Math Antics - Scientific Notation - Free Educational videos for Students in K-12

Math Antics - Scientific Notation - By mathantics

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00:0-1	Hi rob here with Math Antics . Just wanted to
00:02	say hi to all the people who watch our videos
00:05	and say thank you so much for watching and liking
00:08	and subscribing . I know we usually don't do milestone
00:11	videos , but this seemed like a pretty cool milestone
00:14	that we just had to mention it . We have
00:17	one times 10 to the six YouTube subscribers . That's
00:20	crazy . That's awesome . Thank you so much .
00:24	Oh wait , You don't know what numbers like ?
00:26	One times 10 to the six even mean ? Well
00:28	, you're in luck . We have a video that
00:30	explains it . In fact , it's this video that
00:32	you're watching right now . Uh huh . Hi ,
00:40	I'm rob . Welcome to Math Antics . Have you
00:43	ever heard people use numbers like that ? One times
00:46	10 to the sixth ? Or maybe 3.4 times 10
00:48	to the negative eighth . Well , those are examples
00:51	of a way of writing numbers called scientific notation .
00:55	Huh ? That didn't sound quite right . Let me
00:57	try that again . Scientific notation . Uh Yes ,
01:02	that's better . Numbers can be really big or really
01:06	small , right ? Like if you wanted to count
01:08	up all of the cells that make up your body
01:11	, it would be a really big number , Something
01:13	like 35 trillion cells . But if you wanted to
01:17	measure the diameter of one of those cells using meters
01:20	, you'd get a really small number , something like
01:23	0.000005 m . Not only are really big or small
01:30	numbers , a lot of work to write down because
01:32	of all the zeros , they're hard to quickly evaluate
01:35	and compare at a glance . It's not easy to
01:37	tell just how many number of places there are in
01:40	these really big or small numbers . And that's where
01:43	scientific notation can really help us out , instead of
01:46	using a long sequence of decimal digits to represent numbers
01:50	. Scientific notation uses a shorter number , multiplied by
01:53	a power of 10 . And it's always in that
01:56	form some number times 10 to the some exponent .
02:00	Here's an example of a really long number . 125
02:04	million . And here's the equivalent number written in scientific
02:07	notation . 1.25 times 10 to the 8th . Want
02:11	to see how these two numbers are . Just different
02:13	ways of writing the same thing . Let's start by
02:16	making a copy of our big number and messing with
02:18	its decimal point a bit . Where's the decimal point
02:20	? You ask ? Remember that ? It's always right
02:22	here , immediately to the right of the ones place
02:25	. We just don't need to show it if there
02:27	aren't any decimal digits . Okay , so what would
02:30	happen if we shift the decimal .1 place to the
02:32	left ? Well doing that would change the number right
02:36	. By definition , the decimal point is always immediately
02:39	to the right of the ones place . So shifting
02:42	the decimal point shifts the ones place and all the
02:44	other number of places to . And if we line
02:47	up the ones place of our new number with the
02:49	ones place of the original number , you see that
02:51	the new number is 10 times smaller . That means
02:54	shifting the decimal .1 place to the left is equivalent
02:58	to dividing the number by 10 . But do we
03:01	want a number that's 10 times smaller than before ?
03:03	Well no , we don't want to change the value
03:06	of the number at all . We just want to
03:08	write it in a different way . Since shifting the
03:10	decimal resulted in a number that's 10 times smaller than
03:13	before to compensate and keep the value the same .
03:16	We need to multiply the new number by 10 ,
03:19	making the number smaller and then compensating for that might
03:22	seem like a weird thing to do but it will
03:24	make more sense in a minute . Let's do that
03:26	process again . Let's make a copy of the new
03:28	number and shift the decimal point to the left again
03:31	. Since that shifts all the number of places ,
03:34	we can align the ones places and see that the
03:36	same thing happened . The new number is 10 times
03:39	smaller than before . So to keep it the same
03:41	value as the original number we need to compensate by
03:44	making it 10 times bigger . We need to multiply
03:47	by another 10 . And if we repeat that process
03:50	again if we make another copy and shift the decimal
03:53	point again we'll see that the number gets 10 times
03:55	smaller . So we need to compensate by multiplying by
03:58	another 10 . Alright time out we seem to have
04:01	a little problem here . Each time we shift the
04:03	decimal point to the left . Our number gets smaller
04:06	but since we have to compensate with a factor of
04:09	10 each time it's making kind of a mess .
04:12	This part is getting shorter but this part is getting
04:14	longer . No problem . Exponents can fix that .
04:18	Do you remember that ? Exponents are a way of
04:19	writing repeated multiplication . If you don't then be sure
04:23	to watch our videos about them before moving on .
04:26	Instead of writing 10 times 10 we can write 10
04:29	to the second power . And instead of writing 10
04:31	times 10 times 10 we can write 10 to the
04:34	third power . That's much better now we can continue
04:37	on will shift the decimal point again and multiplied by
04:40	10 again . But this time instead of writing another
04:43	times 10 we can just increase the exponent by one
04:47	. Since there would be a total of four tens
04:49	being multiplied together . Let's keep going with this process
04:52	of shifting the decimal point to the left and multiplying
04:55	by tim for each number place we shift and we'll
04:58	stop when there's only one digit remaining to the left
05:01	of the decimal point , wow that's quite a pattern
05:05	. Each time we shifted the decimal the number got
05:07	10 times smaller . So each time we had to
05:09	multiply it by another 10 to keep the value the
05:12	same . And because we did that each one of
05:15	these lines represents the same value . So even though
05:18	it looks a lot different , this last line has
05:21	the exact same value as the first one . In
05:23	fact it's just the original number written in scientific notation
05:28	. But why is it only this last line and
05:30	not any of the others ? I mean they all
05:32	look pretty scientific to me . That's a good question
05:36	for a number to be in proper scientific notation ,
05:39	it's supposed to have only one digit to the left
05:41	of the decimal . There can be more than one
05:43	digit to the right of the decimal depending on the
05:45	accuracy of the number , but just one digit to
05:48	the left . But why , I mean that rule
05:51	sounds kind of arbitrary . It's not arbitrary at all
05:55	. If there's more than one digit to the left
05:57	of the decimal point , that would mean that we
05:58	didn't get out all of the factors of 10 that
06:00	we could have . And factoring out all of the
06:03	tents helps us quickly determine a numbers , order of
06:06	magnitude , Order of magnitude . What in the world
06:09	is that ? That sounds kind of scary . Order
06:12	of magnitude is basically just how many tens you need
06:15	to multiply to get a certain number . And when
06:18	a number is in scientific notation , the order of
06:20	magnitude is just the exponent . Because that's telling us
06:23	how many tends to multiply together . In this example
06:27	the scientific notation says that if we take this small
06:30	decimal and multiply it by eight tins will get our
06:32	original number . So scientific notation is a way of
06:36	taking a really big number and reducing it down to
06:39	a value that's less than 10 . But keeping track
06:42	of how many tens we need to multiply together to
06:44	get the full number . You can think of it
06:46	as basically just extracting its order of magnitude and storing
06:50	an exponent form . But why would we want to
06:53	do that ? I mean it seems kind of complicated
06:56	. Well yeah but did you see how much writing
06:59	it saved us ? When we rode out 125 million
07:02	we had to write 11 characters including commas . But
07:06	when we wrote the same number in scientific notation we
07:10	only had to write eight characters . What ? Not
07:13	convinced that it's worth the savings . Well how about
07:16	this number ? That's a lot of Zeros to right
07:19	isn't it ? But in scientific notation this number is
07:22	just 8.4 times 10 to the 31st power . That's
07:26	much better . So scientific notation is very useful when
07:30	it comes to writing down really large numbers or really
07:34	small ones . For example , this number is really
07:37	small . 0.0000095 . It's much less than one but
07:44	it's not zero and here's the same number written in
07:47	scientific notation . Again it consists of a number that
07:51	has only one digit to the left of the decimal
07:53	point which is being multiplied by 10 to a certain
07:56	power . But do you notice anything different about the
07:59	exponent , yep . It's negative . So what does
08:02	that mean ? Well the short answer is that positive
08:05	? Exponents show repeated multiplication or negative exponents show repeated
08:10	division . And since this is a negative exponent with
08:13	10 as the base it means to repeatedly divide by
08:16	10 to see how that works . Let's copy our
08:19	original number and do that decimal point shift thing again
08:22	. Only this time we're going to shift the decimal
08:24	point to the right . What happens if we shift
08:27	at one place to the right ? It makes the
08:29	number 10 times bigger than it was before . There
08:32	used to be six zeros between the decimal point in
08:34	the night but now there's only five again we don't
08:37	want to change the value . So what can we
08:39	do to compensate for shifting the decimal point In this
08:42	case ? Since shifting one place to the right made
08:45	the number 10 times bigger . We need to compensate
08:48	by dividing the number by 10 . And because the
08:51	way multiplication and division are related , dividing by 10
08:54	is the same as multiplying by 1/10 or 1/10 .
08:58	So we can just multiply by 1/10 . Or we
09:01	can multiply it by 10 to the -1 Because 10
09:04	to the -1 is just another way of writing 1/10
09:07	. That may seem odd if you haven't learned about
09:09	negative exponents before and we explain it in more detail
09:13	in our video about the laws of exponents For now
09:16	, all you really need to know is that multiplying
09:18	by 10 to the negative one is the same as
09:20	dividing by 10 . So it compensates for shifting the
09:23	decimal .1 place to the rate Continuing on . If
09:27	we shift the decimal point , another place to the
09:29	right . The same thing happens , we make the
09:31	number 10 times bigger . So to keep the value
09:34	the same we have to multiply by another factor of
09:36	10 to the -1 . And if you're wondering whether
09:39	we can combine these exponents , you're on the right
09:41	track 10 to the negative one times 10 to the
09:44	negative one . Combined to become 10 to the negative
09:47	too . Which makes sense because we shifted the decimal
09:50	point a total of two places to the right .
09:53	And if we shift the decimal .3 places to the
09:55	right we need to multiply by 10 to the negative
09:58	three to compensate . And if we shift four places
10:01	then we need 10 to the negative four to compensate
10:03	get the idea . And if we continue doing that
10:06	until the decimal point is positioned so that there's only
10:09	one digit to the left of it . That gives
10:11	us the number in scientific notation 9 , 5 times
10:15	10 to the -7 . And can you figure out
10:18	what the order of magnitude of this number is ,
10:20	Yep , just like before the exponent tells us it's
10:23	-7 . Being able to quickly identify a numbers .
10:27	Order of magnitude is pretty handy . For example ,
10:30	if the order of magnitude is a big positive exponent
10:33	, then you know right away that you're dealing with
10:35	a really big number . But if it's a big
10:38	negative exponent then you know , you're dealing with a
10:41	really small number . And if you're comparing two really
10:45	big numbers like these two or two really small numbers
10:48	like these two , it's hard to tell at a
10:50	glance which is actually bigger or smaller , but if
10:54	you see them in scientific notation , it's easy to
10:56	see that this numbers order of magnitude is bigger than
10:58	the others , which means that it's bigger and this
11:01	number is order of magnitude is less than the others
11:03	, which means that it's smaller . So now that
11:06	you've seen how scientific notation works and you realize that
11:09	it's just a shorthand way of writing . Really big
11:12	or really small numbers . Let's break down the procedure
11:15	for converting back and forth between numbers written in regular
11:18	form and scientific notation . Starting with these two examples
11:21	in regular form . First count how many number of
11:24	places you need to shift the decimal point for there
11:27	to be only one digit to the left of it
11:29	. For this number we would need to shift eight
11:31	places and for this number six places , the number
11:34	of places you need to shift will be the exponent
11:37	in the scientific notation for . But the sign of
11:39	that exponent is determined by the direction you shifted .
11:43	If you shifted to the left because you started with
11:45	a big number then the exponent will be positive .
11:48	But if you shifted to the right because you started
11:51	with a small number , then the exponent will be
11:53	negative . So that gives you the times 10 to
11:56	the something part of the scientific notation . And to
11:59	get the number that's multiplied by that power of 10
12:02	, you just take the shifted decimal number and remove
12:04	any zeros that don't really need to be shown there
12:08	. That wasn't too hard was it ? But what
12:10	if we start out with numbers that are in scientific
12:12	notation and want to convert them into regular form ?
12:15	Let's do that with these two examples . The first
12:18	step is to look at the exponent which is the
12:20	order of magnitude of the number . It tells you
12:22	how many tins you'll need to multiply or divide by
12:25	to get the number in regular form . If the
12:27	exponent is positive , it means that you'll need to
12:29	multiply by that many tents in this example , that
12:33	means that we need to multiply it by a total
12:35	of seven tens to get the number in regular form
12:38	. That would be a lot of work . But
12:39	we can also just shift the decimal point . That
12:42	number of places . Which direction do we need to
12:44	shift it ? Well , since we're multiplying by factors
12:47	of 10 , we need to shift it in the
12:49	direction . That will make the number bigger . That
12:51	is . We need to shift it to the right
12:53	. So we'll just shift the decimal .7 places to
12:56	the right . But as you can see there aren't
12:59	seven digits after the decimal . So any places that
13:02	don't have a digit would just be filled with a
13:04	zero . There are number and regular form is 41
13:08	million , 650,000 . In the second example , the
13:12	exponent is negative , which means that we'll need to
13:14	divide by that number of tens to get the number
13:16	in regular form again we could just do that division
13:20	or we can shift the decimal point to save time
13:23	. Since dividing by factors of 10 , make a
13:25	number smaller will need to shift the decimal point in
13:28	the direction that results in a smaller number that is
13:31	to the left . Since our exponent is negative ,
13:34	five will shift the decimal .5 places to the left
13:38	and any places that we shift past that don't already
13:40	have a digit in them will get filled with zero
13:43	. We'll also put a zero in the ones place
13:45	. Since that's always good form for decimal numbers .
13:48	There are a number in regular form . 0.0000 109
13:54	. All right . That's the basics of scientific notation
13:58	. It may seem a little confusing at first ,
14:01	but as you get more experience with it , it
14:03	makes a lot of sense . And when it comes
14:05	to writing really big or really small numbers , it's
14:08	totally worth it . And even if you understand how
14:11	scientific notation works , it may take some practice to
14:14	get good at converting back and forth between it and
14:17	regular form , so be sure to practice on your
14:19	own as always . Thanks for watching Math Antics and
14:22	I'll see you next time . Learn more at Math
14:25	Antics dot com .

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