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.
Tutorial for Assignment 8
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Del curso dictado por Stanford University
Introduction to Mathematical Thinking
1298 calificaciones
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De la lección
Week 6
This week we complete our brief look at mathematical proofs
Conoce a los instructores
Dr. Keith DevlinCo-founder and Executive Director
H-STAR institute
Well, Simon's eight was really quite difficult, I think.
On the other hand, looking at the description on this forum, I think many
people found it difficult because it hadn't yet made this transition to doing
mathematical thinking. They were trying to, to do something that,
that wasn't really required. This was very definitely the, the case
with this one. Remember the, that the things I keep
repeating are, one of the essences of mathematical thinking is that you, you,
you, first of all, you ask yourself, what is this asking me?
What is, what do I know? What does it tells me?
What kind of objects is it talking about? And what do I need to do in, in, in terms
of, what's my, my target. So you have to stop >> You pause, you
reflect, you think about it. What you don't do, at least in the
beginning, is say, does this remind me of a problem that I've solved before that I
can just instantly apply that previous technique, because that can lead you in,
in, in completely the wrong direction. For example, the various words here, you
know? And in high-school, a good strategy was
look for key words, and try to map them into techniques.
But it really isn't a good strategy it, in, in terms of advanced mathematics,
okay? There are many questions, many problems we
have about perfect squares. The two are about the natural numbers.
Okay, so what? This isn't about the natural numbers, this
is about the integers. Completely different set of numbers.
So, well not a completely different set, it's a larger set of numbers.
So this is about the integers and the fact that some of the words are often used in
discussions about the natural numbers, well, so be it.
I mean words get used for all sorts of contexts.
Okay, so this is a question about the integers.
And if it's about the integers, then why don't we just take m equals n equals zero.
Then m squared plus m n, plus m squared equals zero, which is zero squared, and we
are done. Folks, this is not a trick question.
This is a question that gets at the heart of mathematical thinking, and the heart of
mathematical thinking isn't knowing a ton of techniques from mathematics.
The heart of mathematical thinking is, what do I know, and what do I need to do
with what I know? And what am I talking about.
Stop, slow down, reflect, think, mathematical thinking at this level is not
a sprint. All of the techniques you're taught in
high school, were taught in high school were very good for succeeding in high
school. And there's certainly leave you with a lot
of valuable skills. But we've been there, done that.
You know, if we've been out of high school, if we've mastered that.
Then we try to do something else. High school sort of teaches you how to lay
bricks. What we're doing now is seeing how can I
take all of those bricks, which are valuable things to have, and use them to
build a house. We're now going from being bricklayers to
architects. We certainly need all of those bricks, you
know, I'm definitely not knocking high school, right.
We're use this stuff all the time. But that was providing us with the basic
tools. Now we're learning how to make use of
these tools. Okay.
That's what mathematical thinking is about.
And sometimes we need a lot of tools and sometimes we can get by with something
very simple because we just ask ourselves, what am I trying to do now.
And we're not getting seduced into thinking that this is just another variant
of something we've met before. Think of every problem is a new problem.
Then you'll find that things get much more doable because you'll be focusing on
thinking not applying techniques. Okay that's the end of my sermon.
Well, it's not really the end of my sermon because that's what the whole course is
about. But let me just move on now to question
two. And in this case you probably haven't seen
anything quite like this before. So you have to start by asking yourself,
how could it happen that the answer being was a perfect square?
I mean, just how could that come about? What, what kind of thing must happen?
Well, let's just write it this way. How can we have mn+1 equal to p-squared
for sum p. Well at this point, one of those bricks
that I was taught in high school becomes really useful.
Because if mn plus 1 equals p squared, that means mn equals p squared minus 1.
And one of the bricks I had drilled into me in high school, was that p squared
minus 1 is P minus 1, P plus 1. So, what we're about to do is take
something that was drilled to us in high school and make clever use of it.
Okay? Because, if A minus P minus 1P plus 1,
then we could have M equals P minus 1 and N equals P plus 1.
In that case, p would be m plus 1. Now, m is the number we're given.
We're trying to find an n. And I'm just using this to say, what could
that n be? What must that n look like?
Go given an m, we'll be able to take p equals m plus 1.
And then, from this one, n equals p plus 2.
And then we found the n. That is m plus 2.
Oops! M plus 2.
Okay? P plus 1.
P is m plus 1 and n is p plus 1. So n is m plus 1.
So given the m, we've already found the n. So given n, let's just summarize it now.
Given m, take n equals m plus 2. Then mn plus 1 Equals m, m plus 2 plus 1
which is m squared plus 2 m plus 1 which is m plus 1 squared.
So we've found, we've, we've, we've, we've answered the question we've said that
given an m take n equals m plus 2 Then MN plus 1 is M plus N squared which is a
perfect square. Now if I hadn't gone through this.
It would have appeared that this was a rabbit out of a hat trick.
And unfortunately, it's a consequence of the way that mathematicians often write
their papers, that they don't include all of the reasoning, they just give you the
conclusions. This will be an easy simpler relatively
simple example. But if I had simply said, given m, why did
we take m plus 2, you would have said, how on Earth did he come up with that?
What made him think of that? You know?
Then, and, and I'll look as though I've got some kind of magical ability.
No I don't. I just went in and said, well, how could I
possibly get to that answer? And then it was just something I learned
in high school. Okay?
So, you, you know? One of the techniques is, just say, how
could I possibly get the answer that I'm asked to find?
Now, I'm couching this in terms of, of classroom questions.
But the same, the same issue arises when you're dealing with mathematical problems
in the real world. Look at the problem, what does it tell
you? What you have to do, how can I get the
answer that I'm, that I'm going for? This might be a simple, classroom type
example but it has many of the elements, a good mathematical thinking.
And in this case, it was how come that I'm supposedly arrive as soon as Alex, so the.
Start looking at that, it just drops out, okay?
Recognizing that was the entire key to the thing.
Once you're doing that. Straightforward.
Okay let's move on and look at the next one in question three.
Well, question 2 is another 1 of these things, that.
When you first read it, you think. Wow, this is asking for something really
deep. How could I possibly come up with a, with
a quadratic, all of whose values are composite?
How could I guarantee that all of the values are composite?
You know? Sounds like it's going to be really deep,
but not if you, if you sort of just take a breath.
Sit back and say, you know, how could it happen?
That numbers are composite. Well if they're composite, they have to be
the product of two, two other numbers. Okay, wait a minute.
Why don't we just take? If that's going to always be the product
of two whole numbers, let's make it the product of two whole numbers.
And lo and behold that is indeed of the form n squared plus bn plus c, where b and
c are positive. That's all there is to it.
We just wrote one down instantly. We didn't have to prove that one exists,
we just, we just ran that we did prove it, we did it by just writing one down.
And all we have to say to ourselves is, oh this sounds complicated but no, all it
means is that the values are products. And quadratics are products of things.
So we just write it. We explicitly make it a product of two
things. The value will always be the product of
two numbers, n plus 1 and n plus 2. N plus 2.
And these are all positive integers. So, we'll have maybe 2 and 3, or whatever.
So these are always greater than 1. So this thing is composite.
N squared plus 3n plus 2 is always composite.
That's it. Didn't involve any advanced machinery.
Just thinking about what the problem asked us to do.
Very similar with this 1. You know it sounds deep, well it's about
the Goldbach Conjecture. A problem that's been around for hundreds
of years, hasn't been solved, how on earth can we do anything with the Goldbach
Conjecture. Well, the answer is, just look at what it
tells us and ask us what we can do with it.
And for 1st of all, we observe that this is talking about numbers n Bigger than 5
is odd. Well, if n is an odd number bigger than 5,
then n is of the form 2k plus 3, where k is Bigger than 1.
I mean, normally, we think about numbers as being of the form, 2k plus 1.
But because I'm, I'm looking for prime numbers here.
I'm going to write it as 2k plus 3. And I can do that, because n is bigger
than 5. So for numbers bigger than 5, any odd
number bigger than 5 is of the form, 2k plus 3 where k is bigger than one.
Well, in that case, since 2k is bigger than 2, because k is bigger than 1, 2k is
an even number bigger than 2, so by Goldbach's Conjecture 2k equals p plus q
where p and q are primes. In which case, n, which is 2k plus 3, is p
plus q plus 3. The sum of three primes.
We're done. Actually, it wasn't difficult at all.
Once we sort of thought about what it says.
All of the complexity is in this unsolved problem, Goldbach's Conjecture.
If we assume Golbach's Conjecture, which we're allowed to, 'because it says, if
that's true, then every even number bigger than 5 is the sum of three primes, one of
which, in the case of our proof, is 3. That was it!
Okay? Actually it wasn't difficult at all, it
just looked as though it might be when we first met it.
So, another lesson we can learn is, don't get put off because something looks
complicated. Until you think about it, you don't really
know whether it's complicated or not. Okay.
Couple more on this on this assignment sheet, and then we're done with it.
The question five, the question 6, the last question, they're both induction
proofs, and so what I'm going to do is I'm going to do example 5, and then I'll leave
you to do number 6 if you've already tried it and failed and you came here looking
for an answer. You're not going to find it explicitly.
But hopefully, by, by watching me go through another example, namely, number 5,
you'll be able to go back and do number 6. Because these things are all very similar.
At least the ones I'm giving you are all very similar.
Not all induction proofs are similar. But these, the ones I'm giving you from,
from number theory are all very similar. Okay.
Now what the first thing you have to do is, is express this is for some kind of an
equation you know, with a formula in it. Because otherwise we're going to deal with
this expression, the sum of the first n odd numbers.
So we're going to have to write down a formula for the sum of the first n odd
numbers. So, what it asks us to do is show that 1
plus 3 plus 5 plus and then the nth one is 2n minus 1.
Now we have to show that equals n squared. Okay.
The hardest part of this whole thing is figuring out what the last term is.
Okay. So, that's what we have to prove and we're
going to do it by induction. And we taught to use by induction so, we
have to begin by looking at the first case.
For n equals 1, this becomes just 1 equals 1 squared, which is true.
So it's true for n equal 1. Okay, so now let's assume the result.
Let me call that star. So let me now assume star.
And now I need to prove the same, corresponding result with an n plus one in
place of n. So why don't I just add the next term to
both sides. The next term would be 2n plus 1.
So both sides of star, in which case, I get 1 plus 3 plus 5 plus, plus 2n minus 1,
plus 2 n plus 1. This is the first n plus 1.
Okay that's the first, n plus 1, odd numbers.
This is the next case in the induction. And now, now when I add, 2 n plus 1 to the
right hand side it becomes that. Wait a minute, I can now use another of
those bricks that I learned in high school.
Thank you to my high school math teachers. Because now, I can just get this one
straight out. This was drilled into me.
Boy, it's good to have these things at my fingertips.
That's n plus 1 squared. We're done.
It's the same formula with n plus 1 in place of n.
Wasn't difficult at all. In fact, all of the machinery I needed to
solve this one, I learned in high school. The one thing I didn't learn in high
school, not, at least, not very well, was how to strategize about a novel problem.
And that's what this course is about. This is, how do you take all of that great
stuff you learned in high school, and strategize, and use it, and get new
results, and think about new problems. But before you do that, you have to ask
yourself what is the problem I'm given, what do I know about it, and what am I
having to prove. And if you misread a sentence, you going
to end up doing the wrong problem. And there were a lot of, a lot of people
writing on the forums where what they were really seeing was, you know, I
demonstrated all of my high school skills in great virtuosity, and solved a problem
that I wasn't asked to solve. Well you know in this course there's very
little penalty for doing that, except you feel bad.
But you know, if this wasn't a course, if you were working for a large corporation
you might be out of a job or demoted or moved to a less interesting job.
So this is a low penalty experience for how to think mathematically.
Which is why these MOOC's are great. Okay?
Well, as I say, I'm going to leave you do question 6 and, and, and having seen
another example, I hope you're not put off by the formulas and the complexity in
question 6. It's really just logical thinking, okay?
Mathematical thinking. Okay.
So much for assignment 8. Seemed hard at the time, actually was
hard. But, hopefully it doesn't seem quite so
hard now. It's not hard, if you become a good
mathematical thinker, and that's achieving, that is what the whole thing's
about. Okay.
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