Hey everybody, and welcome to Math 098!
My name is Lain Tomlinson. I'm one of the
math instructors here at Cumberland
University,
and these videos are just going to kind
of give you an overview
of every section as we go throughout the
course. We made these videos
to be supplemental to the course,
so always remember to go to your
instructor if you have any questions, but
we are going to provide a few examples
from each topic in the section. 
For Section 1.1 we're
going to be talking about the
introduction to real numbers,
and so the first thing that we need to
figure out is
what are the subsets of the real numbers?
So, we have this nice little box here
that's also in your textbook.
If we look at the smallest box inside
here we have what are called the natural
numbers.
Those are your accounting numbers they
start with 1 and they just increase, so 1,
2, 3, so on and so forth. Above that we have
what are called the integers. Now the
integers are represented by this bold
faced Z. The integers are your natural numbers-
your counting numbers-
their negatives (so if you've got one
you've also got negative one and so on)
for each of the natural numbers,
and then you also have zero.
Above that we have what we call our
rational numbers. Those are represented
by the boldface Q.
Your rational numbers are any number
that can be represented as the ratio of
two integers. So if you look at our
examples up here
we have negative three-fifths, that's
negative three over five, which are both
integers.
We have four which can be represented as
four over one,
and we also have 17 over 22 right? So
those are all represented
as a ratio of two integers or a fraction
with two integers.
On the other side kind of by itself we
have the irrational numbers.
Now these are numbers that cannot be
represented as a ratio of two integers. Another way
you can tell the difference between the
two:
if you make a rational number into a
decimal
it's either going to stop or it's going
to repeat,
right? So if you think a negative
three-fifths if we put that
in decimal form that would be negative
0.6. If we think of the number
one third, we write that as a decimal, it would be 0.333 with that 3 repeating
to infinity. For the irrational numbers
if you turn those into decimals
they do not terminate and they do not
repeat. So
pi is a good example of that, the square
root of 2 because it's not a perfect
square,
and if we put any of our irrational
numbers into a fraction form,
those are also still going to be
irrational. All of these numbers that
we've talked about so far
are part of the real number system. So
the real numbers
include all of these, and that's what
we're going to be working with
throughout this course. So let's do an
example: let's classify
some numbers and figure out what sets
they are in. One thing that we want to
point out is if
it is a natural number that means it's
also going to be an integer
a rational number and a real number so
whenever we're classifying these we want
to find the smallest subset that it fits
into
then we're going to list everything on
top of that. So let's start off with
negative 9.
So if we look at negative 9, the smallest
one that that's going to fit in is the
integers, right? It's not a natural number because
it's negative, but it is an integer.
So that means it's going to be an
integer, a rational number,
and a real number, and that's our answer.
If we look at the square root of four
-now this one's a little bit tricky
because can we simplify the square root
of four?
We can. The square root of four is equal to two, so when you see
two, we know that that is a natural
number. It's actually listed here as one
of our
examples. So this is a natural, it's an
integer,
it's a rational, and it's a real. We have
one more example, and that's
pi over three. Now as we mentioned
earlier on
pi is irrational. Since pi is irrational
even if it's in fraction form, it's still
going to be considered an irrational
number.
So this is the smallest subset that it
can go in, so this one is irrational
and real. That's our answer. The next
topic that we're going to discuss
is the real number line. The real
number line a lot of times you'll see it
looks something like this. It doesn't
have to have zero in the middle.
We move the scale around wherever we
need to, but
a lot of times you'll see zero in the
middle. As we go to the right you will
see that the numbers increase.
As we go to the left, we see that the
numbers decrease, so they become more
negative.
And so we need to be able to plot
numbers on this number line. It's really easy if we
ask to plot negative three. You just put a
closed circle right there on the tic
mark above negative three, but
what if we ask you to plot a fraction
like nine fourths? That's going to take
a little bit of extra work. There are a
couple of different ways to solve this
you can try to find the two integers that it's
between by looking for your multiples of four.
What I like to do,
though, is I say how many times does four
go into nine?
Four goes into nine two times with some
left over, right? So it's gonna be two
with one fourth left over if we divide
that, and so we know that it's going to
be in between two and three because of this
because you've got two and some change.
So we're gonna do two, and we're gonna
plot our dot right there, and that's
how we plot fractions, and this also
works on the negative side. You just have
to remember to put it between negative two and
negative three if it was something like
negative
nine fourths. Finally we're going to talk
about absolute value.
Absolute value is the distance from any
number to zero. So when we think about
distance, it's always going to be positive. We want
positive numbers when it comes to
distance. So if we put this in symbols
we have these two straight lines that
surround our value,
and that's how we denote absolute value.
So if your absolute value
is a positive number, if x is greater
than or equal to zero,
you're just going to write that number
down again. If x is a negative number, if
x is less than zero?
We're going to change the sign and we're
going to make it positive. So
let's do a couple of examples. Example a
here says the absolute value of negative
42.
well like we said it's the distance from
negative 42 to 0 so that's going to be
a positive 42. Example b's got a little bit of
extra work to it though.
You see, we have a negative on the
outside and a negative on the inside.
The absolute value only affects the
negative that's on the inside,
so this negative on the outside is just
going to hang around, and then we're
going to work this problem out. So the
absolute value of negative 70 is
positive 70, but our final answer is
negative 70. Finally, we have a fraction
the absolute value of negative
9/17. The same principle applies even if
it's a fraction
so this is going to be positive 9/17.
That's it for Section 1.1.
If you have any questions, remember to
reach out to your instructor, and keep on
keeping on!
