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0:05
in this video we're going to talk about different types of numbers if you have to memorize the different types of numbers for a class then this is a great place to start if not then we're going to get on the same page by just quickly reviewing these different types of numbers so we start with the natural numbers which are sometimes called the counting numbers because we start at one and we just count up from there one two three four
0:27
five six so on and so on until we get to 100 a million a trillion and it just keeps going
0:34
if we want to add in zero we can do that and sometimes we write that little subscript of a zero
0:41
sometimes we call these the whole numbers the natural numbers with zero
0:46
that's not too important now the integers have all the natural numbers they have zero and they have these negative numbers so an example of an integer is something like negative seven
1:03
now negative seven is an integer but it's not a natural number
1:08
because there's no negatives in the natural numbers on the other hand something like 20
1:15
is both a natural number and an integer we can clearly count up to 20 in the natural numbers and if you look at the integers the integers include all the natural numbers 1 2 3 4 5 and so on
1:31
now what's an example of a number that's not an integer well how about something like a fraction
1:39
now when we move to the when we if we want to include all the fractions then we move to the rational numbers so for instance one-half is a rational number but it's not an integer now one half is a rational number because if you look one is an integer in the top and two is an integer in the bottom so when we make a fraction with integers that's a rational number now can you think of any examples of a number that is both of a rational number and an integer well five is both
2:20
why think about it 5 is definitely an integer and a natural number for that for that matter but it's also a fraction because 5 is the same thing as five over one now five is an integer and one is an integer so we can write five as a fraction of integers
2:45
so five is a rational number and an integer in fact all the integers are rational numbers
2:55
now what about things that are not rational numbers well the square root of 2 is not a rational number but it is something called a real number
3:07
so real numbers are all the rational numbers all the fractions and there are these mysterious things like the square root of 2 and the square root of 3 and numbers that we call irrational so for example five halves is a fraction so it's both a real number and a rational number but the square root of two is only a real number we'll talk more about the real numbers at the end of the video but
3:34
they're really really really important they're the numbers we'll be using most often in this course
3:42
finally we get to the complex numbers which are just real numbers with imaginary numbers things of the form a plus bi where a and b are real and i is the imaginary number square root of negative one
3:58
now if this all seems if the complex numbers right now seem mysterious and weird to you don't worry about it we'll talk about it later but let's revisit the real number line
4:08
for the real numbers so this is the real number line and it's just a continuous line of numbers you can see we have all the integers 1 2 3 and so on we also have everything in between
4:27
so for example right where that green dot is that's perhaps the square root of 2 because the square root of 2 is approximately 1.414 and it keeps going on forever so it's just a little less than 1.5
4:44
now the interesting thing about this is that if we were to draw a number line with only rational numbers so only fractions well we already talked about the fact that the square root of 2 is not a rational number it's not a fraction so if we drew a number line with only fractions there would be whole holes in it there would be a hole where the square root of 2 goes
5:12
so the real number line is really special it has this property that every number is there there's no holes or jumps or gaps and we'll we'll use the real number line in that sense that uh basically when we can think of it as all the numbers uh on that line the the whole line is continuous and filled with numbers okay see in the next video
5:44
we have two groups of marbles on screen five blue marbles and two red marbles and if i ask you to tell me how many total marble marbles are there some of you might write well there's five blue ones plus two red ones so there's seven marbles in total but some of you might start counting with the red and you'd say there's two red ones plus five blue ones which still gives you a total of seven marbles so the important thing here is that five plus two is equal to two
6:16
plus five it doesn't matter which order we add in because we're counting the same number of marbles
6:25
and this is a a really important property of addition it's called the commutative property of addition
6:37
and it says that if we have any two numbers a and b then a plus b is equal to b plus a so it doesn't matter which order we add two numbers in and we have something similar for multiplication
6:54
so let me write some dots on screen here and i want you to count how many dots there are on screen or tell me how many you don't have to count them but tell me how many dots there are
7:11
some of you might say well there's five groups of three here's a group of three and there's five of them so that's five groups of three or in other words five times three
7:29
and so there's 15 dots on screen
7:35
others of you might look at it a little bit differently
7:42
instead of seeing five groups of three maybe you see three groups of five
7:51
so here's a group of five and there's three of those so that's three times five
7:59
which still gives us 15 total dots on screen so again the important thing here is that 5 times 3
8:09
is equal to 3 times 5. and there's nothing special about five and three this works for any two numbers so this is the commutative property of multiplication
8:28
and we write a times b
8:33
is equal to b times a so given any two numbers you can multiply them in either order you like these two properties seem simple the commutative property of addition and the commutative property of multiplication and they are simple but they're really important so they'll be good to know okay see in the next video
8:58
in the last video we learned how addition and multiplication have this nice property called the commutative property now we're going to learn a new property that addition and multiplication both have and it's the associative property so addition is associative let's start with addition here so a plus b as a group plus c is the same thing as a plus the group b plus c so really it tells us something about how we're grouping addition where we
9:28
write these parentheses and really it's saying it doesn't matter where we write them because they're they're the same let's see an example 9 plus 3 plus 5. and we'll do 9 plus 3 as a group
9:45
well this is equal to 9 plus the group 5 plus three or three plus five
9:52
so nine plus three is twelve plus five and then on the other side we're gonna get nine plus eight
10:02
twelve plus five is 17 9 plus 8 is 17 and so we can see the associative property in accent action doesn't matter where we put the parentheses
10:12
the both sides are equal since it doesn't matter where we put the parentheses what we're going to do is just drop them the parentheses don't tell us anything interesting so we don't need them it doesn't matter where we put them so why put them anywhere
10:31
okay so that was addition multiplication is the same multiplication is associative
10:42
so we write that as a times b as a group times c is equal to a times the group b times c
10:56
so it's the same thing and let's see an example of this
11:06
five times three as a group is equal to five times the group three times two so five times three is fifteen times two and then on the other side we'll get five times six
11:22
and fifteen times two is thirty five times six is thirty so you can see that both sides are equal
11:30
if you're wondering if anything isn't associative in the next video we'll see that subtraction is not associative and we'll see how that works okay see you then
11:45
i want to talk to you about subtraction subtraction is kind of nasty it has it doesn't have very many good properties that we like for instance six minus two is not equal to two minus six why because six minus two is four and two minus six is negative four so obviously four is not equal to negative four so subtraction is not commutative
12:17
and the example we just gave is 6 minus 2 is not equal to 2 minus 6. there's other things wrong with subtraction for instance subtraction is not associative so an example of that would be 9 minus 4 grouped minus 1 is not equal to nine minus parentheses four minus one
12:47
and we can go through this just to prove it to ourselves nine minus four is five minus one
12:53
and then on the other side we have nine well four minus one is three so nine minus three those are not equal because five minus one is four and nine minus three is six
13:05
so subtraction is not associative
13:12
and there's another reason subtraction is not that great but first let's just write our example down so our example here was 9 minus 4 parentheses minus 1 or parentheses 9 minus 4 and parenthesis minus 1 is not equal to 9 minus parentheses 4 minus one
13:35
okay another reason that subtraction is not that great is because sometimes it can be confusing with the order of operations so for instance ten minus six plus two maybe we're not sure should we do subtraction first or should we do the addition first so we think of the order of operations parentheses exponents multiplication division addition and subtraction and that looks like we're supposed to do addition first
14:04
addition becomes or it comes before subtraction so this would be 10 minus 8 which would be 2. but that's wrong that is the wrong answer
14:17
what we forgot is that what the order of operations really says is then what is when you have only subtraction and addition left do everything from left to right
14:31
so this is really 10 minus 6 which is 4
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plus 2 which is equal to 6 and that's the right answer so the third reason subtraction isn't that great is because subtraction can be confusing with the order of operations if we're not careful
14:55
in our example for this one is ten minus six plus two
15:02
but all is not lost there's something we can do we can actually turn subtraction into addition because a minus b is an abbreviation for a plus negative b
15:17
so let's look at our first example how would that work
15:21
well 6 minus 2 is equal to 6 plus negative 2 and now since we're doing addition we know this is commutative so we can change the order so now we have negative 2 plus 6. notice in this example the negative sign stays with the two in our original example the negative sign went to the six that was the problem the negative side was on the wrong number
15:50
okay so when we think of subtraction as addition of a negative number we get commutativity back
15:59
let a little note about the notation here why do we put parentheses around the negative sign it's really because doing it that way is ugly and unclear and you might even confuse somebody
16:12
if you're if you're not very neat so instead what we do is we put parentheses around the negative sign so that everything is really clear
16:24
if we look at this it's very clear what we mean we mean six plus negative two and now anybody looking at that will know exactly what you mean okay so that's just a note about why we put parentheses around the negative sign let's talk about uh what happens it with associativity when we think of subtraction as addition of a negative number
16:52
well subtraction really does become associative again because we're turning it into addition so let's just look at our example here nine minus four from the parentheses nine minus four and parenthesis minus one is equal to nine plus negative four plus negative one and now this is all addition and since addition is associative it doesn't matter where we put the parentheses
17:23
we could put them around the nine plus the negative four
17:28
or we could put them around the negative four plus the negative 1. and why don't you try doing that at home see if you get the same answer you should because addition is associative
17:40
okay and finally let's look at our third example
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does this fix the problem so 10 minus 6 plus 2 if we think of this as 10 plus negative 6
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plus 2 now if we do the negative 6 plus the 2 first we would get negative 4 from that so 10 plus negative 4 which is equal to 6. so even though we did the negative 6 plus the 2 first we got the right answer okay so the important thing to take away from this video is that subtraction is not commutative and it's not associative but we can think of subtraction as addition of a negative number and then we get all those nice properties back okay see in the next video
18:38
well let's talk about fractions what information does a fraction convey
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so if i have the fraction one fourth what type of information does that convey
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well it it it conveys two pieces of information
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and they relate to uh it's always related to one whole thing so for instance we have a whole circle here you can think of this as a pie it's it's usually the easiest way to think about it
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so we have one whole pie the bottom number known as the denominator so let me write this out
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denominator
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the denominator tells us how many pieces to cut this pie into
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so in this case we're going to cut the pie into four pieces now it's important that you remember that the pieces are equal sized they're always equally sized pieces
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so there we go we've cut this pie into four equally sized pieces
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now the top number
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it oops its name is the numerator
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and it tells us how many pieces of of that pie we have so in our case we have one of the four pieces
20:36
okay if this one was not a one but was instead a two well then we would have two of the four pieces and so on and so forth if it was a three
20:59
well then quite simply we would have three of the four pieces
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oops sorry about that
21:10
and if it was a four well then we'd have four of the four pieces or in other words we'd have one whole pie
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okay
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fair enough what if we instead
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or well let's look at another example i should say what if we have now
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one third the fraction one over three well first of all how many pieces of pie are we going to have how many pieces are we going to cut this pie into
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we're going to cut it into three equally sized pieces so again the denominator
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the number on the bottom tells us how many equally sized pieces to cut our pie into
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how many pieces uh how many of these three pieces are we gonna have
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well we're gonna have one out of the three that's what the numerator tells us and you can do this for any numbers you could cut a pie for instance into
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eight pieces so you could have you could have five out of eight pieces so this would be a pie cut into eight equal pieces and you would have five of them
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and and again you could do that with any two numbers so
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the idea behind a fraction is is really really simple
22:58
things get a little bit more uh involved when we start talking about adding fractions together or multiplying fractions and things like that
23:09
so let's let's go ahead and look at doing things like that in the next few videos see you then
23:23
all right let's learn how to multiply fractions it's really the easiest thing you can do with fractions so what do we do all we have to do is multiply the top two numbers so one times three
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which is three and then divide by the bottom two the multiplying the bottom two numbers so four times five is twenty
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and that's it you multiply across the top and then you multiply across the bottom so if you can multiply numbers you can multiply fractions it's it's just that easy how about uh 3 7 times 4 11.
24:11
well hopefully you've already done this in your head by the time i finish writing 3 times 4 12
24:19
7 times 11 77 and that's our answer easy as pi okay what happens if we have just a single number times a fraction so let's do like 5 times 2 over 17.
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so now we don't have a fraction we have just a number times a fraction
24:45
well let me pull you aside for a second and just tell you that five is equal to five over one
24:58
and if you remember what we talked about that should make sense this tells you how many pieces to cut the pie into well if you only cut it into one piece you haven't really cut it at all it's just still one whole pie and you have five of them you have five whole things so you just have the number five you have five
25:19
one other way to look at this in case you're still curious is this is the same thing as five divided by one which you should know is simply five okay so you got that so so coming down here
25:36
if we wanted to we could rewrite this as five over one times 2 over 17
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and now we know how to do this 5 times 2 is 10 and 1 times 7 is 17 so
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5 times 2 is 10 1 times 7 is 17. so we could write this intermediate step where we write 5 as 5 over 1 but i recommend that you don't even do that you don't even don't even waste the ink doing that unless it really helps you because we can easily just know
26:17
that when we encounter a number times a fraction what happened here
26:25
a number times a fraction you just always multiply the number to the numerator for this reason okay that's multiplying fractions there's really nothing else to say
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it's the simplest thing you can do with fractions see in the next video
26:52
i want to talk to you about simplifying fractions or what's sometimes called canceling fractions so let's start with the basics 5 divided by 5 is what it's 1 right any number divided by itself is just 1. so negative one thousand one hundred thirty one divided by itself again is just one
27:14
what about fractions something like three sevenths divided by three sevenths
27:22
well a fraction is just a number and so we're dividing it by itself so we get one
27:29
what about zero zero divided by zero well this is not equal to one this is the only exception and it's because we can't divide by zero
27:42
if we tried to divide by zero the the universe would split open and your mother would cry it just would be terrible so 0 divided by 0 is undefined and that's the only exception
27:54
so we end up with this nice little rule if a is not equal to 0 then a divided by itself is equal to 1. so what does this have to do with canceling fractions well let's see a really simple example 2 divided by 4 is the same thing as 2 times 1 divided by 2 times 2.
28:17
well because we know the way fractions multiply that's the same as two over two times one over two
28:25
but two over two we just talked about that's just one so we're left with one times one over two or in other words one half and one times anything is just that thing so we're just left with one half
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so it took us quite a few steps but we simplified 2 4 to one-half let's see another example
28:48
six over twenty-seven well six is three times two and twenty-seven is three times nine
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and again we can split this up into two fractions
28:58
so we end up with three divided by three times two divided by nine
29:05
the three over three that just becomes a one so we're left with one times two over nine which of course is just a two over nine so now this is taking a lot of writing it takes quite a few steps so maybe we want to abbreviate this we know how everything's going to work out we're going to end up multiplying by one so instead of writing out those steps we're going to do something like this
29:29
let's see it let's see this again only with an abbreviation so 6 over 27 is 3 times 2
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over 3 times 9 and we know that 3 is going to cancel to 1 so we'll just cross it out
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and that's an abbreviation for those steps underlined in orange and we end up with 2 9.
29:57
okay so this can actually be really useful let's combine simplifying fractions uh what we're doing in this video with some things we've already learned
30:07
what if we had something like 3 times 3 over 47 times 2 over 13 sorry 13 times 3 over 47 times 2 over 13 times 47 well it looks like the 13 in the denominator there and the the 13 on the left maybe will cancel and the the 47 in the denominator and the 47 on the right might cancel so since multiplication is commutative something we've already learned we can re rearrange this any way we want so we'll rearrange it this way let's put the 13s together and the 47s together
30:48
also multiplication is associative which means we can group it any way we want so not let's not only put them next to each other but let's actually group them
30:59
and now we know the way fractions multiply that's something we've already learned 3 times 2 over 13 is just sorry 13 times 2 over 13 it's just 13 times 2 over 13.
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we get something similar over on the right hand side and now we can just cancel but let's since this is our first example let's just go through one more step so 13 times 2 over 13 times 1 right 13 is the same as 13 times 1 so we'll just put that 1 in there and 3 times 47 is the same as 47 times 3 and 47 is the same as 47 times 1 and now we're all set to cancel
31:39
so we're just left with 2 over 1 times 3 over 1 well 2 over 1 is just 2 and three over one is just three so this is two times three which is equal to six and it took us quite a few steps because i wanted to write everything out so we can see how this works but the the main point here is that
32:02
we can use all of our rules to to simplify that that multiplication
32:10
okay what if we have addition instead of multiplication
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can we cancel then something like this well the answer is definitely no doing something like that is wrong you don't want to do that why not well let's see if we cancelled we'd be left with one-third so instead of canceling let's see what happens if we just do this addition and try and simplify so 5 plus 1 is 6. 5 plus 3 is 8. and then we're left with 6 is 2 times 3 8 is 2 times four and now we know those twos cancel and we're just left with three fourths
32:48
well three-fourths certainly is not the same thing as one-third if we had canceled the fives originally we would have gotten one-third but of course three-fourths does not equal one-third so that's just an example of why this doesn't work out
33:05
it really it doesn't work out because we need to be multiplying by something over itself because when we're multiplying by something over itself we're really just multiplying by one and that's where the simplification comes from but when we have addition we don't have any multiplication so we're certainly not multiplying by one okay see you in the next video
33:31
i think it's pretty clear when we think about it that one whole pie is the same thing as having four out of four pieces of pie so here's a whole pie
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oops sorry about that fill that in that's the same thing as having all four pieces of this pie
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so here's the whole pie
34:00
here's four out of four pieces of pie it's the same you have the same amount of pie
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and this would be true no matter what number we chose so we could have four out of four pieces we could have
34:19
seven out of seven pieces of pie we'd still have the whole thing we could have a thousand and one out of a thousand and one pieces of pie
34:30
if we have as many pieces as we as the num we well let me put it this way if we cut the pie into a thousand and one pieces and then we take all thousand in one of them well we we have the whole pie
34:46
so that works for any number that works for any number and we can erase the the pi part
34:58
and just think of these numbers so one is equal to four over four just like just like we were just saying it's also equal to 1001 divided by a thousand and one or any number divided by itself so a number divided by that same number
35:25
okay knowing that what happens if we multiply
35:35
uh let's say we have 2 over 2
35:41
times 1 4. well we know how to multiply fractions that's easy it's just 2 times 1 which is 2 and then we have 2 times 4 which is 8. so let's see what two-eighths looks like so here's a pie cut into eight equal pieces and we're gonna take two of them
36:14
now let me go back
36:19
and do something here oops there we go
36:31
okay so all i wanted to do was show us again what 1 4 looks like
36:39
so one-fourth is is this exact same amount of pi as two-eighths
36:52
why is that
36:56
well because of this right here we multiplied by two over two but we just just just talked about the fact that that's the same thing as one this is the this is like all the uh two over two is the same thing as one so it's like we multiplied by one one times one fourth is equal to one fourth
37:27
so these two equations are giving us the same information really 1 4 and 2 8 are the exact same amount it's just that we kind of disguised one so to speak by mult by by calling it two over two but we just just talked about the fact that that really is the same thing as one
37:52
make sure you understand this this is a this is something that will come up throughout your math career the fact that you can multiply by a number over itself and that is the same thing as multiplying by one
38:12
okay i i hope this helps i'll see in the next video
38:24
all right here we have a very simple uh two very simple fractions that we're adding together we have one fourth one out of four pieces plus two fourths two out of these four equally sized pieces well it seems like when we add them together we should simply get three out of the four pieces one two three out of four pieces and that is exactly right that's exactly what we get we get three fourths
39:01
that that should make sense to and notice that when we added the fractions what happened
39:08
only the numerators added we did one plus two and the reason is just like we said we're adding one out of four places plus two out of four pieces well that should give us three out of the four pieces
39:27
okay let's just look at another quick example
39:35
so let's say we have here
39:39
5 out of 12 pieces and we're adding 3 out of 12 pieces
39:50
well it seems like if we if we take 1 2 3 4 5 of the pieces plus one two three we should just end up with eight out of the twelve pieces we should end up with eight of those twelve pieces
40:07
and indeed that's exactly what happens
40:11
when we add them together we just now have a total of 8 out of the 12 pieces
40:19
now the thing that's allowing us to add is the fact that we're talking about the same sized pieces notice that that that corresponds to the fact that the denominators are the same
40:36
so we had 12 pieces 5 out of 12 and 3 out of 12. so in other words we were cutting our hole into 12 pieces each time so every piece is the same size so we can add them up no problem
40:56
we have 5 out of 12 plus 3 out of 12 we get 8 out of 12. okay simple enough
41:03
what happens though if we have something a little bit different
41:12
what happens if we try and add something like
41:18
sorry about that one-fourth to one-third
41:26
so i'll write that out
41:31
one-fourth
41:34
plus one third
41:40
well visually it's not too hard to imagine
41:45
sorry again it's just going to look something like this
41:52
that's the total amount we have but what would that look like as a fraction
42:07
well it turns out we can't write this as a fraction the way that it is and that's because these pieces are not the same size
42:20
we have three different sized pieces and so this cannot be represented as a fraction
42:28
remember one of the things that we we stressed is that a fraction represents equally sized pieces
42:35
so for instance four equally sized pieces here we have three equally sized pieces
42:43
but in this last example we don't have equally sized pieces so we can't write it as a fraction
42:52
so what can we do
42:59
well maybe we can cut our pie into smaller pieces so that they're all the same size
43:13
and we can do that using multiplication
43:18
so i'm just going to do something i know you'll probably have questions about it
43:23
you know how did i know to do this or you know what's the technique we'll talk about that in the future but just follow along for a minute
43:32
let's say i multiply this by three over three and let me make some room for myself here
43:45
i'm just going to bring that up a little bit well we know how to do that multiplication it's just going to be 3 times 1 is 3 and 3 times 4 is 12. so this is
44:00
uh shoot i really want this to be smaller so i have more room i apologize so we have three over three times one over four three times one is three three times four
44:21
is 12. now we already talked about in a previous video when we multiply by a number over itself that's the same thing as multiplying by 1 so it doesn't change the fraction and you can see that three-twelfths is the same as one-fourth it's the exact same amount here we have three out of twelve pieces here we have one out of four pieces but we're still talking about the same amount of pie or the same amount of the whole
44:56
it's just there are smaller pieces and we have more of them okay same thing for one-third i want to do something similar what i want to do is i want to do one third times by four
45:14
over four and now we have four times one that's four and three times four that's twelve
45:24
and you can see that four twelfths is the same as one third let me show you
45:30
here now we have four of the 12 pieces but that's the same amount as one-third
45:38
they're smaller pieces but we have more of them
45:44
well hey look it what happened now now we have 3 out of 12 equally sized pieces plus 4 out of 12 equally sized pieces this should end up being 7 out of 12 equally sized pieces
46:01
and you can see that here if you count these up there's going to be 7 out of the 12 pieces
46:12
now this is the idea behind finding a common denominator so notice our denominator
46:20
ended up being the same a 12 and a 12.
46:27
and we needed to do that in order to add before we did that we had something like this we had one-fourth plus one-third we couldn't add those then we found this common denominator in other words we split the pi into the same same number of pieces and then we could add them
46:49
okay so this is the concept behind finding a common denominator we will talk about the techniques of finding a common denominator in the next few videos see you then
47:07
okay here we want to add two-fifths plus one-sixth and we've already talked about why you need a common denominator in order to add now we're gonna talk about how how do we get a common denominator one way is to find a least common multiple of five and six or i should say the least common multiple of five and six so what does that mean and how do we do that
47:36
well what we're going to do is make a list of multiples of 5 and multiples of 6.
47:48
so 5 times 1 that's 5 5 times 2 is 10 times 3 is 15 times 4 is 20 times 5 is 25 times 6 is 30 times 7 is 35 times 8 is 40. and maybe i'll stop there so these are the first eight multiples of five
48:08
now we'll do some multiples of six six times one is six times two is twelve times three is eighteen times four is twenty-four times five is 30. and hey we can stop because now we have a common multiple meaning these multiples are the same
48:30
and not only that but this is the least common multiple because if we look at this list it's the smallest number that's the same in both lists in fact right now it's the only number that's the same
48:45
okay so that means we want our denominator to be 30.
48:52
so what we've done so far is we looked for
48:56
we looked for an lcd a least common multiple or lcm the least common multiple and that's going to be our new denominator so we want 30 to be our new denominator denominator
49:20
okay well over here we did 5 times what well it was 5 times 6 was equal to 30. here
49:31
it was what it was six times five the same two numbers
49:38
okay how does that relate to our original problem well that means to get the denominator to be 30 we need to multiply by six six times five that was
51:36
okay let's try and add these two fractions together by finding a common denominator and the way that we're going to find the common denominator is by finding a least common multiple between the two denominators we're given so we have 25 and 15 we want to find the least common multiple so multiples of 25. 25 times 1 is 25 25 times 2 is 50 times 3 is 75 times 4 is 100
52:09
times 5 is 125 times 6 is 150 times 7 is 175 and i'll stop there so we got a 7 multiples
52:20
now let's do 15 times 1 is 15 times 2 is 30 times 3 is 45 times 4 is 60 times 5 is 75 and now we can stop because we notice that there's a 75 in both lists
52:41
and so 75 is a common multiple it's also the least common multiple because it's the smallest number that's the same in both lists lists
52:53
okay so now it's time to figure out how did we get 75 well we did 3 times 25 was equal to 75 and over here we did this is equal to 5 times 15.
53:13
okay so we know what we need to do now we're going to multiply this by 3 over 3 because we want this denominator to be 75 but we don't want to change the fraction so we have to multiply by 3 over 3 because that's the same as multiplying by 1. so we don't change the fraction we're just changing the way it looks okay over here we need to multiply by 5 over 5 because again we want this denominator to be 75 and 15 times 5 is 75
53:52
okay let me get rid of that and what's our result well 4 times 12 this sorry 3 times 4 is 12.
54:05
3 times 25 is 75 like we said plus 7 times 5 is 35
54:15
and 15 times 5 is again 75 just like we said and now we have a common denominator and so we can add these two fractions together so this is going to be 12 plus 35 which is 47
54:35
and we just keep that same common denominator 75.
54:57
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