Today we're representing
real numbers on a number line,
and then we're rounding
and finding absolute value, also.
These are all related topics,
because it all represents
how numbers work on number lines.
A number line is a line which numbers
are marked on increments,
used to illustrate simple 
numerical operations.
An increment is the equal spacing
between each tick mark on a number line.
Notice I said equal spacing,
and each number when 
you're marking on a number line
needs to be on its own tick mark.
So plot each group of numbers 
on the same number line.
So we have -5, -1, 2, and 3.
We can't just write -5,
-1,
2,
and 3, because the space
between here to here
is 4.
The space between here to here is 3,
and the space between 
here and here is 1.
That means our increments are not the same,
so we can't do that.
Instead, we need to mark it,
and find our increments first.
These are all integers, so we're gonna make
our increments worth 1.
So we'll start at 0,
and then go 1,
2,
3,
4.
It's a good practice to go one place further
than your highest number,
and one place further
than your lowest number, if you can.
Then we go -1, -2, -3, -4, -5, and -6.
So we can write, I don't want to make it 
too crowded, so that will be -2.
This will be 2, and then you can see 
then this is 4.
This is -4 and -6,
and now we want to mark up
where our numbers are
on our number line.
So we have -5
is right here.
-1
is right here.
2 is right here,
and 3 is right here.
We count 1-2-3.
-1, -2, -3, -4, -5.
It gets a little bit more complicated
when we have decimals.
So this starts at 3.05,
and we end on
our largest number is 3.2,
so here we're gonna start our increment,
our number line at 3.00.
And we're ending at 3.2.
Let's end at 3.2 just because otherwise
we'd have to go 
quite a bit further,
so we'll have 3.00
and 3.20.
This is the same as 3.2,
we just can add another 0
to make it easier for us to count.
In between, if we ignore the 3's now,
in between 0 and 20 is 10,
so we have 3.10.
Now we need
to split up in between 
each of these.
We're gonna have
to divide into 10s,
because this is in hundreds.
So we're gonna divide.
This would actually be 3.05
because if we ignore the 3s,
it goes 0, 5, 10.
This one would be 15.
And then 20.
So 3.00, 3.05, 3.10, 3.15, and 3.20.
Notice how I kind of started 
splitting things in half first.
That's because it's a lot easier 
for your eyes to see half
than it is to divide into 10s,
and you're gonna get
much better equal increments
when you divide into half first.
Now, we have this 3.14,
so we do need to divide
every single bar into fifths.
1-2-3-4, so then this would
be 3.01,
3.02, 3.03, 3.04, and 3.05.
So we're gonna keep going.
2-3-4.
Notice to get five spaces, 
I need four tick marks.
1-2-3-4.
1-2-3-4.
Now, we can start
putting down numbers
on our number line, 
so we have 3.1,
which is the same as 3.10,
so this is 3.1.
3.14, so 3.10, 3.11, 
3.12, 3.13, and 3.14 is right here.
So that's 3.14.
3.2 is the same as 3.20,
so we have 3.2 here,
and 3.05 is already marked
on our line,
but we will mark it again.
Notice when I marked what 
problems I was plotting,
I used the same values 
we had up here.
To mark
two odd numbers, 
all we have are two numbers here.
We don't have anymore numbers,
so we can actually
just mark 54
and 421.
Previously, this one, our increment
was equal to 1.
This one, our increment
was equal to 0.01.
This one,
our increment
is going to be 421–54.
So that's 367.
So here, our increment is 367.
So, our next number,
if we wanted to make another tick mark,
would be 788.
If we wanted to make
another tick mark this way,
our next number would be -323.
So, if you're only plotting two numbers,
all you need to do is
write those two numbers
because that can 
be what our increment is.
Increments can be any amount
you want it to be,
as long as every tick mark you have,
every tick mark you have, 
has the same increment.
So our numbers plotted
are 54 and 421.
Our increment was 367, so we can plot -323,
positive 54, positive 421,
and positive 788.
Now we wanna plot
some fractions on a number line.
We do have two negative numbers here
and two positive numbers,
so we need to have 0,
and then our denominators
are 4 and 8.
Eight is a multiple of four,
so we can divide
our number line into eighths,
just like our equivalent fractions.
1/4
is the same as 2/8.
When we multiply the denominator
by two and the numerator by two.
So, we're gonna divide things 
into eighths.
So our largest number is 3/8,
and our smallest number is -5/8,
so the numbers range from -1 to 1.
Now we're gonna start dividing.
We divide this one in two
and we get halves.
If we divide again, we get fourths.
Same thing here.
There's four spaces
between -1 and 0.
1-2-3-4.
At this point, because we have fourths,
we can actually plot
our denominators with 4 first.
Here's 1/4,
so there's 1/4,
and then we have -1/4.
So we can plot -1/4,
and now we still need
to divide into eighths,
so let's divide into eighths.
That's just dividing
the fourths in half again.
And then we can count.
3/8 we count 1-2-3.
And this is 3/8.
For -5/8, we go to the left 1-2-3-4-5, 
and that's -5/8.
So, for negative numbers
we go to the left,
positive numbers
we go to the right,
and once we have
the right number of divisions,
we can just count 1-2-3-4-5.
When we round,
it's a way to approximate numbers
to make it easier
 to perform operations.
Sometimes you round to the same place
value for multiple numbers.
Sometimes you only want to round
to one non-zero digit.
It depends on what operation
you're performing.
So this one says
we want to round
4,638 to the hundreds,
4,638 to the hundreds place.
So, if we look at a number line,
we have 4,600
and we have 4,700.
In the middle is 4,650
and 4,638 is somewhere in here.
That means it's closer to 4,600 
than it is to 4,700.
So when we round 4,638,
we're gonna use a squiggly equals
because it's approximately equal to.
We're rounding to the hundreds,
so it's closer to 4,600.
On the other hand,
let's erase.
When we round–
so this one was approximately
equal to 4,600.
Now we want to round
to one non-zero digit.
So, 4,638, our last non-zero 
digit is 4, the 4,
so we want to look at
is it closer to 4,000
or 5,000, because now 4,000
has one non-zero digit
and 5,000 has one non-zero digit.
4,638 is closer to 5,000
than it is to 4,000 on a number line.
So, when we round to one non-zero digit,
this one is approximatlely equal to 5,000.
So, it really does depend on where,
what we're rounding to.
If we wanna round 33,578
to the thousands,
we wanna look and see
if it's closer to 33,000
or 34,000.
So 33,500 is right in the middle,
but this is a little bit more 
than the middle,
so it's closer to 34,000.
So it rounds to approximately 34,000.
If we wanna look at one non-zero digit,
we have to look at
the largest digit we have,
so that's the 30,000.
So now we're looking to see 
if it's closer to 30,000
or 40,000.
33,000 is closer to 30,000
than it is to 40,000,
so this rounds to approximately 30,000.
When there's a 5,
so we wanna round 55
to one non-zero digit,
so we're looking at 50 or 60,
but 55 is directly
in the middle of 50 and 60.
When it's directly in the middle, 
you always round up.
So, 55 is approximately equal to 60.
When we wanna round 
0.0543 to one non-zero digit,
our one non-zero digit,
our largest non-zero digit is .05.
So we look and see is it 
closer to 0.05 or 0.06.
This is 0.054, so that means 
we're closer to 0.05.
So when we round,
that's approximately equal to 0.05.
Notice after we round,
we don't put any more zeros after there.
That has a lot to do
with how we work in science,
because that talks about how precise 
our measurements are.
So once we round, no more zeros
after a decimal point.
If we wanna write, round 0.543,
0.0543 to the thousandth,
that's our third.
The thousandth spot is this spot,
so now we wanna look to see
if its 0.054 or 0.055.
The 3 is closer to–
43 is closer
to 40 than
it is to 50.
So that's what it rounds to
is the 0.054.
The opposite of a number,
we do need a formal definition of this,
and we do talk about the opposite
of a number quite a bit.
It just flips the sign
to the other side of the number line.
The number flips over to the other side
of the number line.
In general, we change the sign
of the number.
So the opposite of 45 is -45.
The opposite of 36 is positive 36.
It flips from
this side of the number line.
So we have 0 and 45 over here.
It flips us over here to -45,
or for instance this one.
We have 0 and -36.
The opposite flips us to this side 
and makes it positive 36.
To write finding the opposite mathematically,
we write a negative
in front of the entire number.
So that means
we're writing negative of 45.
So that's -45.
That's what we just found out.
Negative OF -36 is positive 36,
and when we talk about negatives,
if they're in front of parentheses like this,
we're  gonna say negative OF 45.
Negative OF -36.
Absolute value is the distance from 0.
Distance is never negative.
So, whenever we have
an absolute value,
the result is
always gonna be positive.
Absolute value is written
with two vertical bars
surrounding the expression.
So for instance, this says
the absolute value of 4.
So, if we look on a number line
and we have 0 here
and 4 here,
the distance from 0 to 4
is 4, so the absolute value of 4 is 4.
This one,
we have
0 and -4.
The distance from -4 to 0
is also 4 units.
So the absolute value of -4 is 4.
This expression is saying the opposite
of the absolute value 4,
the opposite of what we found
for the absolute value of 4.
So the opposite of 4
is -4.
The absolute value of 4
we just found was 4,
so the opposite of 4 is -4.
This is saying the opposite,
opposite of the absolute value of -4.
We found that the absolute value
of -4 was 4,
and the opposite of that is -4.
When we compare numbers,
we can think about how
they're positioned on a number line.
With fractions, sometimes it's easier 
to make a common denominator.
Sometimes its easier to make
a  common numerator.
So first, we're gonna talk
about 5.67, 5.672, and 5.793.
If we think about how those 
are positioned on a number line,
we can say
that we have 5.670,
and we can add a 0
because we're not rounding,
and because these will
all have three places
after the decimal point.
Then we can see that .670
is actually smaller than .672,
so then our 5.672 would be next,
and then finally
our 5.793.
Our directions say to list them
from greatest to least,
so we're gonna say 5.793, 5.672,
and then 5.67.
Again, I'm writing them how
they were written in the original problem,
not how I modified 
to figure out what to do.
I'm gonna erase this,
and then we're gonna talk 
about this one.
This one, the numerators are all the same,
because remember 4 is the same as 4/1,
but the denominators 
are all different on this one.
So, let's start thinking
about how this is.
So, if we draw our fractions,
the first one is 4 whole amounts.
So that's 4.
If we divide,
let's try dividing into 3.
So we have the same size wholes, 
and then we're dividing into 3
and we're shading 4 of them.
Notice that's smaller
than this one.
The next, we have 4/7,
so we're gonna divide,
we're gonna take another whole
about the same size,
and we're dividing it into 7.
We'll check the 8 and see if we can't
see what's going on.
So here's 4/8.
Notice as we keep going,
these pieces are all getting smaller
as the denominator gets bigger,
and therefore, if we're shading
the same number of pieces
and the pieces are smaller,
as the denominator gets bigger,
the values get smaller.
So, to list this from least 
to greatest,
we have 4,
and then 4/3 is the next one,
then 4/7
because the pieces
of sevenths are smaller
than the pieces of thirds,
and then we have 4/8,
4/11, and 4/13.
This is different than this one.
This is the same size piece.
They're all divided into 4,
but now we have
a different number of pieces.
So here, we have the largest
would be 13/4
because we have 13
of something divided into 4,
and then 11/4,
and then 8/4,
and then 7/4,
and then 3/4,
and then 1/4.
So this is
number of pieces is the same.
Number of shaded pieces is the same.
The size is different.
So when the size is different,
we think about
the size of the pieces.
This is the
size is the same.
And this is the
different number of pieces,
different number of shaded pieces.
So then we look at how many 
pieces are there that are shaded,
so the size is different,
the number is different,
and that's how
you're gonna differentiate
if you need to look at the numerators
or the denominators.
