next Fall. I wish to major in physics. I took algebra and geometry in high school and did fairly well, but that was many years ago.
Calculus in Context
I love math but due to Parkinsons, tend to have memory issues. Anyway, in preparation for this endeavor, I wish to re-educate myself to prepare for college calculus. It has been very interesting for me as a teacher to use a little calculus gadget to teach them a new way of seeing things. For example, with the formulas you write perimeter, area, volume of a sphere of radius, I always get the most surprised faces when I show them how the derivative of the volume function leads to the surface area function, and the derivative of the area function leads to the perimeter function.
It is quite exciting for them to see that and they start asking themselves questions, which is great. The counting of syllables, like numbers in math, is critical to the practice and appreciation of verse. Will be using this site more in the Fall when I start my first of several Calculus classes for my physics degree. Very glad I found this site. Can we find the function from the integral?
If so, how can we find the integreal with so little data? Hi Tim, great question. If we only have 2 data points the start and end , then we have to assume a linear progression from 0 to 60mph over the course of 6 seconds i. In this case, integrating to figure out how much distance was traveled may not be accurate. As we gather more data points, we can get a better idea of the actual shape of the acceleration curve. I seriously love you man. I have hated math my whole life and failed calculus miserably.
I felt like something was always missing, and that was insight. You nailed everything on the hammer and gave such a helpful guide. I actually know what is going on in class now, instead of starting at the board and zoning out at the giant mass of information. It is like studying a language before you can speak it, or study the physics of art before art itself. But this is honestly the best thing ever, and thank you. In college, I enrolled in calculus and dropped it within my first week. I was lost by the end of lesson 1 and drowning by day 3. Everything is within our grasp when explained properly.
Very, very happy the approach is clicking for you. Dear friend, Your text was so fascinating. I am a student at Engineering faculty in Afghanistan. Please help me up. Really appreciate your great work and intuitive to help people understand things better. Will you try to explain some concepts in Linera Algebra in future? I think that is a weakness in some colleague students, i am one of it honestly: Hi Keith, thanks for the note.
Yep, I have a quick intro to linear algebra http: I thought I hated math for a long time, but as with many other things, it turned out I just hated how humans were approaching it. Hi Kallid, I love the way you derived the formula for the area of a circle using the circumference of a circle. My question is on how to derive the surface area of a sphere in a similar manner. I also have ADHD which makes the classroom setting a nightmare, especially in math.
Explanations like this are greatly appreciated because it takes the horror away from math and helps me to understand it in a logical and practical way. Wow, thanks for the thoughtful comment! I always knew I was smart but never could get even the basics of math. Well, until we got to the end of the unit, or the next math up. I needed to actually see the practical application to appreciate what I was doing. Even graphing calculators stunk because those curves served no practical purpose.
I found his website in eighth grade and it was really helpful in teaching me to love math, even if calculus seemed like a kind of fascinatingly foreign idea at the time. Beautifully explained i like the way…. I am a ninth grade student, trying to learn stuff ahead of our syllabus, and calculus was my first pick. The way you have given an intro to calc is just epic. I understood everything as well as possible.
I will surely continue on your series. This is actually very interesting Im 11 and Dad is trying to teach me calculus and I can understand you, unlike my prealgebra book. Hope you enjoy the rest of the calculus series! Both grown adults, are exceptional in mathematics, to this day. Love the way you have explained the fundamentals of calculus. I love the insights. Not understanding the essence of mathematics makes the majority of people not appreciate it. In order to understand what an abstract word really means, one must get a hold first of its manifestations in the concrete world, and then how the abstract thereafter relates to the concrete.
I feel moved to share some facts, inferences and insights regarding its validity. Our scientific formulae are so predictive only because each scientific formula represents a scientific generalisation that has been based on factual observations. We keep on observing sets of phenomena in this way. However, that does not explain how they can be consistent. Therefore one is left with two general categories to explain the consistency of each of them: What do we call these certainties in the universe?
What intuition do you think drove us to call physical laws laws? Nothing comes from nothing. The law of conservation of energy signifies this. This therefore makes us conclude that the universe has always existed from eternity past. However, the universe began. Our universe is characterised by cosmic expansion. The second law of thermodynamics indicates that the longer time has elapsed, the greater the overall entropy of the universe shall be.
Given that the universe is currently not at a state of maximum entropy, the first and second laws of thermodynamics indicate that the universe must not have always existed from eternity past. Matter, energy, space and time, which constitute the universe, have not always existed. Therefore, because the universe began to exist, either some Being or something must have caused it.
This cause of the universe must be immaterial, because the cause of the universe cannot be the universe itself, which is the totality of all material things, as nothing can cause itself that has not arisen from nothing. In other words, something causing itself is like saying that it appeared out of nowhere. Something arising out of nothing can only be true if that thing is not under the law of conservation of energy, or, if some Being xor some other thing caused it that, being able to create energy, is above the law of conservation of energy. Because of laws such as the laws of thermodynamics, only the Creator can and will create the universe from nothing.
The theory of evolution holds that millions and millions of years ago, fish began evolving by means of little cumulative changes over long periods of time. Over approximately years, fish managed to evolve to amphibians. Over approximately years, amphibians evolved to reptiles. Some of these reptiles evolved to nonmonkey mammals, still over a long period of time, which then evolved to monkeys—simply put, our ancestors.
Of course, fish came all the way from a common ancestor. This is what Darwin has proposed. After the discovery of DNA, however, the theory of evolution itself evolved to include nonliving chemicals that happened to live by time and incredible luck. There is no substantial evidence, however, to support this.
The assertion that genus evolves to another genus over a very long period of time is contrary to science genome is the total of all the genetic possibilities for a given species, and should not be confused for genotype. I understand that, in order to appear as though it was falsifiable, and thus be convincing, this assertion depends on natural selection.
One purpose of natural selection is to eliminate the abnormal mutations cause abnormalities. Too much of this and extinction would occur. Living beings adapt to their surroundings because of the way they were designed — not because of natural selection; without design in the first place, natural selection would be meaningless. No one has ever observed actual evolution happen naturally.
One only sees supposed evolution in some man-made books with pictures and in man-made realistic 3D animation movies. All proponents of the theory of evolution can show are some fossil remains with similarities, which have already undergone decomposition. The lips, the eyes, the ears, and the nasal tip leave no clues on the underlying bony parts. You can with equal facility model on a Neanderthaloid skull the features of a chimpanzee or the lineaments of a philosopher.
These alleged restorations of ancient types of man have very little if any scientific value and are likely only to mislead the public… So put not your trust in reconstructions. The fact that one language was used to design, and to dictate all the functions of, all living beings on Earth is just undeniable. After all, all living beings on Earth have one thing in common— life. If one has ever used a programming language before, one would understand the necessity of reusing a set of specific codes to a number of different programs.
Computer programmers though have a way of converting lengthy codes to just a short one by saving codes in header files because it would be tiring for humans to retype lengthy codes over and over again. Information is contained in our DNA, and our bodies were designed, and functions, as well, according to the specifications of this information.
What happens when a living being is exposed to harmful things such as radiation? Mutations are alterations that take place in the DNA—damaging the information in it. Information never originates by itself in matter; it always comes from an intelligent source. The outdated microscopes of their time made the very complex structure of the cell look so simple. However, if we would subscribe to the current scientific discoveries, as well as the technologies, of our time, we would begin to apprehend that the indications never really pointed to the theory of evolution.
As science progresses, intelligent design becomes more evident. What else is the meaning of evidence? Everywhere we look, the more attentive we are to the details, the more evident intelligent design becomes. Mathematical formulae are symbolic representations of mathematical ideas, and ideas can only be conceived by the mind. We experience this whenever we conceive mathematical ideas. The Fibonacci numbers is one such idea. Fibonacci numbers and golden section often occur in nature, even in our bodies, and this repetition goes against mere coincidences.
It seems to me then that just as we humans can make something only out of that which has already been created, we can not conceive mathematical ideas other than that which has already been thought by an immaterial intelligent Being prior to the universe, as we humans rely upon the universe to derive our conclusions and mathematical ideas from. All mathematical ideas that we know of are embedded throughout the whole universe. As a matter of fact, mathematics is so pervasive it even permeates science.
This does not contradict intelligence prior to the universe, but rather, proves it. I understand that not all religions can be trusted to teach one what is true, but lies exist not only in religion. One should learn upon the insights of the reasonable, rather than calling the untaught ignorant without even educating them. I love math because to me there is nothing more beautiful than the truth, and math to me is also the realisation of the quantitative objective aspect of the truth algebraic logic counts truth value — 0, 1.
The Internet Archive has a copy, though. You could replace that link with this one: The base is just the x axis in that graph…. This diagram may help: When limits ranges and domains met my math brain I thouht I was done and I and I amost lost hope in maths! Finally a site that explains the understanding and ideas behind maths rather than just robotically regurgitating rigid formulae.
I have a degree in mathematics a singular noun from an American university that I earned in the s. We did have superscripts for exponents and normal algebraic notation for products. Maybe a sentence of explanation would be in order. Hi Steve, thanks for the great feedback!
Glad you enjoyed this post! Thanks for a great explanation, Kalid. Coming from an evolutionary biologist background it was very easy to follow. Just wanted to let you know that fat actually has the most calories.
Mathematics
Professors rarely teach the bigger concepts. They teach you the nit picky problems that will be on the test. THIS is why I hate math or at least learning it in college. Mr azad you Can blame me for My lagard intuition but i want to correct myself the. Newton did understand the mathematical rigor behind the calculus but only intuitive part became popular and Cauchy got the credit for mathematical rigor. It is difficult to believe that a robust theory like calculus originated purely out of intuition.
For people struggling with math I created powerful derivative calculator in addition to steps derivative can be evaluted at point and integral calculator with steps shown. Also, I looked for good free online graphing calculator, but all they lack customization, so I created another one. You can check them out here: Graphing calculator — http: Still, I love those moments where I get mind blown at how he math all comes together. So, thank you for this incredibly helpful site. Thank you for your help!
I am about to take calculus and was really discouraged about it until I read this. Though I have not the slightest clue on what calculus is. That notion scared me as I hate to be not prepared for anything, and more importantly I hate not being able understand how something works, let alone how to do it. This little introduction cleared my worries on the fact. I wish you had been my math teacher all of my school days and maybe I would have loved it instead of fearing it. Its actually called Maths but as you are probably american and the article is very good I will just add the s on as i read through your stuff.
Amazing, your explanation is easy to understand and just like calculus itself you explain the little things which helps to work with the bigger things. If you are not a teacher already then you should consider teaching it. By far the best explanation I have ever seen. Kalid, thanks so much for this and your other articles. At school I enjoyed maths but lost interest when it became a mass of meaningless formulas. I came across your articles while digging into neural networks and discovering horror that exponentials, logarithms and calculus lay at their heart.
After 30 years of avoidance the subject has become tangible again: And one think i really appreciate that you replied to every single comment you possibly could. Could tell you that how awesome this site is. I really wished I had discovered it before. I would never know such site which is solely based on intuitive exist.
Consider i am a engineering student in India you know that how bad condition the education system here is. I really had to depend on internet for learning. You can also email me the list Thanks. Thanks Rohit, really glad it helped! For resources, I like anything by Richard Feynman the famous physicist. There are a bunch of books and lectures of his online and a great place to start.
Never even heard of such things as Algebra, calculus and all the other things until I was an adult. So I thought; can a 86 year old learn these things and that led to your article and no, I got as far as your coloured in circle then I was lost. I hope that beside being an author you are an up front teacher, for if you are your students must be absorbed. PS, excuse the English spelling. Can we Go Further????. Are They Satisfied NOw?????. JUst Trying To Help???. Because programming always starts with a known objective i. I now teach higher mathematics at degree level.
The reason is that I understand the algorithms, rather than just knowing how to mindlessly repeat them like a parrot. Your gentle introduction to Calculus is very good. Anyhow, thank you, and well done. Is it fine for me to want to learn this? Why does Calculus insist on functions, which are limited to a single output, e. I enjoyed math from a very young age. When the math classes focused on memorizing steps of the process, I struggled and intimidation settled in. I realize now that I enjoyed it when it was intuitive. I look forward to learning more your work.
Thanks for this help for me and others like me.
You Seem To Understand Then???. Your argument is faulty in several ways. For example, poetry is not about the form meter, stanzas, etc. Nobody — appreciate what the writer and do not try to nick pick on things. May be you can counter him by writing better stuff on Calculus? Kalid, thank you for this article. The way you used rings and made a right triangle of circumferences is very imaginative and helpful in understanding how the actual formula for area relates to circumference. I had one question for you. I am returning to school after graduating with a bachelor of science degree in criminal justice in I want to further my education and pursuing an engineering degree to go along with my criminal justice degree.
I also took trigonometry in high school but that was even farther back. I just wanted your opinion about what you think my best options are going forward. Thank you in advance. It was invented centuries earlier by an 11th century Indian mathematician Bhaskara, and later refined by the scholars at Kerala school in the 14th century notably Madhava. Like a true snake western scholars presented works already introduced by others as their own.
In actuality the albino race is relatively stupid and dull. BetterExplained helps k monthly readers with friendly, insightful math lessons more. Calculus is similarly enlightening. Imagine studying this quote formula: But calculus is hard! Calculus does to algebra what algebra did to arithmetic. Arithmetic is about manipulating numbers addition, multiplication, etc.
Using calculus, we can ask all sorts of questions: How does an equation grow and shrink? How do we use variables that are constantly changing? Heat, motion, populations, …. And much, much more! Realize that a filled-in disc is like a set of Russian dolls. Here are two ways to draw a disc: We get a bunch of lines, making a jagged triangle.
But if we take thinner rings, that triangle becomes less jagged more on this in future articles. For each possible radius 0 to r , we just place the unrolled ring at that location.