 
In this lecture,
we're going to talk
about how to visualize
numbers on a number line.
Understanding and having
a solid foundation
in how to plot numbers
on a number line
will allow you eventually
to be more proficient, when
we do rectangular
coordinate systems,
also solving equations
and inequalities.
So let's start with
plotting a line.
And we pick a point on the
line as our reference point.
So here, I have picked a
0 as my reference point.
It does not have to be 0.
It can be any number.
And then, based on that point,
once you plot a second point,
that determines your scale.
So in this particular case,
we are using increments of 1
as our scale.
And that will dictate how you
plot the rest of the numbers.
The convention is that all
the numbers to the right of 0
are positive, and all the
numbers to the left of zero
are negative.
So let's plot some more points.
If you keep going in
increments of 1 to the right,
you just keep going up by 1.
If you keep going to the
left by increments of 1,
then you're going to have
smaller and smaller numbers,
like negative 1,
negative 2, negative 3.
Once you place your
numbers on a number line,
it's also easier to then see
the hierarchy of numbers.
For example, here negative 3
is smaller than negative 2.
Negative 2 is smaller
than negative 1.
So 1,00 3 is smaller
than negative 1.
So you can see how
ordering of numbers
becomes easy, just by
looking at the number line.
So always remember that
numbers to the left
are smaller than
numbers to the right.
Numbers on the right are bigger
than numbers on the left.
All right.
So suppose I mark
this coordinate
as 0 and the next one as 5,000.
Once you have an
increment of 5,000,
that means your counting
everything in 5,000s.
So the increment doesn't
always have to be 1.
Or it doesn't have to be in
groups of 10s or 100s either.
You can also have
fractional increments,
which we'll see in a little bit.
So if I go in 5,000
increments, then that's
how my number line would look.
All right, let's take a look
at, what if my reference point
is negative 2,001.
And the next tick
mark I plot is 10,000.
Then, I'm choosing an
increment of 12,001,
and I will have to go
accordingly left and right.
So see if you can
plot a tick mark
to the left of negative 2,001.
You also don't have to have a
number line always horizontal.
You could have a vertical number
line and plot numbers there.
In fact, we see
vertical number lines,
in real life, lots of places,
like in measuring cups,
thermometers.
We use horizontal and vertical
number lines together,
so two number
lines perpendicular
to each other to plot points
in our rectangular coordinate
system.
So convention is that, if you
mark a 0, then to the left
is negative.
To the right is positive.
Also look at the number of tick
marks that are shown over here.
That can also dictate how
your scale is going to be.
So for example, if I
labeled this tick mark as 0,
then I have 1, 2, 3,
4, 5, 6, 7, 8, 9, 10.
I have 10 equal spaces between
0 and the next darkened tick
mark.
So that means I'm going
to be able to plot
decimal numbers on here.
So for example, I can make the
left-hand tick mark negative 1,
right-hand 1.
I can also do 0.1
and negative 0.1.
I can also do negative 0.01,
0.01, and 10 and negative 10.
So for example, if I
have 0.1 as my tick mark,
which do you think
this tick mark will be?
That will be a 1/10 of the
way in, between 0 and 0.1,
1/10 of 0.1.
That would make it what?
Good, 0.01, very nice.
So this will be 0.01, 0.02,
0.03, 0.04, 0.05, 0.06, 0.07,
0.08, 0.09.
And 10 would make it 0.1.
Does everybody understand?
All right, so what
if I had 0.01?
Tell me what this
tick mark will be.
See if you can write
it down on your paper.
What will this tick
mark be, right here?
So if this is 0.01, this
is point is 0.02 here.
So this would be 0.015, good.
All right, if this
was 10, then this
is 20, negative 20, and so on.
So we can keep
plotting like that,
if you know what the scale is.
So if you have, say, negative
3, you can think of negative 3
as a debt.
You owe something.
And the debt of 3 is
shown by plotting it
to the left of 0, 3 units.
Depending on the scale will
decide where negative 3
happens to be.
So that means we can plot our
decimal and rational numbers
on number line now.
So if I say that the
increment is 1 over 10,000
and I end up with this tick
mark to be negative 2.135,
then 10,000 later,
which is right there,
will be negative 2.134.
Why is that?
Because this is tenths,
hundredths, thousandths.
So if is thousandths, then
the next little tick mark
is going to be 10 thousandths.
And so if you have
tick marks and you're
plotting negative numbers,
look what happens.
So this is negative 2.134.
This will be negative 2.1342,
43, 44, 45, 46, 47 48, 49.
And that will become the 5.
So when it's negative numbers,
remember, the order switches,
because we're going to the
left instead of to the right.
If you took these
two tick marks,
so that would be negative
2.1358 and negative 2.1357.
I can't show 10
tick marks in here.
So I'm going to expand
that or zoom in.
If you have graphing calculators
or graphing software,
you can zoom in on
a smaller scale.
And so if I divide
this segment here
into 10 equal parts, which
is what I've done here,
then this tick mark will
be negative 2.13576.
So look.
The 7 is this.
And the 6 is 1, 2, 3, 4, 5, 6.
 
If you're not sure what's
happening, just look again.
So this tick mark here,
this is negative 2.135.
This would be 51, 52.
53, 54, 56, 57, 58.
 
So once you have that, you
can expand that and have
10 more tick marks
and then plot numbers
and figuring it out that way.
So if I plot numbers, and I
have fractional increments,
like say, 1/13.
Then, if I label one of my
tick marks as negative 10/13,
I'll have a negative 9, negative
8, negative 7, negative 11, 12.
So you see, the numerator
behaves just like integers
then.
The only difference is
that the bottom tick mark,
the denominator , you have
a denominator that you have
to carry with you.
So really, plotting fractions,
positive or negative,
is nothing different
than plotting integers,
as long as you understand what
the denominator stands for.
So that's an
important connection
that you want to make
with plotting integers
versus plotting
rational numbers.
So let's take a look at
this number line here.
So I have 0 here.
How do I know what
fraction represents
each of the tick marks?
You have to look at how
many divisions there
are between the two
dark tick marks.
So in this case,
we have 1, 2, 3, 4.
So that means each
tick mark is 1/4-- 1/4,
2/4, 3/4, 4/4 or 1,
5/4, 6/4, 7/4, 8/4.
So what do you think this
tick mark will be here?
So 9/4, 10/4, 11/4.
That's what this
tick mark will be.
So each tick mark
here is 1/4 of a unit.
All right.
How about this one?
So now, I'm making
my increment of 1/6.
So increment of 1/6-- so if
you look over here, from here
here, I have 24/6 is 4.
So this would be 4 and
1/6, 4 and 2/6, 4 and 3/6.
Now, you can see this
tick mark is also halfway
between those two, so
that's also a 4 and 1/2.
Or you can think of how--
1, 2, 3-- this is 3.
1, 2, 3-- so three
increments of 1/6 make 1/2.
Two increments of
1/3 will make what?
So look-- 1, 2, 3,
so that makes 1/3.
So two increments
off 1/6 makes 1/3,
because you can see that
you have 3 equal pieces.
So two is 1.
Here's another other two.
That's 2.
Here's another two.
That's 3.
So depending on how many
groups of the increments
you're taking, you can,
on this number line,
be able to plot
sixths, halves, thirds.
So basically, if you look at
denominators of different size,
as long as you have
a common denominator,
you can plot all three or
four or how many fractions
you have on the same number.
 
So now that we know how to plot
real numbers on a number line,
many of you have asked us, well,
what about complex numbers.
We introduced
complex numbers, so I
think it is important
that we tell you
quickly how to plot it.
It's not very important
for this particular class
that you know how to
plot complex numbers.
But it's not that hard either.
It uses number lines.
But now, because we have a real
part and an imaginary part,
we need two number
lines to graph that.
The convention is that
the horizontal axis
is the real axis.
The vertical axis represents
the imaginary axis.
You can think of complex numbers
as points in the complex plane.
Or you can also plot
them as vectors.
A vector is something that has
a magnitude and a direction.
And the direction
of it goes from 0,0
to the coordinates of
that point right there.
So if I have a number,
say, negative 5 plus 3i.
The 3 represents
the imaginary part.
The negative 5
represents the real part.
So negative 5 on the
horizontal number line,
because that's my real
axis-- so negative 5,
and then 3 on the
imaginary axis,
which is my vertical
axis, so 3 up.
So negative 5 to the left
and 3 up-- so that point
there is your complex point.
Or you can also think of this
vector as a complex vector.
So there are two
ways to plot it.
So what do you think, if I
change this, let's say, to a 7.
So now, I'm going at 7.
Let's change the
3 to a negative 3.
So 7, negative 3
will be down there.
So that's how you
plot complex numbers.
 
