Integers and Rational Numbers

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Integers and Rational Numbers

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Let's do some integer and rational number problems.
So this first one here,
the tick marks on the number line
represent evenly spaced integers.
And just in case you forgot what integers mean,
that essentially is the counting numbers,
including 0 and the negative counting numbers.
So 1, 2, 3, 4, those are all integers.
0 is an integer.
Negative 1, negative 2, negative 3,
and so on and so forth.
Those are all integers.
You might imagine the whole numbers,
those are also another way of saying the integers.
So they say find the values of a, b, c, d and e.
So these are evenly spaced, and this is 0 right here, and this is 21.
Let's see how many tick marks it takes us to go from 0 to 21.
We have one tick mark, 1, 2, 3, 4, 5, 6, 7 tick marks
So 7 tick marks take us to 21 and they're evenly spaced--
they tell us that -- evenly spaced tick marks.
So if 7 of these take us to 21, there must be 3.
Each must be a jump of 3.
This must be 3, 6, 9, 12, 15, 18, and then bam, 21.
And the way I figured that out, I said 7 tick marks gets us to 21,
21 divided by 7 tick marks is 3.
So each of those increment by 3,
so this goes to 0
and then this must be 3 to the left of that.
So it's negative 3 right there.
So what are a, b, c and d? a is negative 3.
a is equal to negative 3.
Right there, that's a 3.
b is equal to 3 right there.
c is equal to 9 right there.
d is equal to 12, and e is equal to 15.
Not too bad.
All right.
Determine what fraction of the whole
each shaded region represents.
All right. Well let's see.
Here we have 3 sections of this pie.
So in our denominator, which is the bottom of our fraction,
we'll write there's 3 possible pieces,
and only one of the possible 3 sections is shaded.
So 1/3 is shaded. So this first one is 1/3.
The second one is how many sections do we have,
and all the sections are equal sized,
so we have 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 sections.
How many are shaded?
1, 2, 3, 4, 5, 6, 7.
So it is 7/12.
That's what this shaded region represents.
And then here, let's see, how many squares do we have?
We have 1, 2, 3, 4, 5, 6, 7 along this side.
And then we have 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 along that side.
So we have a total of 70 squares, 7 times 10.
And let's count how many are shaded in.
Well this is going to be 7 right there.
This is 7 right there, that is 7 right there, that is 7 right there.
So let me add this up.
So so far I have 28, that's 7 times 4.
28 plus, let's see, I have 2-- this is 2, 2, 2, 2, 2, 2, 2 and 2.
So how many 2's are there. That's 1, 2, 3, 4, 5, 6, 7, 8 2's.
So that 8 times 2, that's 16.
So 28 plus 16.
28 plus 10 is 38 plus another 6 is 44.
So we have 44 of the 70 squares filled in.
And we can write this rational number in simpler form.
We can simplify it.
Let's see, the numerator and the denominator both are divisible by 2.
So this is equal to 22 over 35.
And that's about as simple as we can get.
This is divisible by 2 and 11.
This is not divisible by either 2 or 11. So there you go.
That's the fraction in simplest form.
Problem 3.
Place the following sets of rational numbers in order
from least to greatest.
So let's do part a.
I might not do all of these for the sake of time.
1/2, 1/3 and 1/4.
So if you think about a pie,
going on, if you take a pie and you divide it into just 2 pieces.
1/2 is half of that whole pie.
That right there is 1/2.
This right here, 1/4, 1/4 is--
you just take the same pie --1/4 is just that.
So clearly 1/2 is greater than 1/4 or 1/4 is less than 1/2.
I'll write it like that. And 1/3 is right in between.
If you did 1/3-- it's maybe harder to visualize here --
but if this is 1/2, if we divided this into thirds,
1/3 is less than this.
So if we divided this into thirds, 3 equals sections,
1/3 would be just like that right here. 1/3 is more than that.
OkŁŹThere's multiple ways we could visualize this.
So 1/4 is less than 1/3, which is less than 1/2.
In general, the larger your denominator the smaller the fraction.
I have only one 1 of 4, that's less than 1 out of 3,
and that's even less than 1 out of 2,
which would be even less than 1 out of 1.
So that's our first one. We put that in order: 1/4, 1/3, 1/2.
Let me just do part to c, just for the sake of time.
So we have 39 over 60, we have 49 over 80,
and we have a 59 over 100.
So the easiest way to do this, if you had a calculator
you would actually just divide
each of these numbers out and you would get a decimal
and it's very easy to compare decimal numbers.
But let's if we can do this a little bit more artfully.
So we can approximate it.
We could say that this is a little bit less.
So 39 over 60, it's almost--
I'll make those squiggly equal signs for almost --
it's approximately 40 over 60, which is equal to 2/3.
So this is almost 2/3. A little bit less than 2/3.
This right here, 49 over 80, it's almost 50 over 80,
so it's almost 5/8.
And then this right here, 59 over 80-- 59 over 100, sorry.
It's almost equal to 60 over 100, which is equal to 3/5.
And just to get a sense of,
at least now we might be able to do it in our head,
2/3, that's approximately equal to 0.66 repeating.
You could say 0.667, roughly equal to that.
That's the 2/3 right there.
Let me do it in that color so you know what I'm talking about.
So this is approximately 0.667, not exactly, we're estimating.
This right here, 5/8--
So the way I always think of eighths is 1/8 is .125.
So 5/8-- Let me just divide it right here, let me just do the work.
So if we take 8 goes into 5, put some decimals right here,
8 goes into 5 zero times, 0 times 8 is 0.
5 minus 0 is 5. Bring down this 0.
8 goes into 50 six times. 6 times 8 is 48.
50 minus 48 is 2, bring down a 0. 8 goes into 20 two times.
2 times 8 is 16. Subtract, you get a 4.
Bring down this 0.
40.
8 goes into 40 five times. So this right here is 0.625.
Not exactly, it's a little bit less than this.
This is a little bit less than that.
And 3/5 is pretty straightforward.
3/5 is the same thing as 6/10, which is approximately equal to 0.6.
So this is a little bit less than 0.6,
this is a little bit less than .625,
this is a little bit less than .667.
So if you put them in order, this is the smallest,
then this one, then that one. So let me write it down.
So the smallest is 59 over 100, followed by 49 over 80,
followed by the largest, which is 39 over 60.
Which might be a little unintuitive because this
actually has the smallest number in the numerator,
but it also has the smallest number in the denominator.
So you actually had to work it out.
They're actually reasonably close to each other.
Let's do problem number 5.
Find the opposite of each of the following.
And when they say opposite, I think they mean find the negative,
or find the reflection on the number line.
So we essentially just have to
multiply all of these numbers by negative 1.
So for part a, the opposite of 1.001 would be negative 1.001.
For part b, the negative of 5 minus 11-- well we could write it --
it's going to be negative 1 times-- Well, what's 5 minus 11?
5 minus 11 is negative 6.
This number right here is negative 6.
So the opposite of negative 6-- negative 1 times negative 6 --
is going to be equal to 6.
Part c.
x plus y.
The opposite's going to be negative 1 times x plus y,
or we could have just written negative times x plus y
You distribute the negative 1.
This is equal to negative x,and then negative 1 times plus y minus y
Finally, part d.
Negative 1 times x minus y.
I'm just taking the negative of everything.
So I'm taking the negative of this,
and so this is going to equal to negative 1 times x, negative x.
Then negative 1 times minus y or negative y, that's plus y.
Or we could write this as y minus x.
And that's actually a neat thing to remember later on
when you're manipulating things algebraically.
The negative of x minus y is y minus x.
We just did that right there.
The negative of a minus b is b minus a.
You just swap the two when you take the negative,
and we just proved it for you right there.
All right.
Let's do a couple of these.
Simplify the following absolute value expression.
Absolute value is essentially telling you the distance from 0.
So for example, if that's 0, if I have the number 4.
How far is 4 from 0? The absolute value of 4?
Well it's 4 from 0. So you just say it's 4.
So the absolute value of positive numbers is just the number.
Let's say I had negative 3.
How far is negative 3 from 0?
Well, I don't care whether it's to the left of the right of 0,
it's just 3 away from the 0.
It's 3 to the left. So this is going to be 3.
So one way to think about absolute value is,
whether it's positive or negative, it becomes positive.
It's that far from 0.
So let's do these.
So 11 minus the absolute value of negative 4.
This is equal to 11 minus--
the absolute of negative 4 is just 4, which is equal to 7.
I'll do every other one of these problems.
The absolute value of negative 5 minus 11.
So that'll be equal to the absolute value of
negative 5 minus 11 is negative 16.
The absolute value of negative 16.
the absolute value of negative 16 is just 16.
It's 16 to the left of 0.
You could view it that way, you could just get rid of its sign.
And then finally here, let's do part e.
The negative of the absolute value of negative 7
is equal to the negative of--What's the absolute value of negative 7?
It's just 7.
So it's equal to negative 7.
This right here is 7.
Anyway, hopefully you found that useful.