Because the real numbers include both the
rational numbers and the irrational numbers
sometimes it can be kind of a challenge to
show or express all the real numbers at once.
Especially in any sort of like logical order
because we have fractions and decimals and
whole numbers and sometimes its hard to tell
what's bigger than what. And to show or layout
these numbers in any meaningful sense can
be a challenge so one way that's a great way
to show the real numbers is on whats called
a number line specifically
a real number line we use these all the time
in algebra. So it would be good for us to
get comfotable using these guys. So basically
it looks like this, basically you have zero
in the middle of your number line and then
all the positive numbers are listed to the
right of zero, and all the negative numbers
are listed to the left of zero. And just as
points of reference, we usually put tick marks
on all the integer values. So we'll have zero,
then we'll have one, and then two, and then
three, four, five, six, etc... And then that's
not saying that these are the only numbers
on the number line, these are just simply
points of reference. And to the left of zero
we would have negative one, negative two,
negative three, negative four, negative five,
negative six, etc... And it extends out beyond
six, but at some point you just run out of
space and so if you see these arrows on the
end right here, that indicates that the number
line continues out father than I am really
drawing. And again just to be clear, I am
not saying that these are the only numbers
on the number line. There are numbers here,
and here, and everywhere. The ticks marks
are just meant to be simply points of reference.
So if you want plot a number on a number line
that's showing a reader where it is in relation
to the other numbers on the number line.
For example, if we plotted a number at five,
we would go out to five, we would put a dot
and then we plotted five. We've shown the
number five on the real number line especially
showing where it is in relation to other numbers.
For example I know that it's greater than
zero, but its less than six because its after
zero, but is before six as you read the number
line from the left to the right. So just for
some practice, let's practice plotting a few
points. So I've labeled these points A, B,
C, D, and E. Let's quickly just go through
and plot them real quick. To plot the point
A, we would go to three, so we'll start at
zero on the number line, and would go right,
one, two, three units. We go to the right
because three is a positive value, if we want
to plot B so this is A right here. So if we
want to plot B we'll go left two and a half
units that's negative two point five.
So we will start at zero and go left one,
two point five, that's half a unit. So you
can certainely plot points between tick marks.
So this would be B. C is a zero, you can have
points plotted at a zero that's totally fine
as well. Sometimes you will get fractions
like thirteen-halves. Thirteen-halves if you
just use long division or a calculator and
divided thirteen by two, you would get a decimal
of six point five. Six point five is halfway
between six and seven. So you would go to
the right six and a half units and this would
be where D is, and then E, you can even have
very intricate decimals like negative four
point eight seven one now this kind of strains
the limits of capabilities. There is definately
a point - negative four point, eight, seven,
one, on the number line. But the best we can
really do with just pencil and paper is that
we'd know this is just shy of negative five.
So I know it's between negative four, negative
five, probably a little closer to negative
five, because negative four point eight seven
one is a little closer to negative five than
it is to negative four. So about the best
we can do just eyeballing it is putting it
just short of negative five, and we will call
that point E. So anyways the real number line
we use this all the time to plot real values
and especially
to show them in relation to one another.
