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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.

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