Lecture Note
University
Stanford UniversityCourse
Introduction to Mathematical ThinkingPages
9
Academic year
2023
Murphyal
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46
Introduction to Mathematical Thinking. Introductory Material ● Mathematics is the study of quantity, structure, space and change. Mathematicians use logic and reasoning to explain patterns in the world around us. Math can beused to solve problems in every area of science, technology, and society — fromphysics, to medicine, to education. ● The Science of Patterns is the study of patterns in nature, science, and mathematics. The patterns are studied for their aesthetics, simplicity, andcomplexity. ● Arithmetic is related to counting and calculation, which can be performed using concrete numbers. Counting can be viewed as a kind of measurement whilecalculation refers to solving problems by performing arithmetic operations such asaddition, subtraction, multiplication, and division. ● Number Theory is a branch of pure mathematics that deals with the properties of numbers in general. It is used to solve problems in other branches of mathematicsand to study topics such as prime numbers, factorization, cryptography and thehistory of mathematics. ● Geometry is the study of shapes and spaces, as well as the properties of angles, lines, curves and surfaces. Geometry is one of the oldest branches of mathematics.Geometry has many practical applications in medicine, architecture, technology andother fields such as mechanical engineering. Geometry can be used to solvereal-world problems. ● Calculus is a branch of mathematics focusing on limits, functions, derivatives, integrals and infinite series. This subject constitutes a major part of modernmathematics education. ● Logic is the study of valid reasoning and argumentation. Logic is important because it is used to make decisions, solve problems, and communicateinformation. It is a type of reasoning that helps us come to conclusions, or answers,based on the information we have. ● Probability theory is a branch of mathematics concerned with the analysis of random phenomena. It studies the behaviour of stochastic processes, and it can beused to describe and predict how likely certain events are to occur. ● Topology is the study of shapes and spaces. It is a branch of mathematics that focuses on the properties of figures and the relationships between them. It studiesthings like whether they are connected or separate, whether they have holes, howthey look without altering their size, etc. ● Fractal geometry is a branch of mathematics concerned with the study of fractals. Fractals are irregularly shaped geometric structures that often exhibit similarpatterns at all scales, therefore having an appearance of complexity andrandomness. ● Banach-Tarski is a mathematical theorem that states that you can take a 3-dimensional solid and cut it into 5 pieces, reassemble the pieces into twoseparate 3-dimensional solids, one of which is the same size as the original. ● Paradox is a term used to describe a situation which seems contrary to common sense or is self-contradictory.
The quizzes serve to assess your progress, allowing you to determine if you are moving atan adequate pace. The questions focus on the most important things. If you find thosetests simple, it means you are doing good. Quiz What was the first use of numbers? ● Money● Measurement of land● Tracking the seasons Which patterns are studied in topology? ● Geographical terrain● Sounds● Closeness ! Weekly problem sets, unlike quizzed, involve substantial work and stimulate real learning. What became the primary focus in mathematics in the nineteenth century,? ● Dynamical systems● Concepts and relationships● Fractals According to Keith Devlin, the most valuable mathematical ability in advanced nationstoday is? ● Use of technology● Mastery of basic skills● Adapt old methods or develop new ones It is important to be precise in your language. One American dies of melanoma almost every hour. Almost every hour, an American dies of melanoma. 1. There are infinitely many prime numbers. 2. For every real number a, the equation x 2 + a = 0 has a real root. 3. is irrational. 4. If p(n) denotes the number of primes less than or equal to the natural number n, then asn becomes very large, p(n) approaches n/log e n. We can show that if we list the primes p1, p2, p3……, the list continues forever. If we have reached some stage n= p1, p2, p3……, pn, is it possible to find another primethat can be added to the list? Here is a number N=(p1*p2*p3.....*pn)+1 Obviously, we can tell that N is bigger than pn.
If N is prime, we have found a prime number larger than pn and can continue the list. If Nis not prime, then it's divisible by a prime number, say p. It is impossible for З to be any of the numbers p1, p2, ….. , pn. If any of these numbers isused as a divisor for N, the remainder is 1. So p>pn. We have found another primenumber greater than pn. Thus, the list can always be continued. 1. There are an infinite number of prime numbers. True. 2. With every real number a, the equation x2+1=0 has a real root. False. What if a is equal to 1? We know that x2+1=0 does not have a root. 3. is irrational. True. 4. As n becomes very large, the number of primes less than or equal to n tendsasymptotically toward (n)/log e n. Quiz Euclid's proof that there are infinitely many primes relies on the fact that if p1, ……, pn arethe first n primes, then N=(p1, ……, pn)+1 is prime. TRUE FALSE July is a summer month. The man saw the woman with a telescope. Sisters reunited after ten years in checkout line at Safeway. Large hole appears in High Street. City authorities are looking into it. Mayor says bus passengers should be belted. Object a has property P. 3 is a prime number Every object of type T has property P. Every polynomial equation has a complex root. There is an object of type T having property P. There is a prime number between 20 and 25 If statement A, then statement B. If p is a prime of the form 4n+1, then p is a seen of two squares.
andornotimpliesfor allthere exists Let’s start with a question. What is mathematics? It might seem strange, given that you have probably spent several years studying the subject, but relatively little time has beendevoted to explaining exactly what mathematics is. Instead, the focus is on learning and applying strategies and methods to solve mathproblems. This is a bit like saying that soccer is a series of maneuvers you execute to getthe ball into the goal. Both accurately describe all those key features, but they miss thewhat's and why of the big picture. If all you want to do is learn new mathematical techniques and apply them in differentcircumstances, then you can probably get by without knowing the theoretical basis behindmathematics and its principles. But, this is not how it should work. It is important to note that a great deal of the mathematics taught in schools dates back tomedieval times, with most of the rest coming from the 17th century at the very latest.Virtually nothing from the last 300 years has transferred into the classroom. In the last 10 years alone, the world we live in has changed dramatically. Most of the majorchanges in mathematics over the centuries were just expansion—of ideas and conceptsthat had been around for a long time. But in the 19th century, mathematics began to change its focus. It became much moreabstract and focused more on analyzing relationships rather than simply followingprocedures. The change in emphasis from procedures and computation to understanding did not comeabout arbitrarily. Rather, it was a result of the increasing complexity of the world we knowtoday. Procedures and computation are still important; however, they are not enough tounderstand this new world. The change in emphasis in mathematics usually occurs when you transition from highschool to university. The science of patterns is defined according to that description as the application ofmathematics to discover and study patterns. Pattern theory concerns identifying andanalyzing abstract patterns. There can be numerical patterns, patterns of shape, patternsof motion, patterns of behavior, voting patterns in a population, patterns of repeatingchance events, and so on. Patterns can be either real or imaginary, visual or mental, static or dynamic, utilitarian orrecreational. They can arise from the natural world around us, from the pursuit of scientificinquiry or from within our own minds.
A variety of patterns provide the basis for branches of mathematics. For example,arithmetic and number theory study the patterns of counting and number; geometrystudies the patterns of shape; calculus allows us to handle patterns of motion; logic studiespatterns of reasoning; probability theory deals with patterns of chance; topology studiespatterns of closeness and position; and fractal geometry studies the cell-like similarityfound in many natural objects. It can be continued. One consequence of the increasing abstraction and complexity of mathematics in the 19thcentury was that methods developed to solve important real-world problems seemedcounterintuitive. An example for you. The Banach-Tarski Paradox shows how, in theory, you can cut asphere into pieces and rearrange the pieces to create two identical copies of the originalsphere. As a result, mathematicians had to learn to trust their math above any intuitions they mighthave. This was true not just for the math itself— it happened in physics as well whenrelativity theory and quantum mechanics were discovered. Therefore, if you are going to rely on mathematics and disregard intuition and commonsense, you must be sure that the math is correct. This is why mathematicians in the 19thand early 20th centuries developed a precise way of thinking, reasoning and callingmathematical thinking. Now about the quiz questions. First one, the correct answer is money. It appears thatpeople certainly measured land and they used various kinds of yardsticks, but they did notuse numbers. They certainly counted seasons, but you can do it without numbers. People have developed many different ways of counting over time. At one time, peoplecounted by notches on sticks, then by pebbles, and eventually by abstract numbers.Scientists believe that people invented abstract numbers about 10,000 years ago in orderto count wealth, to count money. Now for the second question, topology is the study of patterns of closeness. If you thoughtit was geographical terrain, you were confusing topology with topography. 19th century mathematics was characterized by a focus on concepts and relationships.This revolution took place in Germany, and according to many mathematicians, the abilityto adapt old methods or develop new ones is most important today. Yes, mastery of basic skills is important. Yes, you need to be able to use technology. Butthe crucial ability in today's world is adapting old methods or developing new ones. It is intended that you should understand the answers to the quizzes immediately. Theyshould seem obvious. If you find them easy, this means that you are sufficiently engagedin the material and not trying to move too quickly. If you find yourself spending time on aquiz question or going back to review lecture material, then this indicates that you are notengaging with the material closely enough and need to slow down a bit. The secret is reflection, not completion. As professional mathematicians, we are dismayed by school systems that impose tighttime limits on completing mathematics tests and discourage slow, careful work. Real
mathematics takes time. And sometimes you think even more than write. You should oftentry to think in another way from before. We begin with topic that covers precise usage of language. The American MelanomaFoundation, in its 2009 fact sheet, states that one American dies of melanoma every hour.As any mathematician can attest, it is easy to be amused by such claims. We are notinsensitive to the tragic loss of life that melanoma causes, but we find it preposterouswhen people claim that one American dies from melanoma every hour. This claim that oneperson dies from melanoma every hour does not mean what the Association for MelanomaResearch intended it to mean; rather, it means that there is an American who dies frommelanoma every hour. The writer should have written "Almost every hour, an Americandies of melanoma." The difference between these two sentences is the difference between a correct use oflanguage and a figure of speech. Except for mathematicians and other professionals who must be precise in their use oflanguage, no one ever notices that the first sentence above actually makes an absurdclaim. When people speak in everyday contexts about everyday circumstances, they relyon a common knowledge of the world to determine what they mean. When mathematicians use language in their work, there often is no shared commonunderstanding. Moreover, the need for precision is paramount in mathematics. That meansthat when mathematicians use language in doing mathematics, they rely upon the literalmeaning of words. Because of the precise nature of mathematical language, beginning students ofmathematics in college are generally given a crash course on the proper use of language.This crash course is often hundreds of pages long, but it is necessary due to the limitednature of mathematical language. Task actually turns out to be relatively small. Modern pure mathematics is primarily concerned with developing precise definitions,statements and proofs about mathematical objects. Mathematical objects are things likeintegers, real numbers, sets, functions and so on. Here are some mathematical statements. There is an infinite number of prime numbers.For every real number a, the equation x2 + a = 0 has a real solution. The square root of 2is irrational. If p(n) denotes the number of primes less than or equal to n, then as nbecomes very large, p(n) approaches nlog e of n. Mathematicians are interested in statements such as these, but they are above allinterested in demonstrating which statements are true and which are false. This is done byconstructing a proof rather than through observation or measurement or experiment, as inthe natural sciences. We will look at some different methods of proving statements. In those four examples, one,three and four are true but two appeared to be false. Here is a proof to the first statement. It's due to the ancient Greek mathematician Euclid.We show that if we list the primes p1, p2, p3, etc., the list continues forever. Well supposewe've reached some stage n, so we've listed p1, p2, p3 up to pn. Can we find anotherprime to continue the list?
If we can always find another prime, then the list goes on forever and we've shown thatthere are infinitely many primes. Well, can we do this? Euclid described a clever proof inhis famous book Elements in 350 BC.\ We define the number N as follows: N= (p1 x p2 x p3 all the way up to pn), with themultiplication of positive integers. Clearly, if N is prime, it will be bigger than any previouslyfound prime number, so we can keep adding more factors to our equation until we reach aprime number that is larger than any previously found prime. If n is not prime, then it must be divisible by a prime number. If p is this prime number, thenp cannot be any of the primes between p1 and pn. This is because if you divide any ofthese numbers into n, the result will either be a remainder of 1, or a multiple of all thoseprime numbers. Thus, p can't be any of those numbers. Why? Because dividing them leaves a remainderof 1, so p is greater than pn. That means we found a prime number greater than pn. Eitherway, if N is prime or if it's not prime, we've shown that there is a greater prime number thanpn, which means the list can always be continued. This proves that there are infinitely many prime numbers. Let's just take another look atwhat we've done. We start with a list of all prime numbers, or we try to list all the primes.We need to show that we can do this and keep going. Let us begin with a list of the primes. Let us assume that we have reached stage n, wheren could be 10, a million, a billion or a trillion. We shall show that at each stage, there isalways at least one prime number which is greater than the last prime number found. Howdo we do this? We can do smartly. We look at this number, which we call N. N is the product of twonumbers: a little n and a big prime. We add 1 to that product, which makes N bigger thanthe last number in that sequence. Since big N is prime, it is greater than pn. We are not saying that big N would be the nextprime after pn; in fact it almost certainly would not be because big N is a lot bigger thanthese numbers. So this number is a lot bigger than Pn. So this will not be the next prime,almost certainly. However, we are sure that there is another prime and whatever the nextprime is, we will put it on our list. The alternative was that it was not prime, in which case we would call it P. Now, P cannotbe any of these primes. Why? Well, this is why we defined N the way we did: if you divideN by any of these primes, you are left with a remainder of one. The prime that divides N is not any of those listed and must be a different one. If it is adifferent one, it must be larger than Pn. The point is that we found a prime larger than Pn.Is this prime P the next prime after Pn? Well, it might be, but there's no reason to assumeit is, and it doesn't matter. The point is we found a prime larger than Pn so once again, thelist can be continued either way. This is the clever trick that makes the proof work. Defining N that way, we define N in sucha way that if there is any prime dividing N, it won't be equal to any of those numbers.Proof: There are infinitely many primes; thus we have proved the first of our fourexamples, that there are infinitely many primes.
The first statement is true. However, the second statement is false because it says thatevery real number has a real root. To prove this statement false, one only needs to findone number that doesn't have a real root. We can just take a minus 1. This way we know that the equation x squared plus 1 equals 0does not have a root. Because the square root of any real number is positive, and you takea positive number and add 1, there is no real number whose square root is 0. A positive number is needed up. The equation for every real number, there's a root doesnot have a solution, which shows that the statement for every real number, there's a root isfalse. Number three is true. Then we will prove it, and learn the cause. The false one is a rathercomplicated looking statement about the distribution of the prime numbers. This result wasproved just about a hundred years ago at the end of the 19th century, and is known as thePrime Number Theorem. The answer is false. The proof requires looking at the number, but it doesn't requiremaking the number prime. It was a rather different. If you assumed that the answer was true here, I would strongly advise you to go back andlook at that proof again. We did not presume that the number was prime. In order to prove whether a statement is true or false, we must first be able to understand itprecisely. Mathematics is a very precise subject, where exactness of expression isrequired. This one already creates a difficulty, since words tend to be ambiguous. And in real life, our use of language is often precise. For example, when we use languagein an everyday setting, we rely on context to determine what our words convey. For example, an American can truthfully say that July is a summer month, but anAustralian cannot make the same statement. The word summer means the same in bothstatements, namely, the hottest three months of the year. However, it refers to one part ofthe year in America and another in Australia. In everyday life we use context to understand general knowledge of the world, and of ourlives, fill in missing information in what is written or said, and eliminate false interpretationsthat can result from ambiguities. For example, we need to know something about the context in order to correctlyunderstand the statements, "the man saw the woman with a telescope." Did the man havethe telescope or did the woman? Sometimes, ambiguities in a newspaper headline can result in unintended but amusingsecond readings. Among my favorites are the following: Sisters reunited after ten years incheckout line at Safeway. Large hole appears in High Street. City authorities are lookinginto it. Mayor says bus passengers should be belted. The process of making the English language precise so that people can communicateeffectively by defining exactly what each word is to mean is an impossible task;furthermore, it would be unnecessary since people generally do just fine by relying oncontext and background knowledge. However, in mathematics, precision is crucial, and ambiguity cannot be assumed to beremoved simply by all parties having the same contextual and background knowledge.
However, mathematics is a language used in science and engineering. The cost ofmiscommunication through an ambiguity can be high, possibly fatal. At first, it might seemlike a herculean task to make the use of language in mathematics sufficiently precise. Butfortunately, it turns out to be very doable, though a bit tricky in places. And it is the highly restrictive nature of mathematical statements that makes thempossible. Almost every key statement in mathematics, the axioms, conjectures,hypotheses and theorems, is a positive or negative version of one of four linguistic forms:categorical propositions, hypotheticals, conditionals and disjunctions. All objects of type T have property P. There exists an object of type T having property P. Ifstatement A, then statement B, or else the statement is a simple combination ofsub-statements of these forms, using the connecting res, which we call combinators (and),(or), and not. For example, three is a prime number—that is, it's divisible by only one and itself. Ten isnot a prime number, because it's divisible by two and five. Every polynomial equation hasat least one complex root; that is, such an equation can be factored into quadratic factors.There is always at least one prime number between 20 and 25. There are no evennumbers beyond 2 that are prime. However, those are just variants. For example, the last statement about the primes of theform 4n + 1 is a celebrated theorem of 19th century mathematician Carl Friedrich Gauss. The ancient Greeks were among the first to notice that all mathematical statements can beexpressed using one of these simple forms. They made a systematic study of the keylanguage terms involved. Namely, and, or, not, implies, for all and there exists. The Greeksprovided universally accepted meanings of these key terms and analyzed their behavior. In a mathematical formal way, we see that study connects with a formal logic.Mathematical logic is the branch of mathematics that deals with the formal properties ofmathematical proof and systems of inference. The study of mathematical logic is a well-established branch of mathematics that is studiedtoday in university departments of mathematics, computer science, philosophy, andlinguistics. The work of ancient Greek philosophers such as Aristotle and his followers, and of Stoiclogicians, was more complex than is widely understood. Quiz now. Mostly the answer is clear from the beginning. If you have some troubles –review and practice the material. The ancient Greeks were the first to formalize the study of language and reasoning, whichled to the development of the branch of mathematics known as formal logic. It is a good idea to discuss the material with your colleagues. Working alone can beharmful. Group thinking can result in a nice progress.
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