Math Antics - Integer Multiplication & Division - Free Educational videos for Students in K-12

Math Antics - Integer Multiplication & Division - By Mathantics

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00:03	Uh huh . Hi , welcome to Math Antics .
00:08	In our last video we learn how to add and
00:10	subtract integers . In this video , we're going to
00:13	learn how to multiply and divide integers . The good
00:16	news is that multiplying and dividing integers is easier than
00:20	adding and subtracting them because even though it involves both
00:24	positive and negative numbers , the multiplication and division still
00:28	works basically the same way as we learned in our
00:31	video about negative numbers . The negative numbers are like
00:35	a mirror image of the positive numbers on the number
00:37	line . Each number on the positive side has a
00:40	negative counterpart on the negative side . There's a two
00:44	on the positive side and there's a negative two on
00:46	the negative side . There's a five on the positive
00:49	side and there's a negative five on the negative side
00:52	. And so on realizing that can help us understand
00:56	that negative numbers are just like positive numbers but they
00:59	have a negative factor built into them . For example
01:03	, Think about the number 3 . 4 minutes .
01:05	Do you remember when we learned about factors , we
01:07	learned that one is always a factor of any number
01:11	and that's because multiplying by one doesn't change a number
01:14	three and one times three are the same value .
01:19	So if you have the number three you can factor
01:21	out one and you have one times three . On
01:24	the other hand , what if you have negative three
01:26	instead ? We could still factor out of one from
01:29	it and nothing would change . You'd have one times
01:31	negative three . But because the number is negative ,
01:34	we could also factor out a negative one . Doing
01:37	that would give us -1 times three . So one
01:41	way to think about negative numbers is to imagine that
01:44	they're just like positive numbers but they always have a
01:47	factor of negative one built into them . That means
01:50	if you want to change a positive number into a
01:52	negative number , all you have to do is multiply
01:55	it by a factor of negative one . five times
01:58	negative one is negative five , seven times negative one
02:01	is negative seven , 10 times negative one is negative
02:05	10 and so on . And one way to visualize
02:08	this is to see that multiplying by a factor of
02:11	negative one just switches a number from the positive side
02:15	of the number line to the negative side . Ah
02:18	But that raises an interesting question . What happens if
02:21	you multiply a number that's already on the negative side
02:24	by a factor of negative one Like -1 times -3
02:29	. Is that going to make it extra negative ,
02:32	nope . In fact it's going to do just the
02:34	opposite multiplying a negative number by another negative factor is
02:40	actually going to switch the answer back to the positive
02:42	side of the number line , multiplying by negative one
02:46	acts like a switch , no matter which side of
02:48	the number line you start on . If you start
02:51	with a positive then multiplying by negative one switches it
02:55	to negative . But if you start with a negative
02:57	multiplying by negative one switches it back to positive and
03:01	you can keep switching back and forth between the positive
03:04	and negative side of the number line by multiplying by
03:07	another negative one as many times as you want .
03:12	So multiplying by one negative gives us a negative multiplying
03:19	by two negatives gives us a positive multiplying by three
03:23	negatives gives us a negative multiplying by four negatives gives
03:27	us a positive multiplying by five negatives gives us a
03:30	negative Multiplying by six negatives gives us a positive and
03:34	so on . Did you see the pattern in a
03:37	multiplication problem . If you have an even number of
03:40	negative factors , they'll form pairs that will balance each
03:43	other out and we'll give you a positive answer .
03:46	That's because the pair negative one times negative one just
03:50	equals one , which has no effect on the answer
03:54	. But if you have an odd number of negative
03:56	factors that are being multiplied together after you balance out
04:00	all the pairs , there will always be one negative
04:03	factor leftover . That will give you a negative answer
04:06	in a minute . We'll see some examples of how
04:09	knowing this will help us when multiplying and dividing lots
04:12	of integers . But first , let's look at the
04:14	simple problem three times five . We know that the
04:18	answer is 15 . But now , thanks to negative
04:21	numbers , we know that there's three more variations of
04:24	this problem that we need to learn how to do
04:26	. This might seem a little complicated , but it's
04:29	really not . That's because we're going to get the
04:31	same number for the answer for all four problems .
04:34	It's just that the sign of the answer , whether
04:36	it's positive or negative will be different depending on how
04:39	many negative factors were multiplying . Basically when doing integer
04:43	multiplication and division , you can just pretend that the
04:46	negatives aren't there while you multiply or divide and then
04:50	you count up how many negative factors you have to
04:53	figure out the sign of the answer . If there
04:55	is an even number of negative factors , the answer
04:58	will be positive . But if there's an odd number
05:00	of negative factors , then the answer will be negative
05:03	. So in the first problem we don't have any
05:06	negative factors . So our answer is just going to
05:09	be positive . 15 , which we already knew In
05:12	the second problem , we have only one negative factor
05:15	being multiplied , which means our answer will be negative
05:18	15 . In the third problem , we also have
05:21	just one negative factor , which means that her answer
05:24	will also be negative 15 . And in the fourth
05:27	problem we have to negative factors being multiplied . That's
05:30	an even number of negative factors . So they'll balance
05:33	each other out . They'll switch and switch back and
05:35	they'll give us an answer of positive 15 . That's
05:39	pretty easy . Right ? And the really great news
05:42	is that because multiplication and division are so closely related
05:46	their inverse operations , the rules about negative factors are
05:49	exactly the same for division problems . For example ,
05:53	if you have the problem eight divided by two ,
05:56	shown here , infraction for him , the answer is
05:58	positive for But if you have eight divided by negative
06:02	two , then there's one negative . So the answer
06:05	is negative for likewise , if you have negative eight
06:09	divided by two , then there's still just one negative
06:12	. So the answer will also be negative for But
06:16	if both of the numbers are dividing our -8 divided
06:20	by -2 , then there's two negatives in the division
06:23	problem . So the answer will be positive for And
06:26	one way to see that a negative divided by a
06:28	negative gives you a positive is to realize that if
06:31	you factor out the negative one on the top and
06:34	you factor out the negative one on the bottom ,
06:36	then you have a pair of common factors that you
06:38	can cancel just like you would if you were simplifying
06:41	a fraction . Okay , are you ready for some
06:44	more complicated examples in these problems ? We're going to
06:47	be combining both multiplication and division . And the good
06:51	news is that it doesn't matter whether a negative factor
06:54	is being multiplied or divided . You still get to
06:57	count it when figuring out if you have an even
06:59	or an odd number of factors . How about this
07:02	one ? Negative three times eight over negative too .
07:06	Well on the top negative three times eight is going
07:09	to give us negative 24 since three times eight is
07:13	24 there's only one negative factor And then -24 divided
07:19	by -2 is going to be positive 12 . Since
07:22	24 divided by two is 12 and we have two
07:26	negatives , which means our answer will be positive .
07:29	Here's another good example negative one times negative eight times
07:34	negative six over the quantity negative three times negative four
07:39	. All five of the numbers are negative . And
07:41	since that's an odd number of negative factors , we
07:44	know that the answer is going to be negative watch
07:47	and see on the top negative one times negative eight
07:51	is positive eight Positive eight times -6 is -48 .
07:56	And on the bottom we have negative three times negative
07:59	four which is positive 12 . And then as the
08:03	final step -48 divided by 12 is -4 . So
08:08	it worked . Our final answer is negative because we
08:11	had an odd number of negative factors . But what
08:15	if we made one slight change to that same problem
08:18	? What if we made one of the numbers on
08:19	top positive instead of negative ? Then we only have
08:23	four negative numbers which is even so the answer should
08:27	be positive for let's see if it is On the
08:30	top negative one times positive eight is negative , eight
08:35	-8 times negative six is positive 48 . And on
08:40	the bottom we have negative three times negative four which
08:44	is positive 12 . And then as the final step
08:48	, 48 divided by 12 is positive for just as
08:52	I suspected . So do you see what I mean
08:55	? All the multiplication and division works the same .
08:58	It's just that you have to figure out whether the
09:00	answer is going to be positive or negative and to
09:03	do that , all you have to do is figure
09:05	out if you have an even or an odd number
09:08	of negative factors in the multiplication or division , if
09:12	there's an odd number of negative factors , then the
09:14	answer will be negative . And if there is an
09:16	even number of negative factors , the answer will be
09:19	positive . And remember this process only works for integer
09:24	multiplication and division . If you have a problem that
09:27	also contains integer addition and subtraction , you need to
09:31	do those operations using the rules we learned in the
09:33	last video as always , the way to get good
09:37	at Math is to practice So be sure to work
09:39	some integer multiplication and division problems to make sure you've
09:42	got it . Thanks for watching Math Antics And I'll
09:45	see you next time . Learn more at Math Antics
09:49	dot com

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