Last topics of the week were divisibility.
I know that some people didn't get to divisibility this week, that's okay.
Here's an interlude to next week.
We say that m divides n and write m vertical bar n,
we read this as m divides n,
if and only if there exists an integer k
such that m k equals n.
So I told my class this “and only if” part,
we'll get to that next week, okay?
Usually, mathematicians don't write this,
they're generally sloppy
with their definitions in this sense, but
it should be “if and only if”. I'll try to do that throughout the course for myself,
but note that other books might not do this.
Otherwise, we write m does not divide n so
m vertical bar slash, this is
“slash not slash mid” inside LaTeX if you're interested.
That is when there's no integer k satisfying m k equals n.
That's just the definition of divisibility.
From here…divisibility is a very core concept in this course.
We kind of understand divisibility from an intuitive standpoint, right,
like 3 divides 9, negative 6 divides 36,
the only ones that are weird are things like 0 divides 0…
remember 0 times 3 is 0, right? So it satisfies the definition of divides, right?
Same with things like 3 divides 0
because 3 times 0 equals 0, right?
So just keep that in mind that 0’s can be in here,
just remember which way they go.
So if m is 0 and this is true, then n also has to be 0.
I'll let you think about that for a minute.
Finally bounds by divisibility, this was the last major thing we did.
This is the first look at an acronym in the textbook.
I also like to use DeMorgan's Law, DML, as an acronym but
BBD is the first one in the book, bounds by divisibility.
How do we read this? So we have a divides b and b does not equal 0 implies that
the absolute value of a is less or equal to the absolute value b.
So one thing I did here intentionally...
unlike before...one thing we do here intentionally is if
we don't specify the domains of these symbols,
then take them to be as big as they can possibly be.
So because we defined the vertical bar for integers,
we're gonna take a and b as integers. That's really what I want here anyways,
but I want to make this note because it is possible that I will forget
during class, or I will forget during…something…
during an exam, hopefully the questions are written crystal clear,
but you know mistakes do happen.
But the exam should be clear otherwise, take that to mean that.
Same with other textbooks as well. This is usually a golden rule.
What's the proof of this? Well okay, if I take two integers a and b
satisfying the two requirements,
then the definition of a divides b, remember that's all we really have, right?
So when all you have are hammers and nails, then you should probably nail hammers…
or hammer nails… I got that right…
You should probably hammer nails, right?
All you have is a hammer, use the hammer.
So all we have is the definition, so let's use the definition.
So we know there's an integer k such that a k equals b.
Since B does not equal 0, what do we know? We know that
K is non-zero and so we have the following
sequence of statements.
So the absolute value of a is less than or equal to
the absolute value of a times the absolute value of k.
that’s true because k is non-zero and k is an integer, so
the absolute value of k is at least 1,
so therefore this holds.
Remember absolute values, you can take them in,
so now I have the absolute value of a k.
That's just equal to b, as required.
Last thing I want to mention,
a lot of you probably already know this but, I do have
symbol and theorem cheat sheets on the Math 135 Resources Page
so, for example, I didn't mention here but
Z being the integers,
Z being the first letter of the word integers in German,
there exists is this backwards E,
we’ll see this a lot this upcoming week,
but I want to introduce it now just so that
it's a little bit easier to cope with these symbols.
You're gonna see a lot of symbols over the next couple of days,
the beginning of this course does have a lot of symbols, but
once you get them down it's a lot easier to write things mathematically.
There are theorem cheat sheets.
So the first couple of weeks the theorems aren't too too bad,
but I did include them.
After week five or six,
the theorems start to pick up very heavily,
and there's where the cheat sheets will probably be more useful to you.
But, you know, for the moment,
the symbol cheat sheet might be more useful.
Again though, there are theorem cheat sheets to help you out.
Okay, so again thank you very much for listening.
Hopefully this gave you a little bit of an insight as
to what we did, where we're going,
why these things are important.
Again if you have any feedback, email me at
cbruni@uwaterloo.ca
post on Piazza,
and that's it. Thank you very much.