Crazy math proofs pdf

The proof they found was quite complex and relied in part on a computer, but it states that in any political map say of the states only four colors are needed to color each individual state so that no states of the same color are ever in contact. Example b prove this proposition using a proof by contradiction. In the second group the proofs will be selected mainly for their charm. They have not yet realized that a proof is nothing more than an explanation of why a carefully. Riemanns rearrangement theorem stewart galanor, 4 west ninetythird street, new york, ny 10025 mathematics teacher,november 1987, volume 80, number 8, pp. The amount of detail that an author supplies in a proof should depend on the audience. Why do we have to learn proofs university of south carolina. A mathematical proof is an inferential argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. Then ask, when are other times in math that we need to justify or show our work. We start with the language of propositional logic, where the rules for proofs are very straightforward. Henning school of mathematical sciences university of kwazulunatal. This follows from the the cue lies in a powerful mathematical result about curved surfaces invented by the german mathematical genius carl friedrich gauss in the 19th century. A mathematical proof of a statement strongly depends on who the proof is written for. The logic and set theory are presented in a naive way.

We hope you enjoy our collection of funny math proofs, favorite math jokes and jokes about numbers. If stuck, you can watch the videos which should explain the argument step by step. But after seeing so many geometry haters i think there can be a problem with teaching methods or the proofs are a bit difficult for majority. That is a bizarre though sometimes useful invention of mathematics. If your exposure to university mathematics is limited to calculus, this book will probably seem very di erent from your. In most of the following examples the underlying \theorem may be a fact that is well known to you. Outline godels incompleteness theorems continuum hypothesis.

This view fails to explain why it is very often the case that a new proof of a theorem is deemed important. And even textbook proofs look different depending on the level of the audience high school vs. This is a list of unusually long mathematical proofs. Why do we have to learn proofs university of south. The trouble with this is that, sooner or later, mathematics becomes suf.

Mathematical method and proof carnegie mellon university. Robert was a famous mathematician who has just died of a heart attack in his fifties. Chaitin although randomness can be precisely defined and can even be measured, a given number cannot be proved to be random. He has made contributions to riemannian geometry and geometric topology. Anyone who has ever fought with a proof will love this list. Math 109 is an introduction to proofs and some mathematical concepts. This book is excellent for an introduction to mathematics proofs course. Mar 21, 2009 i had no idea that so many people hate geometry proofs. We include results in almost all areas of mathematics. Alternatively, one could maybe make a case that the statement of problem 1 is obvious. This textbook is designed to help students acquire this essential skill, by developing a working knowledge of. You wont need formal training, unlimited time, or a genius iq.

However some results to follow require a technique that is less natural, mathematical induction. In this document we will try to explain the importance of proofs in mathematics, and to give a you an idea what are mathematical proofs. May 18, 2018 mathematical proofs are concrete and have to be accepted by every individual on our planet. Since it is a rare occasion when we are able to get by with one of these two methods of proof, we turn to some we are more likely to need. You will nd that some proofs are missing the steps and the purple. It has been approved by the american institute of mathematics open textbook initiative. In this book gibilisco will gently guide you through the nuts and bolts of how to write, read and construct mathematical proofs.

Introduction to mathematical arguments background handout for courses requiring proofs by michael hutchings a mathematical proof is an argument which convinces other people that something is true. Mathematicians were not immune, and at a mathematics conference in july, 1999, paul and jack abad presented their list of the hundred greatest theorems. Great teachers introduced us to the arts of mathematics and writing. The art of proof basic training for deeper mathematics august 8, 2011 springer. Krantz1 february 5, 2007 amathematicianisamasterof criticalthinking,of analysis, andof deductive reasoning. Writing and proof is designed to be a text for the. Proof that math \not \exist math math \ a, b, c \ math math \in \n2 math such that abc is an equilater triangle. Introduction to proof in analysis 2020 edition steve halperin with contributions from elizabeth hughes cc. In mathematics, we are constantly dealing with statements. I personally was a great fan of them in high school. A rule of inference is a logical rule that is used to deduce one statement from others. This is an example, or test, of the theorem, not a mathematical proof.

Each theorem is followed by the \notes, which are the thoughts on the topic, intended to give a deeper idea of the statement. What are some of the most ridiculous proofs in mathematics. How to write a proof leslie lamport february 14, 1993 revised december 1, 1993. The history and concept of mathematical proof steven g. This enigma establishes a limit to what is possible in mathematics. An additional point is that the actual presentation of the proofs themselves were instructive in their elegance. Proof that math ot \exist math math \ a, b, c \ math math \in \n2 math such that abc is an equilater triangle. What are some cool math theorems for example, benfords law. The purpose in presenting them, however, is not to surprise you with new mathematical facts, but. Ask, are there other times in math that you know order matters. On a traditional view, the primary role of a mathematical proof is to warrant the truth of the resulting theorem. Induction results in the prior section need only proof techniques that come naturally to people with a mathematical aptitude. The changing nature of mathematical proof pdf, berlin, new york.

Pure mathematics consists entirely of such asseverations as that, if such and such a proposition is true of anything, then. You may want to check out our algebra math jokes, calculus math jokes, geometry math jokes etc. By the time you finish this book, youll be proving math theorems with confidence. Randomness and mathematical proof scientific american 232. Jun 24, 2015 5 grade school math problems that are so hard, youll wonder how you ever made it to high school. John paulos cites the following quotations by bertrand russell.

In math, cs, and other disciplines, informal proofs which are generally shorter, are generally used. The first handout is for anyone learning to write up solutions to math problems in complete sentences. Proofs and mathematical reasoning university of birmingham. In math proofs demystified, bestselling math and science writer stan gibilisco provides an effective and painless way to overcome the intimidation most students feel when venturing into math theory. Mat231 transition to higher math proofs involving sets fall 2014 11 11. Nov 20, 2017 9 mathematical facts that will blow your mind and a delicious bonus many people believe math is boring, but in reality its anything but. Most of the proofs i think of should be accessible to a middle grade school student. An introduction to logic and proof techniques michael a. Basic order of operations requires multiplication before addition. We want to study his arguments to see how correct they are, or are not. Mathematical statements and proofs in this part we learn, mostly by example, how to write mathematical statements and how to write basic mathematical proofs. Not only does it cover all of the main topics for such a course, but it also discusses mathematical writing, which is key when it comes to making mathematical concepts clear. Writing mathematical proofs is, in many ways, unlike any other kind of writing.

Turner october 22, 2010 1 introduction proofs are perhaps the very heart of mathematics. Multiplying both sides of x1 by the positive number x, we see that x2 x. More than one rule of inference are often used in a step. This can occasionally be a difficult process, because the same statement can be proven using. In 2003, he proved thurstons geometrization conjecture. The 11 most beautiful mathematical equations live science. Math trivia, math jokes, math quotes we hope you enjoy our collection of funny math proofs, favorite math jokes and jokes about numbers. Some remarks on writing mathematical proofs uw math department. Many algebra proofs are done using proof by mathematical induction.

Mathematical works do consist of proofs, just as poems do consist of characters. The argument may use other previously established statements, such as theorems. More than 200 books, videos, software, posters, and research reports are available. The hundred greatest theorems seton hall university. Covers basics of truth tables and implications, as well as some famous hypotheses and. To demonstrate the power of mathematical induction, we shall prove an algebraic equation and a geometric formula with induction.

Basic proof examples lisa oberbroeckling loyola university maryland fall 2015 note. Simplicity being a source of beauty, selection of proofs into the second group is hard and, by necessety, subjective. Mathematical equations, from the formulas of special and general relativity, to the pythagorean theorem, are both powerful and pleasing in their beauty to many scientists. Advice to the student welcome to higher mathematics. Hence, these proofs are an important part of mathematics. There are four basic proof techniques to prove p q, where p is the hypothesis or set of hypotheses and q is the result. Becoming familiar with a new language can be a frustrating. Godels incompleteness theorems a set of axioms is complete if every sentence that can be written can be proved or disproved from axioms a set of axioms is consistent if there is no. Magnus university at albany, state university of new york preliminary version 0. An introduction to writing proofs, the basic types of proofs, and an introduction to important mathematical objects such as functions and relations.

This is a prove i came out with during a series of extremely boring lectures i had to attend. This section is a pause for an introduction to induction. Each theorem is followed by the otes, which are the thoughts on the topic, intended to give a deeper idea of the statement. If you are not familiar with with proofs using induction, carefully study proof by mathematical induction given as a reference above. To make it more interesting, we can even change the laws to create new systems and then study them. Introduction to mathematical arguments uc berkeley math. Like many of my students, perhaps you are asking yourself or me, why do i have to learn proofs. It is an interesting to observe that the converse of this theorem is false. We may have heard that in mathematics, statements are. So id like to know what mathematical proofs youve come across that you think other mathematicans should know, and why.

Mathematics teacher is a publication of the national council of teachers of mathematics nctm. Theres an absolute certainty in math proofs which makes mathematicians crazy about them. Math isnt a court of law, so a preponderance of the evidence or beyond any reasonable doubt isnt good enough. A transition to advanced mathematics, chartrandpolimenizhang, 3rd ed. The history and concept of mathematical proof department of. After experimenting, collecting data, creating a hypothesis, and checking that hypothesis. An indirect proof uses rules of inference on the negation of the conclusion and on some of the premises to derive the negation of a premise. Lee university of washington mathematics department writing mathematical proofs is, in many ways, unlike any other kind of writing. Math an introduction to abstract mathematics uci math.

A statement or proposition is a sentence that is either true or false both not both. He is already dead when the play begins, but he appears in the first scene in catherines imagination and returns in two later scenes, which flash back to earlier years. Proofs for a research audience are quite different from those found in textbooks. An example for such a new system is the notion of a group. As a beginner and far from being a mathematician there are two proofs that i have come across that i would say, for me, were symphonic capers of some areas of study showing what can be done with material youve studied. Standards addressed taken from nctm principles and. You will nd that some proofs are missing the steps and the purple notes will hopefully guide you to complete the proof yourself. How to minor or major in math at sjsu, the very short version handouts on writing math in paragraph style and writing proofs revised august 2008. These skills travel well, and can be applied in a large variety of situationsand in many di. The amount of detail that an author supplies in a proof. Sally cockburn, sharon williams, and dave foster10.

Pages xviiixix of my text mathematical methods in artificial intelligence on reading mathematics below why do we need proofs of obvious results below a possible taxonomy of proofs below. Sets we discussed in class how to formally show that one set is a subset of another and how to show two sets are equal. Easy math proofs or visual examples to make high school students enthusiastic about math closed. Unlike the other sciences, mathematics adds a nal step to the familiar scienti c method. A mathematical proof is an argument which convinces other people that something is true. Proofs are to mathematics what spelling or even calligraphy is to poetry. This book is an introduction to the standard methods of proving mathematical theorems. It is the proof concept that makes the subject cohere. Over the years, the mathematical community has agreed upon a number of moreorless standard conventions for proof. Proofs of mathematical statements a proof is a valid argument that establishes the truth of a statement. Proofs and concepts the fundamentals of abstract mathematics by dave witte morris and joy morris university of lethbridge incorporating material by p. Additional topics may be discussed according to student interest.

Since any element x in k is also in s, we know that every element x in k is also in s, thus k s. Crazy theorems and conjectures university of chicago. See more ideas about geometry proofs, teaching geometry and math lessons. The book is admittedly simplistic, after all, its target audience is the curious student who has little background in math and needs everyting spelled out at an elementary level. A proof is an argument intended to convince the reader that a general principle is true in all situations. We sometimes hear students speak of theoretical math, usually in a negative tone, to describe mathematics that involves theorems and proofs rather than computations and applications. Another bit of math you may remember from school is geometry, which is the part of math where doodling in your notes was the point. How can they be so easy and so not at the same time by noelle devoe.