Up next in 10
We've already studied integers, natural and whole numbers. In this lesson, we'll now be introduced to a new set, the set of rational numbers. We'll learn how to write rational numbers as fractions as well as learn how to identify and classify them ( Section 3.2)
Chapters
00:00 Introduction
00:24 Introducing Different Numbers' Set
3:11 What Is A Rational Number ( Definition)
03:48 How do you write rational numbers as fractions
05:02 How to write mixed numbers and integers as fractions
06:52 How to write terminating decimals as fractions
13:40 How to write repeating decimals as fractions
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0:00
all right so we're going to start
0:02
section
0:04
3.2 and section 3.2 is rational numbers
0:08
I need you all to be quiet right
0:10
actually sit right there but before we
0:12
talk about I want to sit right rational
0:14
numbers what I want you to do is is I
0:16
want to talk about uh the set of numbers
0:19
to begin with right
0:21
can so go up there go behind no no no
0:24
not before my thing go behind child oh
0:27
yeah that so we have like you need to
0:30
first talk about the sets right does
0:32
anyone need paper me actually never mind
0:35
I my binder I need
0:41
ER I need to my water bottle looks
0:49
crusty and just and then we also have
0:51
decimals
0:54
right so we're going to talk about these
0:56
sets to begin with and then we're going
0:58
to go deep into lecture before we even
1:01
get into that
1:03
so first we need to talk about natural
1:07
numbers right when you hear the word
1:09
natural natural numbers okay natural
1:13
numbers begin at what at one so one 2 3
1:17
4 all the way to Infinity right those
1:20
are called natural numbers natural
1:23
numbers just natural the word natural
1:26
next is whole numbers now whole numbers
1:28
unlike natural numbers they begin at
1:30
zero okay 0 1 2 3 4 all the way to
1:35
Infinity right so those are whole
1:37
numbers okay what's the difference it
1:41
begin at Z whole numbers begin at zero
1:44
natural numbers begin at one okay and
1:47
then we have integers now the integers
1:51
they begin at negative Infinity so we
1:53
have let's say5 -4 -3
1:57
-21 0 1 2 2 3 4 all the way to Infinity
2:02
those are called integers okay those are
2:05
integers okay and then we have decimals
2:08
now decimals are all of these plus
2:12
decimal numbers
2:14
like
2:18
6.55 4 and all of that right 1 2
2:23
3.5 and all that are called decimals
2:26
okay and then the next set after these
2:29
set is the set of what rational numbers
2:33
rational numbers okay now natural
2:36
numbers are the easiest and integers is
2:38
what we've been used to like so far
2:40
we've talked a lot about integers and
2:42
then yesterday or like last week we
2:44
began to talk about fractions okay a lot
2:46
of you guys don't like fractions because
2:48
you don't like to deal with like lowest
2:51
common denominators and things like that
2:52
but it's not that hard okay it's it's
2:55
not it's really not that complicated so
2:57
I'm going to erase this real quick right
2:59
you need
3:02
so now today we're going to talk about
3:05
rational numbers and before we talk
3:07
about it we need to know what is a
3:09
rational
3:10
number rational right what do you hear
3:13
when I say rational what's the word key
3:14
word there rational ration or what else
3:17
ratio ratio ratio a ratio is
3:21
what a ratio is a fraction right so
3:25
basically by
3:27
definition a a rational number is a
3:30
number that can be written as a fraction
3:33
right a number that can be written as a
3:36
fraction is called a rational number
3:39
does that make sense a number that can
3:41
be written as a fraction is called a
3:43
rational number for example one one is a
3:47
rational number right because can one be
3:49
written as a fraction yes or no yeah how
3:53
would you write one one as a fraction
3:54
one one one over one excellent one over
3:58
one what about can you write zero as a
4:01
fraction no no yes we can you just said
4:05
it what is it Z is z no 0 over Z that's
4:09
not that's in determination in calculus
4:12
one Z no 0 over 1 0 over one you know
4:16
what 1/ Z is actually Infinity yes once
4:19
you get the calculus we're going to talk
4:21
about that right what about -3 can you
4:23
write3 as a fraction yeah so what is it
4:25
going to be I don't know probably like
4:31
33 over what over
4:34
what on right now 23 is already as a
4:37
fraction so we don't need to do that now
4:39
0.87 can also be written as a fraction
4:42
and we're going to learn how to do that
4:43
in a minute right we're not going to do
4:45
that just yet but we're going to learn
4:47
how to do that what about one and 1/2
4:49
can you write that as a
4:50
fraction yes or no what is what do we
4:53
call this
4:54
number um you start with an N mixed
4:58
mixed fraction right mixed fraction now
5:01
here's the thing now we're going to
5:02
learn how to write mixed
5:05
fraction as and integers as fractions
5:08
which we just started to do right so for
5:10
example I have
5:12
six and 16 how do I turn this into a
5:16
fraction from a mix what do I
5:20
do
5:22
yes 6 6 * 6 + 1 + 1 is 37 over six right
5:31
so that is 37 over 6 that's excellent
5:34
right what about -23 how would I turn
5:36
this into a
5:39
fraction put a one under put a one under
5:42
it right just put a one under it
5:44
sometimes the easiest side the hardest
5:46
to to do because it has to be
5:48
complicated no right what about four and
5:50
23 Carly how do I turn this into a
5:53
fraction it's a mixed fraction
6:00
um 4 by that's 12 to two so it's 14 143
6:08
right
6:09
like what about this um Sophie how do I
6:14
turn seven into a fraction so you just
6:16
put a one under put a one under there
6:17
right 7 over one and that's it right so
6:20
to turn mixed numbers into fractions
6:22
that's pretty easy stuff okay it's just
6:25
really not that complicated now it
6:28
becomes a little bit more complicated
6:29
now when you have like decimals right
6:32
and not just any type of decimal here we
6:34
have what you call a terminating decimal
6:37
right do you remember I defined that for
6:39
you guys last week a terminating decimal
6:41
is a number is a decimal that has like
6:43
terminating digit right meaning it's not
6:46
like
6:47
0.645 that it's limited 0.64 is two two
6:53
two digit after the decimal that's a
6:55
terminating decimal meaning it ends
6:57
somewhere right it's number repeating is
7:00
terminating does that make sense what
7:02
does the word terminate means when I say
7:05
terminate what does that
7:10
mean I'm talk the Terminator no terminat
7:13
is to finish right it's the price is to
7:16
finish something so this is a this this
7:18
number this decimal can we call a
7:20
finished number it's finished right
7:22
because the ends is 0.64 terminating
7:24
it's not on ending it's ending it's
7:27
ending somewhere does that make sense
7:29
now how do I turn this
7:32
0.64 into a
7:35
fraction put your thinking caps on and
7:38
figure out how to do this would you take
7:40
it and then you go one two all right and
7:44
then you put the dot there
7:46
uhhuh it's getting there right so 0.64
7:51
right every time I move the decimal what
7:54
I put under the fraction
7:56
one one place is what a 10 and two will
7:59
be what 100 100 right so if I want to
8:02
turn this into a fraction I'm going to
8:04
start from here 0.64 over right I start
8:08
with one and I said okay every time I
8:10
move this I'm going to add a zero every
8:12
time I move this I'm going to add a zero
8:14
so therefore now we have 0.64 turns into
8:17
what turns into 64
8:20
over
8:22
100 does that make sense do we all get
8:25
it any questions no right but
8:30
are we done not yet simplify because
8:33
that's the key word we have to simplify
8:35
because this is saying what write this
8:37
ter decimals in their simplest forms
8:40
that means you have to simplify the
8:42
fraction now that's what you have to now
8:44
remember what goes into 64 that goes
8:47
into 100 two 32 you can start at 3 right
8:51
two and then what do you have 50 32 over
8:53
50 do we stop here no we can go further
8:57
right if you know your multiplication
8:58
table this become
9:00
it walking apart but if you don't then
9:01
you have to go step by step right so
9:03
what goes into 32 that goes into
9:06
50 two two how do we know if the number
9:09
is divisible by two it's 16 and 20 right
9:12
16 and 25 right divide two and that 16
9:16
and 25 can I go any further or do I stop
9:19
here stop here right because there's no
9:21
number that goes into 16 that goes into
9:23
five now here's how you can tell if a
9:25
number is divisible by two any number
9:27
that ends with a zero or a even number
9:31
is divisible by two right anytime you
9:34
see a zero next to a number you know
9:37
that you can divide that by two anytime
9:39
you share a multiple of two you know
9:41
that you can divide that by two or a
9:43
even number any even number is divisible
9:45
by two right for three is easy also for
9:48
three if you have for example how do I
9:50
know if this numbers
9:54
here to know if a number is divisible by
9:57
two by three let this there the easy
9:59
thing that I learned a long time ago I
10:01
was your age probably younger right if
10:03
you add the digit in the number 7 + 2 is
10:06
equal to what uh 10 10 n nine because 9
10:11
is a multiple of three is divisible by
10:13
this number is divisible by two try the
10:15
same here 3 + 1 is 4 4 + 5 is 9 since 9
10:20
is a multiple of three this number is
10:22
divisible by 3 this is how you know and
10:24
if you want to know if a number is
10:25
divisible by five you just have to see
10:27
if it end where zero or five or multiple
10:31
of five okay so if you know these things
10:33
it will help you in solving these
10:35
problems now we have that now how do I
10:37
change this now we have 9.67 5 how do I
10:41
turn this into a
10:43
fraction you got it yeah so you what we
10:47
do you would move the dot to the end we
10:49
do we can do that or what else can we do
10:52
so if you move the dot to the end right
10:54
so if you start here
10:55
9675 what do we do put under 10 10 how
11:00
many times I move this three times three
11:02
times 1 2 three right so a th000 right
11:06
is
11:08
9675 over 1,000 now if you feel like you
11:10
don't want to do this there's multiple
11:12
options right you can also write this as
11:15
what 9 and then
11:18
675 over what 1,000 you can write this
11:23
way as well right this is also a mixed
11:25
fraction last year I remember when we
11:28
when we did the section some people like
11:30
it this way some people like it this way
11:32
I won't penalize you you can choose this
11:33
method or you can choose the other
11:35
method now we don't stop here because
11:37
we're going to try to do what we're
11:39
going to try to simplify right now it's
11:41
easier when you do this to simplify
11:43
because now I'm looking at it how do I
11:45
know what would be the first number that
11:47
goes into 675 that goes into 1,00 five
11:50
because we know that because this ends
11:52
with the zero and this end with five
11:54
they are both divisible by five right so
11:57
I'm going to do that I'm going to divide
11:58
just the fraction by 5 and I have my n
12:02
so 675 over 5 I don't know what that is
12:05
right what by, 5 you guys know
12:10
two what 200 right now what was 675 5 do
12:15
you have a calculator I have it on my
12:18
phone I need some help 675 over 5
12:24
right 675
12:29
be like divided by yeah divided by what
12:32
five that's it's 135 135 right that's
12:38
135 and then I can also div I can divide
12:41
this again by what five five again right
12:44
five on both sides so what do I get I
12:48
get would be 27 27 and this would be 200
12:54
/ 5 is 40 40 right and this is
12:57
37 now can I any further no there's no
13:01
number that goes into 37 that goes into
13:02
40 so therefore 9.6 75 turns out to be
13:06
what 9 and 37 over 40 all right you see
13:12
how we work this out okay so we're going
13:14
to keep moving on now we're going to
13:16
learn how to it's going to get a little
13:18
harder now so now how do you turn
13:20
repeating decimals repeating decimals
13:24
into
13:25
fractions we talk about non repeating
13:28
now we going to have repeating so how do
13:29
I turn this into a fraction last year I
13:32
remember when we did this it was a
13:34
little bit of
13:36
uh so how do you go from that to a
13:40
fraction
13:42
okay now how do I turn this into a
13:44
fraction in the simplest form what do I
13:46
do here what you move the dot over you
13:49
can move it put it under but you'll be a
13:51
little hard to do that right that's
13:52
that's a method to this smartness so put
13:57
so we're going to let like we're going
13:58
to call this like X right x =
14:01
0.6 66 dot dot dot right these and these
14:05
are the same thing aren't they me saying
14:08
this is the same thing it's not no don't
14:09
think this because it's 6660 means
14:11
nothing okay we not we not scared of
14:13
that we not are we scared of it no right
14:16
so 666 like that right we're going to
14:19
let x equals this number are we good on
14:22
that we can can we right so I'm trying
14:26
to turn this into that right these and
14:28
these the same I'm technically saying
14:30
the same thing okay so now what I'm
14:32
going to do is this what if I choose to
14:34
multiply this by
14:35
10 right 10x will give me what 10 x will
14:38
be just me timesing this multiplying
14:41
this number by 10 I'm going to get I'm
14:43
going to move this right it becomes
14:45
6.6 6 6
14:48
6 you agree with me on that right so now
14:52
if I do
14:54
10x - x which means this minus that what
14:59
do I get if I do 10 x - x what is that
15:02
going to give me
15:05
10 it gives you what if you do this
15:09
right watch here pay attention if you do
15:11
that everybody pay attention here
15:13
everyone right if I do 10x minus that
15:18
this portion is going to go away isn't
15:21
it that's going to cancel out that so we
15:23
going to be left what your
15:26
six right
15:29
and that gives me what 10 x - x what 9x
15:33
=
15:34
6 now can I solve for x yeah yes by
15:37
doing what div dividing by 9 right
15:41
divide by 9 divide by 9 so X is 6 over 9
15:45
and I can simplify this what goes into
15:46
six that goes into 9 three three if I
15:49
divide this by three we get
15:51
what two 23 so therefore 0.
15:57
66666 is technically
16:01
23 does that make sense I know this one
16:04
is a little bit complicated but if you
16:06
do more we going to get it yes what
16:08
cancels out the two six oh if I do this
16:11
minus that they have the same link here
16:16
right is a chain the chain is the same
16:18
so this and this chain are the same so
16:20
those are gone so it's basically saying
16:22
60 and that's just six that make sense
16:27
all right okay so now now what I want
16:29
you to do is we're going to work on a
16:30
couple of things here and then we're
16:31
going to close this or we can also learn
16:33
how to
16:35
classify um like numbers to see if a
16:39
number belongs into um let's say
16:43
rational irrational and all of that
16:44
stuff I don't know if I want to do
16:46
irrational numbers that's going to be in
16:47
like the next section so we can stop
16:49
here and then we're going to do some
16:50
work in the book okay no I like
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