Tutorial for Assignment 1

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En provenance du cours de Stanford University

Introduction to Mathematical Thinking

1062 notes

Stanford University

Introduction to Mathematical Thinking

1062 notes

Learn how to think the way mathematicians do – a powerful cognitive process developed over thousands of years. Mathematical thinking is not the same as doing mathematics – at least not as mathematics is typically presented in our school system. School math typically focuses on learning procedures to solve highly stereotyped problems. Professional mathematicians think a certain way to solve real problems, problems that can arise from the everyday world, or from science, or from within mathematics itself. The key to success in school math is to learn to think inside-the-box. In contrast, a key feature of mathematical thinking is thinking outside-the-box – a valuable ability in today’s world. This course helps to develop that crucial way of thinking.

À partir de la leçon

Week 1

START with the Welcome lecture. It explains what this course is about. (It comes with a short Background Reading assignment, to read before you start the course, and a Reading Supplement on Set Theory for use later in the course, both in downloadable PDF format.) This initial orientation lecture is important, since this course is probably not like any math course you have taken before – even if in places it might look like one! AFTER THAT, Lecture 1 prepares the groundwork for the course; then in Lecture 2 we dive into the first topic. This may all look like easy stuff, but tens of thousands of former students found they had trouble later by skipping through Week 1 too quickly! Be warned. If possible, form or join a study group and discuss everything with them. BY THE WAY, the time estimates for watching the video lectures are machine generated, based on the video length. Expect to spend a lot longer going through the lectures sufficiently well to understand the material. The time estimates for completing the weekly Problem Sets (Quiz format) are a bit more reliable, but even they are just a guideline. You may find yourself taking a lot longer.

Tutorial for Assignment 12:30

Tutorial for Assignment 29:21

Tutorial for Problem Set 19:41

Rencontrer les enseignants

Dr. Keith Devlin
Co-founder and Executive Director
H-STAR institute

We're in assignment one, most of the questions are just there for

you to think about and to discuss with your fellow students.

And I'm going to leave you to do that and sort them out on your own,

discuss them on the forums or with whatever group you've put together.

And please do put together a group.

What I will do is say a little bit about question 8.

Well, question 8 is related to Euclid's proof that there are infinitely many

primes.

Another crucial point in that proof,

we looked at this number where p1 though pn enumerates the first n primes.

We multiply them together and we add 1.

And I mentioned during the proof that this number is not always prime.

And question 8 asks how would you prove that it's not always prime?

And the answer is you have to find a number, n,

such that when you do multiply the first n primes and add 1, it's not prime.

Well, how would you go about doing that?

Well, the most obvious way is you start searching.

So you would start by looking at the first two primes, say,

multipy them together, add 1, you get 7.

Well, that is prime.

Try another one.

(2 x 3 x 5) + 1.

That's 31.

That's also a prime.

Let's try another one.

(2 x 3 x 5 x 7) + 1.

That's 211.

That's also a prime.

At this point you're probably starting to lose heart,

but if you keep on just two more primes 2, 3, 5,

7, 11, 13 multiplied together and add 1.

You end up with 30,031, which is not prime

because it's a multiple of 59 and 509.

Okay?

So here we've got an example of a number of this form which is not prime.

And in order to show that not every number of that form is prime, all you have to do

is find a single example of a number in that form which isn't prime.

And we've done it.

So, that's the answer to question 8.

That's how you prove that these numbers are not always prime, okay?

Well, that's really all I wanted to say about assignment one.

Assignment two has a little bit more meat to it in a sense

that there are more things that I'll need to show you.

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