Math Antics - Decimal Arithmetic - Free Educational videos for Students in K-12

Math Antics - Decimal Arithmetic - By

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00:03	Uh huh . Hi I'm rob . Welcome to Math
00:07	Antics in this lesson . We're going to learn about
00:10	decimal arithmetic . But before we get started if you
00:13	don't already know how to do multi digit arithmetic with
00:16	regular whole numbers . Be sure to watch our videos
00:19	that covered those subjects first . That's really important because
00:22	I'm just going to show you how you can modify
00:25	the procedures that we already learned in those videos so
00:28	that they work for decimal numbers . So if you
00:30	don't know how to do those procedures already , this
00:32	video won't make very much sense . Specifically . You
00:36	should make sure you've watched the videos about multi digit
00:38	addition , subtraction , multiplication and long division . If
00:43	you know how to do the problems in those videos
00:45	then decimal arithmetic won't be too hard . That's because
00:48	the procedures for decimal arithmetic are basically the same as
00:52	they are for whole numbers but there's a few important
00:54	differences that you need to know about and that's what
00:57	I'm going to show you in this video . Are
00:58	you ready ? Let's start with an easy one multi
01:01	digit edition when adding multi digit whole numbers . The
01:05	key was to stack the numbers up so that the
01:07	ones place digits line up in a column which ensured
01:10	that all the other number places lined up in columns
01:12	to . Then you just add up the digits in
01:15	each column starting with the ones place and working to
01:18	the left . Well adding multi digit decimal numbers works
01:22	the same way . The main difference is that instead
01:25	of lining up the ones place digits . When we
01:27	stack the numbers we line up the decimal points instead
01:30	. But wait a minute . I mean isn't that
01:33	the same thing as lining up the ones place digits
01:36	? Yes it is . And that's because the decimal
01:39	is our reference mark that always goes between the ones
01:43	place and the 10th place . So lining up the
01:46	decimal points is the same thing as lining up the
01:49	ones places . It makes sure all the number of
01:51	places lined up in columns . Now you've probably noticed
01:55	that decimal numbers can have different numbers of decimal digits
01:59	for example 10.8 has only one decimal digit but 5.34
02:05	has two decimal digits . And what that means is
02:08	that when you line up the decimal points of the
02:10	two decimal numbers , they might not form a nice
02:13	column on the right . Each . Some of the
02:15	digits might be missing but that's no problem . Remember
02:19	if there's not a digit in a particular number place
02:22	you can just put a zero there to help you
02:23	keep track of things . Now that these numbers are
02:26	lined up by their decimal points we can add them
02:29	, call them by column . But instead of starting
02:31	with the ones place like we always did with whole
02:33	numbers , we start with whatever number place column is
02:37	the furthest to the right in this case that's the
02:39	100th place . So we'll start there So we add
02:43	the digits in each column carrying or regrouping as needed
02:47	and we get 1614 . So we're done right wrong
02:52	. There's one last really important step . Remember ?
02:56	We're doing decimal edition so we can't just forget about
02:59	that decimal point . We need to bring a copy
03:01	of it straight down into our answer line . So
03:03	we keep the same reference point for our number of
03:06	places . Now we can see that the answer is
03:09	really 16.14 . That's not so hard is it ?
03:13	And I've got more good news . Decimal subtraction works
03:16	the same way you start by lining up the decimal
03:19	points of the two numbers . Remember that the order
03:22	of the numbers matters in subtraction so be sure that
03:24	the number you're taking away is on the bottom .
03:27	Then starting with whatever column is furthest to the right
03:30	, you subtract the digits column by column , borrowing
03:33	if you need to . After that you just bring
03:36	down a copy of the decimal point and you have
03:38	your answer . Okay so decimal addition and subtraction are
03:42	pretty easy . Let's move on to something a little
03:44	harder . Decimal multiplication Now as you know , multi
03:48	digit multiplication is more complicated because there are so many
03:52	multiplication steps . But the good news is that decimal
03:55	numbers don't really make the procedure much harder than it
03:58	is with whole numbers . That's because there's a clever
04:01	way that we can make decimal multiplication look exactly like
04:05	the multi digit multiplication with whole numbers that you already
04:08	know how to do . The key is to pretend
04:11	that the decimal points are not really there . Hold
04:14	on a minute . I mean I like pretending as
04:16	much as you do but if we just pretend that
04:19	the decimal points aren't even there , we are going
04:21	to get the right answer . Are we well know
04:24	but the only thing that would be wrong with the
04:26	answer is that the decimal point won't be in the
04:29	right spot so we'll need to fix that at the
04:31	end . I know it sounds a little confusing .
04:34	So here's an example that should help you understand .
04:37	I knew you would say that Let's say that you
04:41	need to multiply 3.65 by 2.4 and that seems a
04:45	little tricky . But what if we just pretend that
04:48	the decimal points are not there for now ? In
04:50	other words , what if we pretended that the numbers
04:53	were 365 and 24 ? You already know how to
04:57	do that problem ? You just follow the procedure that
04:59	we learned in multi digit multiplication , part two and
05:02	you get the answer 8760 But that's the answer to
05:08	365 times 24 , not 3.65 times 2.4 . So
05:14	it's time to stop pretending to get the correct answer
05:17	for the decimal problem . We've got to understand what's
05:20	going on with those decimal points and why we just
05:22	pretended they weren't there ? The truth is when we
05:25	pretended that the decimal points weren't there , What we
05:28	were really doing is pretending that they had been shifted
05:31	until both of our numbers became whole numbers . Remember
05:35	the numbers 365 and 24 technically do have decimal points
05:40	there right there next to the ones place , we
05:42	just don't need to show them since there aren't any
05:44	decimal digits . So by ignoring the decimal points ,
05:48	what we were really doing is mentally shifting the decimal
05:51	points to the right , We shifted the top decimal
05:53	.2 places to the right and we shifted the bottom
05:56	decimal .1 place to the right . But doing that
06:00	changed the numbers , it made the top number 100
06:03	times bigger than the decimal version and it made the
06:05	bottom number 10 times bigger . That's because every time
06:09	you shift the decimal .1 number place to the right
06:13	, it's like multiplying by a factor of 10 And
06:16	that means the answer we got is way too big
06:19	. It's too big by three factors of 10 because
06:23	the decimal points in our problem got shifted a total
06:26	of three places to the right , two on the
06:28	top and one on the bottom . So to fix
06:31	that we're going to have to shift the decimal point
06:33	in our answer the same amount in the opposite direction
06:37	. In other words we need to move the decimal
06:39	point in our answer three places to the left which
06:42	will make it smaller by three factors of 10 .
06:46	So starting right here where the decimal point would be
06:49	if our answer was 8760 , We shifted three places
06:54	to the left and we end up with 8.760 or
06:58	just 8.76 . And that is the answer to 3.65
07:03	times 2.4 . That's a cool trick . Huh ?
07:06	It means that you can do decimal multiplication just like
07:09	regular multi digit multiplication . You start by setting up
07:13	your multiplication problem exactly like you would at the decimal
07:16	points were invisible but don't just erase them because you'll
07:20	need them at the end to figure out how many
07:22	places to shift the decimal point in the answer .
07:25	Then keep ignoring the decimal points . While you follow
07:29	the multiplication procedure , once you have an answer count
07:33	up how many places the decimal points are shifted and
07:36	the problem you're working don't forget it's the total shift
07:40	of both the top and bottom decimal points and then
07:44	shift the decimal point and your answer to the left
07:47	. That same number of places . So decimal multiplication
07:51	turns out to be not too bad after all .
07:53	But what about decimal division ? That's got to be
07:56	hard . Right Well multi digit division is always a
07:59	little hard but luckily decimals don't really make it very
08:03	much harder . In fact it's only when there's a
08:06	decimal divisor that the procedure is a little different .
08:09	If you just have a decimal dividend and the divisor
08:12	is a whole number . It's really simple . That's
08:15	because you can just do the long division procedure that
08:18	we learned in the long division videos and the decimal
08:20	point doesn't affect it at all . You just need
08:23	to make sure that you bring a copy of the
08:25	decimal point up into the answer line when you're done
08:28	. So if you have the division problem 12.64 divided
08:32	by four . You would follow the division procedure as
08:35	if the decimal point was not even there and you'd
08:37	get 316 and the answer line . But then you
08:41	need to bring a copy of the decimal point straight
08:44	up into the final answer , making it 3.16 .
08:49	That's all there is to it . If it's only
08:51	the dividend , that's a decimal number . But what
08:54	if both the divisor and the dividend or decimals ?
08:58	Like what if you have to divide 6.45x1.5 ? Well
09:03	the first step is don't panic as you'll see this
09:06	isn't much harder . It turns out that there's a
09:09	very simple trick that we can use to make it
09:12	. So our divisor is not a decimal number .
09:14	We can just shift the decimal point in the divisor
09:17	to the right until it's a whole number . But
09:20	if we do that then we also need to shift
09:22	the decimal in the dividend the same amount to the
09:25	right . So in this case if we want to
09:28	shift the decimal point in our divisor one place to
09:30	the right so that it's 15 , we can do
09:33	that as long as we also shift the decimal point
09:36	in the dividend by the same amount , which will
09:38	turn it into 64.5 . And here's the really cool
09:42	part . If we do this new division problem ,
09:44	64.5 divided by 15 will get exactly the same answer
09:49	we would have if we did the problem 6.45 divided
09:53	by 1.5 . That only works because we shifted the
09:57	decimal point in the divisor and the dividend by the
10:00	same amount in the same direction . And you'll realize
10:04	why that works . If you remember equivalent fractions ,
10:07	think about the fraction 1/2 . That's the same as
10:11	one , divided by two . Right , Okay .
10:14	But what if I multiplied both the top and bottom
10:17	number by 10 ? That would give me 10/20 which
10:20	is equivalent to 1/2 . Even though it uses different
10:24	top and bottom numbers , Both represent the value 1/2
10:28	their equivalent fractions . Well , that's what we did
10:32	in our decimal division problem when we shifted the decimal
10:35	point in both the divisor and the dividend by one
10:38	place , we multiplied each number by 10 . And
10:42	since fractions and division are basically the same , we
10:45	made equivalent division problems but now one of them has
10:49	a whole number . Divisor . Pretty cool . Huh
10:53	? That means if we solve 64.5 divided by 15
10:56	, we get the answer 4.3 which is exactly the
11:00	same answer we would get if we did 6.45 divided
11:03	by 1.5 . And you can use that trick to
11:07	avoid ever having to divide with the decimal device er
11:10	, Even if the dividend is a whole number ,
11:12	for example if you have the problem 148 divided by
11:16	1.6 you can shift the decimal in both the divisor
11:20	and the dividend one place to the right . Remember
11:23	there's always a decimal point , even in a whole
11:26	number , it's just that when you shift it to
11:28	the right , you need to put a zero in
11:30	the place that it shifts past . That gives you
11:33	the equivalent division problem 1480 divided by 16 . And
11:38	since these division problems are equivalent , you'll get the
11:41	same answer for both . All right . So that's
11:44	how you can modify all the traditional arithmetic procedures to
11:47	work with decimal numbers . It can be a little
11:50	tricky at first , since there's a few extra steps
11:52	that you have to keep track of when the numbers
11:54	or decimals . But if you practice a lot and
11:56	check your answers with a calculator , you'll get it
11:59	remember . You can always re watch this video if
12:02	you need to go along with the other videos about
12:04	multi digit arithmetic . As always . Thanks for watching
12:07	Math Antics and I'll see you next time learn more
12:11	at Math Antics dot com .

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