Math Antics - Absolute Value - By mathantics
Transcript
00:03 | Uh huh . Hi this is rob . Welcome to | |
00:07 | Math antics . In this lesson , we're going to | |
00:09 | learn about a math concept called absolute value . That | |
00:13 | sounds pretty intense doesn't it ? But don't worry it's | |
00:15 | actually pretty simple . So simple . In fact that | |
00:19 | it might seem kind of boring at least . That's | |
00:21 | the way I remember when I was in school . | |
00:23 | Okay class . This is an absolute value science . | |
00:30 | And it's like magic . If I put in a | |
00:35 | positive number it comes out unchanged . Okay but if | |
00:51 | I put in a negative number yeah then just like | |
00:58 | magic it comes out positive . Any questions ? Well | |
01:12 | maybe it wasn't quite that bad but absolute value sure | |
01:16 | seemed a lot less exciting than the name suggested . | |
01:18 | It seemed like just a way to turn a negative | |
01:21 | number into a positive one . For example the absolute | |
01:24 | value of two is two but the absolute value of | |
01:27 | negative two is also to the absolute value of five | |
01:31 | is five . But the absolute value of negative five | |
01:34 | is also five . See those vertical lines on either | |
01:37 | side of the numbers . That's assemble use for absolute | |
01:40 | value . So when you see something in between those | |
01:42 | vertical lines it means to find the absolute value of | |
01:45 | it . But what does absolute value even mean from | |
01:49 | these examples ? It seems like it's just the positive | |
01:51 | version of any number . Well sort of but it's | |
01:55 | a little more involved than that absolute value is a | |
01:58 | more general concept in math that actually has more to | |
02:01 | do with the idea of distance than it does with | |
02:02 | the idea of positive or negative numbers . It's usually | |
02:06 | introduced in basic math right after you learn about negative | |
02:09 | numbers . But absolute value is a concept that's even | |
02:12 | more useful and more interesting and advanced math because of | |
02:15 | that , I'm gonna teach you about absolute value using | |
02:18 | an idea that you usually don't encounter until a little | |
02:21 | later on in your math journey . And that idea | |
02:23 | is vectors . The term vector might sound kind of | |
02:27 | technical but they're actually really simple vectors are basically just | |
02:31 | arrows in real life . And arrow can have all | |
02:34 | sorts of different properties . For example , this arrow | |
02:36 | is made of wood and has feathers while this arrow | |
02:39 | is red plastic with around thinking at the end . | |
02:42 | But in math the arrows , we call vectors only | |
02:45 | have two properties . They have a direction and a | |
02:48 | magnitude . What do those two properties mean ? Well | |
02:51 | , direction is pretty obvious . It's just which way | |
02:54 | the arrow or vector is pointing . It could be | |
02:57 | up down , left , right or just about any | |
03:00 | direction you can think of . But what does magnitude | |
03:03 | mean ? Well that word might make you think of | |
03:05 | an earthquake . Who actually magnitude is just a fancy | |
03:15 | word for the amount extent or strength of something like | |
03:19 | how strong an earthquake is or how bright a star | |
03:22 | is , or how heavy an object is . In | |
03:25 | the case of a vector , you can think of | |
03:27 | magnitude as being the length of the arrow to see | |
03:29 | what I mean . Let's use the number line to | |
03:31 | measure some vectors . This factor has a magnitude or | |
03:35 | length of two because it starts at zero and ends | |
03:38 | at two and this specter has a magnitude or length | |
03:41 | of five because it starts at zero and goes to | |
03:44 | five . Okay , so we know the magnitudes of | |
03:47 | these vectors . But what about their directions ? Well | |
03:49 | , they're both pointing to the right on your screen | |
03:51 | . And since we're using the number line as a | |
03:53 | reference there , pointing in the positive direction . Right | |
03:57 | . But what about this factor here ? It starts | |
03:59 | at zero like our other vectors do , but it | |
04:02 | ends at -2 . And the arrow indicates that it's | |
04:05 | pointing in the exact opposite direction from the other two | |
04:07 | vectors . It's pointing to the left or in the | |
04:10 | negative direction of the number line . So what do | |
04:12 | you think its magnitude is for those of you that | |
04:15 | said to your right , even though the vector is | |
04:18 | pointing in the negative direction , its length is still | |
04:21 | a positive number . Its length is to just like | |
04:24 | this factor that's pointing in the positive direction . They're | |
04:27 | pointing in opposite directions . But if you rotate one | |
04:30 | vector around you can see that they really do have | |
04:32 | the same length or magnitude . And another way to | |
04:35 | say that is that they have the same absolute value | |
04:38 | . Ah See why I said that absolute value has | |
04:41 | more to do with distance . In fact when it | |
04:44 | comes to any number that you'd find on the number | |
04:46 | line you can think of its absolute value as its | |
04:49 | distance from zero . That explains why absolute value is | |
04:53 | a little bit boring when you first learned about it | |
04:55 | because the number line is a one dimensional space , | |
04:58 | There are only two possible directions positive or negative . | |
05:02 | If you ask for the absolute value of any positive | |
05:05 | number along that line , you're asking for the distance | |
05:08 | that number is from zero , which is just the | |
05:10 | number itself . And since the negative numbers are a | |
05:14 | mirror image of their positive counterparts , when you ask | |
05:17 | for the absolute value of any negative number along the | |
05:19 | line , the only difference is the direction or sign | |
05:23 | of that number and that's why the absolute value of | |
05:26 | a negative number is just it's positive counterpart . Hopefully | |
05:30 | that seems a little more interesting than just thinking about | |
05:32 | absolute value as a way to turn negative numbers into | |
05:36 | positive ones . In advanced math , absolute value gets | |
05:39 | even more interesting and it actually gets kind of complex | |
05:42 | vectors can point in all sorts of directions besides just | |
05:45 | positive or negative . But hopefully thinking about absolute value | |
05:49 | as the magnitude or length of a vector makes it | |
05:52 | a little more interesting . And it shows you that | |
05:55 | absolute value isn't just some silly rule that someone made | |
05:58 | up to make math even harder and now that you | |
06:01 | know that absolute value is related to distance it will | |
06:03 | help you understand a useful application of absolute value in | |
06:06 | the realm of basic math to see what I mean | |
06:09 | suppose that you and your best friend each have a | |
06:12 | certain amount of money in your pockets and one of | |
06:14 | you has more than the other . Now it doesn't | |
06:17 | really matter to you who has more money but you | |
06:20 | want to know the difference between the amounts . How | |
06:23 | do you find the difference between two amounts , yep | |
06:25 | you subtract them and the answer you get from subtracting | |
06:29 | depends on the order of the numbers because subtracting doesn't | |
06:32 | have the community of property , right ? If we | |
06:34 | subtract the amounts in this order seven minus four we'll | |
06:39 | get the answer three . But if we subtract the | |
06:41 | amounts in this order four minus seven we'll get the | |
06:45 | answer negative . Three . Do you notice something about | |
06:48 | these answers , yep , Even though the sign or | |
06:51 | direction is different , the magnitudes are the same . | |
06:54 | They're both three . That means that the absolute values | |
06:56 | of the answers would be the same and like I | |
06:59 | said we don't really care who has more money . | |
07:02 | We just want to know how much that differences . | |
07:04 | So in this problem we only need the magnitude or | |
07:07 | absolute value of the difference . No matter which way | |
07:10 | we do the subtraction , we just take the absolute | |
07:13 | value of the answer to get what we want . | |
07:15 | And this idea can be really helpful when you're entering | |
07:17 | numbers into your calculator to solve math problems like this | |
07:20 | one , let's say an airplane , we'll call it | |
07:23 | plain A . Is flying at an altitude of 7328 | |
07:27 | m . And another plane plane B Is flying at | |
07:31 | an altitude of 9150 m . And the problem asks | |
07:35 | you to find the difference in their altitudes . You | |
07:38 | know that means you need to subtract . So you | |
07:40 | quickly get out your calculator and start typing in the | |
07:43 | first number . But as soon as you get the | |
07:45 | 7328 entered and hit the subtract button , you realize | |
07:49 | the second altitude is bigger . That means you'll have | |
07:52 | a negative number as your answer . Should you start | |
07:55 | over and type the bigger number first . Thanks to | |
07:57 | absolute value . You don't have to no matter which | |
08:01 | order . You subtract the numbers in the magnitude of | |
08:03 | the answer will be the same . Just the direction | |
08:06 | or sign of the answer will be different . So | |
08:09 | if you continue on and enter 9150 and then hit | |
08:13 | the equal sign the answer you get is negative 1,822 | |
08:19 | . Now all you have to do is mentally think | |
08:21 | of that number as an absolute value and ignore the | |
08:23 | minus sign . The difference in the altitudes of the | |
08:26 | two planes is 1,822 m . If you're not quite | |
08:31 | convinced of that , try the problem for yourself , | |
08:33 | subtracting both ways and see what answers you get . | |
08:36 | In one case you'll get 1,822 and then the other | |
08:40 | you'll get negative 1,822 . So now you know that | |
08:45 | in its most basic form , absolute value is just | |
08:48 | the distance between a number and zero on the number | |
08:51 | line . And you've also seen how it can be | |
08:53 | helpful when you want to find the difference between two | |
08:55 | different numbers regardless of which is greater . In that | |
08:59 | case the absolute value represents the distance between those two | |
09:03 | numbers . The last thing I want to show you | |
09:05 | in this video is how to handle a couple situations | |
09:08 | involving absolute value that you might encounter on tests when | |
09:11 | evaluating mathematical expressions . For example what if you're asked | |
09:15 | to evaluate this expression involving absolute values ? We learned | |
09:18 | how to multiply integers in the last video but now | |
09:21 | these integers are inside absolute value science . So what | |
09:24 | do we do ? Well , when it comes to | |
09:26 | order of operations , absolute value science are similar to | |
09:30 | parentheses , which means that you need to take care | |
09:32 | of them first before you start working on the other | |
09:34 | arithmetic operations . So in this problem before we can | |
09:38 | multiply the integers we need to take the absolute value | |
09:41 | of the numbers . 1st the absolute value of negative | |
09:44 | three is three and the absolute value of five is | |
09:47 | five . So the problem simplifies 23 times five which | |
09:51 | is 15 . That example was pretty easy . But | |
09:54 | what about this one ? Negative Absolute Value of -8 | |
09:59 | . Why is there an extra negative sign outside of | |
10:01 | the absolute value signs ? Well whenever you see a | |
10:04 | negative sign immediately outside and to the left of a | |
10:08 | group like parentheses braces or the absolute value signs , | |
10:12 | it means that you need to negate that group . | |
10:15 | That means you need to multiply that group by -1 | |
10:18 | . So if it helps you can think of this | |
10:20 | problem like this negative one times the absolute value of | |
10:24 | negative eight . And since absolute value signs are groups | |
10:27 | like parentheses . To simplify it , we would first | |
10:30 | need to do the absolute value , The absolute value | |
10:33 | of -8 is eight and then we multiply that by | |
10:36 | negative one and we get the answer negative eight . | |
10:39 | That seems pretty simple to . But I want to | |
10:42 | use this example to point out an important difference between | |
10:45 | parentheses and absolute value science . It's a difference that | |
10:48 | can trick you want to test if you're not careful | |
10:50 | . Let's see the expression we just simplified side by | |
10:53 | side with a similar expression that has parentheses instead of | |
10:56 | absolute value science . In both of these cases the | |
10:59 | negative sign outside the group is telling us to negate | |
11:02 | the group . That means multiplying it by -1 . | |
11:06 | But there's a very important difference between the two expressions | |
11:09 | . The order of operations rules tells us to do | |
11:12 | things that are inside of groups first . Right ? | |
11:14 | But there's nothing to do inside the parentheses . It's | |
11:18 | just a negative eight hanging out inside of them . | |
11:20 | And the parentheses themselves don't tell us to do anything | |
11:23 | . So this expression is just asking us to multiply | |
11:26 | negative one times negative eight , which is positive eight | |
11:30 | . But in the case of the absolute value science | |
11:32 | , while they do function like groups , they aren't | |
11:34 | just for grouping like the parentheses are they're also telling | |
11:37 | us to do something to whatever is inside of them | |
11:40 | . What are they telling us to do ? They | |
11:42 | tell us to use the magnitude of the number in | |
11:45 | sight which would be positive eight . So in this | |
11:48 | expression we multiply negative one times positive eight . Which | |
11:51 | simplifies to negative eight . Do you see how it | |
11:54 | would be easy to confuse these examples on a test | |
11:58 | they look similar at first glance but they simplified two | |
12:00 | different answers . The key is to always remember that | |
12:04 | absolute value signs are not just there to group things | |
12:07 | , they're also asking you to find the absolute value | |
12:10 | of whatever number or expression is inside the group . | |
12:13 | All right . As you can see , absolute value | |
12:16 | is a lot more than just a way to turn | |
12:18 | negative numbers and the positive ones . Although that's basically | |
12:21 | what it does when you're just dealing with numbers on | |
12:23 | the number line . But it has lots of other | |
12:26 | applications in math too . And look this student has | |
12:30 | a negative attitude about math class but not to worry | |
12:37 | , I put him in an absolute value sign and | |
12:41 | presto wow he comes out with a positive attitude about | |
12:49 | math class . Any questions remember ? You can't get | |
12:57 | good at math just by watching videos about it , | |
12:59 | You actually have to do practice problems . So the | |
13:01 | idea is you're learning really sink in so be sure | |
13:04 | to do some absolute value problems on your own . | |
13:06 | As always . Thanks for watching Math Antics and I'll | |
13:08 | see you next time learn more at Math Antics dot | |
13:12 | com . |
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