10 - What are Imaginary Numbers? - By Math and Science
Transcript
| 00:00 | Hello . Welcome back to algebra . The title of | |
| 00:02 | this lesson is called what is an imaginary number or | |
| 00:05 | alternatively what are imaginary numbers . Now ? Every once | |
| 00:08 | in a while I teach a lesson that makes me | |
| 00:09 | really , really excited . And the reason I'm excited | |
| 00:12 | about this lesson is because when I first learned about | |
| 00:14 | the idea of imaginary numbers I didn't understand what they | |
| 00:17 | were , I didn't understand why they were called imaginary | |
| 00:19 | . I kind of thought they were useless actually because | |
| 00:21 | of the word imaginary . And in general it was | |
| 00:24 | just not not obvious to me why we care about | |
| 00:27 | imaginary numbers . Usually you go through Matthew , go | |
| 00:30 | through calculus , you go into college , you study | |
| 00:32 | advanced mathematics and then you understand why imaginary numbers are | |
| 00:35 | so useful . What I'm gonna do in this lesson | |
| 00:37 | is first of all tell you what an imaginary number | |
| 00:39 | is then we're gonna do some motivation , showing you | |
| 00:42 | why we needed why we need the concept of imaginary | |
| 00:45 | numbers to solve certain types of equations in algebra , | |
| 00:47 | the equations you're gonna be using . And then if | |
| 00:50 | you stick with me to the end of this lesson | |
| 00:52 | , after we do a couple of problems with working | |
| 00:53 | with imaginary numbers , I'm going to show you why | |
| 00:56 | we actually care about imaginary numbers . I'm gonna short | |
| 00:59 | circuit about five more years of math education to tell | |
| 01:02 | you why imaginary numbers are so useful and a couple | |
| 01:05 | of examples of how imaginary numbers are are used in | |
| 01:09 | modern day mathematics , modern day science , math . | |
| 01:12 | So stick with me to the end . We're gonna | |
| 01:13 | get through all of that . Now , first I'm | |
| 01:15 | gonna give you the punch line . I want to | |
| 01:16 | tell you what an imaginary number is . All right | |
| 01:19 | ? So this right here I'm gonna write on the | |
| 01:20 | board . It's called the definition of the imaginary number | |
| 01:26 | definition of the imaginary number , which we call I | |
| 01:30 | we use the the the letter I . To mean | |
| 01:33 | imaginary number anytime you see I . And mathematics , | |
| 01:36 | it means the imaginary number . So the punch line | |
| 01:38 | here is the following thing . The uh imaginary number | |
| 01:42 | i is equal by definition . It's defined mathematically to | |
| 01:47 | be the square root of negative one . Now , | |
| 01:51 | I've been telling you since the beginning of talking about | |
| 01:53 | radicals that we cannot take square roots of negative one | |
| 01:56 | . Right ? So when you first learned about imaginary | |
| 01:58 | numbers , first of all , you think they're really | |
| 01:59 | , really scary . Really , really complicated . But | |
| 02:02 | I'm telling you , they're not complicated at all to | |
| 02:04 | work with . Just kind of follow along with me | |
| 02:06 | . Secondly , you're kind of scratching your head . | |
| 02:07 | Why ? How can we do that ? How is | |
| 02:09 | that legal ? We can't take the square root of | |
| 02:10 | negative one . Well , the truth is you you | |
| 02:12 | cannot take the square root of a negative one and | |
| 02:15 | get a real number back . In other words all | |
| 02:18 | of the numbers that you've ever been exposed to your | |
| 02:20 | entire life are what we call real numbers . Those | |
| 02:23 | are the numbers that we can kind of hold in | |
| 02:25 | our hands for lack of a better word numbers like | |
| 02:27 | two and three even negative numbers like negative 10 or | |
| 02:31 | fractions like 3/4 or one half or even non repeating | |
| 02:36 | decimals like pie or like other there's tons of other | |
| 02:39 | ones like square root of two and square root of | |
| 02:40 | three and things like that . Those have decimals that | |
| 02:42 | go on and on forever . Those are all called | |
| 02:44 | to real numbers . The truth is you can't take | |
| 02:46 | the square root of a negative number and get an | |
| 02:49 | answer that's a real number . So mathematicians over the | |
| 02:52 | years have invented a totally new kind of number . | |
| 02:55 | It's called an imaginary number . It's a terrible name | |
| 02:57 | because it makes it sound like it's a fictitious thing | |
| 03:00 | that's useless . The truth is a very very useful | |
| 03:03 | and we define the letter I , we're going to | |
| 03:06 | use it as a placeholder to be the square root | |
| 03:08 | of negative one . Now it is true . I | |
| 03:09 | cannot find a real number that will multiply by itself | |
| 03:13 | . You know like we say the square root of | |
| 03:14 | nine is three because three times three is nine . | |
| 03:16 | We say the square root of 16 is four because | |
| 03:19 | four times four is 16 . We can't say that | |
| 03:21 | the square root of negative one is some number of | |
| 03:23 | times some number because there is no number that does | |
| 03:25 | that . We have to invent something else . It's | |
| 03:27 | called an imaginary number . You're gonna have to get | |
| 03:30 | used to seeing the letter I running around when we're | |
| 03:33 | simplifying expressions you can kind of think of I is | |
| 03:36 | like a variable . You can combine terms like terms | |
| 03:39 | we have to have I involved and so on . | |
| 03:41 | We're gonna be squaring and cubing and dividing and multiplying | |
| 03:44 | by I we're gonna be doing all the same things | |
| 03:46 | we do with all variables and numbers all the time | |
| 03:48 | . But in the back of your mind , anytime | |
| 03:50 | you see the letter I anywhere on a paper and | |
| 03:52 | math , you need to remember that . That is | |
| 03:54 | the square root of a negative number or the squared | |
| 03:56 | of negative one . Right ? And it's defined like | |
| 03:59 | that defined defined like that is a definition . This | |
| 04:02 | is not something that you discover under a rock somewhere | |
| 04:05 | . It's something that's mathematically defined . The reason is | |
| 04:07 | because it's useful . Imaginary numbers we will see are | |
| 04:10 | very very useful . All right . If you define | |
| 04:14 | the letter I . This imaginary number to be the | |
| 04:16 | squared of negative one . It's not a real number | |
| 04:18 | that this is an imaginary number , right ? Then | |
| 04:21 | , because of that , we can say the following | |
| 04:23 | thing . Thus , if you square both sides of | |
| 04:26 | this , if I square this and square this , | |
| 04:28 | then we also know that I squared is equal to | |
| 04:31 | negative one . How ? Because if I square this | |
| 04:33 | , I'll get the ice square and if I square | |
| 04:35 | this , they'll cancel with the square roots and then | |
| 04:37 | I'll have a negative one back . So this is | |
| 04:39 | equally important as this . So I'm gonna circle this | |
| 04:42 | and we're gonna reference it throughout the lesson . It's | |
| 04:44 | so important . I want you to stop for a | |
| 04:46 | second and absorb this . It looks and it is | |
| 04:49 | very simple , but there's a lot of depth into | |
| 04:52 | what's on the board here that we're gonna see as | |
| 04:54 | we work through the problems . Any time you have | |
| 04:56 | the square root of negative one , of course it's | |
| 04:58 | I but the concept of I allows us to take | |
| 05:00 | the square root of any negative number we want . | |
| 05:03 | But we're gonna get I somewhere in the answer . | |
| 05:05 | And because of that , anytime you square I you | |
| 05:08 | get back a real number which happens to be negative | |
| 05:11 | one , it turns out this price , this property | |
| 05:13 | right here is the secret sauce as to why they're | |
| 05:16 | so useful and I'll give you a little bit of | |
| 05:18 | a preview . Imaginary numbers are not something you can | |
| 05:21 | hold in your hand , right ? But if I | |
| 05:22 | have an imaginary number in this hand and I have | |
| 05:25 | an imaginary number in this hand . Oftentimes if I | |
| 05:27 | combine them together in this case , what's written on | |
| 05:30 | the board is by multiplying them . I have an | |
| 05:32 | imaginary thing times an imaginary thing which are not things | |
| 05:35 | I can hold because they're imaginary numbers , but when | |
| 05:38 | I multiply them together , I don't have an imaginary | |
| 05:40 | number anymore . I have a real number back . | |
| 05:43 | This negative one is just a negative one . It's | |
| 05:45 | a real number . And that is probably one of | |
| 05:48 | the biggest punch lines as to why imaginary numbers are | |
| 05:50 | useful because I can work with imaginary numbers over here | |
| 05:54 | and I can work with imaginary numbers over here . | |
| 05:56 | Neither one of which represent real numbers , but I | |
| 05:58 | can combine them together and often my equations will combine | |
| 06:02 | them together so poof they're not imaginary anymore , they're | |
| 06:05 | real . So that's actually one of the reasons why | |
| 06:07 | they're so useful . So this is the punch line | |
| 06:10 | and I want to leave it on the board and | |
| 06:11 | I want to motivate why we actually need imaginary numbers | |
| 06:14 | . Um It turns out that all of the equations | |
| 06:16 | we've learned how to solve all . Up up until | |
| 06:19 | now in algebra have required different kind of numbers and | |
| 06:22 | I'm gonna it's a little bit of history of mathematics | |
| 06:24 | here , a little bit . Okay , some equations | |
| 06:26 | require positive numbers to solve . Some equations require negative | |
| 06:30 | numbers to solve . Some equations require fractions to solve | |
| 06:33 | . Some equations require imaginary numbers to solve which is | |
| 06:36 | what we're going to get to . I want to | |
| 06:37 | go down that with you because I think it's very | |
| 06:39 | important that you understand where this comes from . So | |
| 06:43 | I'm going to talk about real numbers for a second | |
| 06:45 | because real numbers , let me see if I can | |
| 06:47 | write real numbers down . Real numbers are the numbers | |
| 06:49 | that you've been exposed to your whole life all the | |
| 06:51 | way from first grade , all the way up until | |
| 06:54 | now . Real numbers , every number you've ever seen | |
| 06:56 | , even if it was negative , is still a | |
| 06:58 | real number . Even if it's a decimal is still | |
| 07:01 | still a real number . They're all real numbers . | |
| 07:03 | Right ? So all the equations that you've ever learned | |
| 07:05 | how to solve only required real numbers numbers to solve | |
| 07:08 | them . But there are different types of real numbers | |
| 07:10 | . So let's talk about a few of those . | |
| 07:12 | Right ? Let's take a couple of equations . Right | |
| 07:15 | ? Let's take a very simple equation . Let's say | |
| 07:17 | we have the equation X plus the number one is | |
| 07:19 | equal to three . You all know how to solve | |
| 07:21 | that equation . You just move the one over by | |
| 07:23 | subtraction . Three minus one is two . So what | |
| 07:25 | we found here is that X is equal to two | |
| 07:27 | . Now what kind of number is too ? Well | |
| 07:29 | , it's a real number . Of course it's , | |
| 07:30 | remember what kind of number is it ? It's a | |
| 07:32 | real number . I'll put that down but it's a | |
| 07:34 | positive number . So that's a type of number that | |
| 07:38 | was invented a long time ago to solve certain kinds | |
| 07:42 | of equations equations that look like this . Something plus | |
| 07:44 | one is equal to a number where a positive number | |
| 07:46 | is the value of the variable . So we needed | |
| 07:49 | a uh a positive real number two to solve that | |
| 07:53 | type of equation . Let's pick another kind of equation | |
| 07:56 | . Let's say we have X plus the number two | |
| 07:58 | is equal to one . Well this looks the same | |
| 08:01 | , but if we solve it it's slightly different . | |
| 08:02 | If we subtract two , then we'll find out that | |
| 08:05 | X is not equal to a positive number , X | |
| 08:07 | is equal to uh not negative to of course X | |
| 08:10 | is equal to negative one . Right ? Because negative | |
| 08:13 | one plus two of course will give me the one | |
| 08:15 | . Now , this is also a real number , | |
| 08:17 | negative numbers are real as well . However , this | |
| 08:19 | was slightly different . It's a negative number . Now | |
| 08:24 | , I need you to think back to maybe 1000 | |
| 08:27 | years ago or 2000 years ago when Algebra mathematics were | |
| 08:30 | really first started to start to be solidified right ? | |
| 08:33 | The idea of a positive number is easy to understand | |
| 08:36 | . I have five eggs in my hand , easy | |
| 08:38 | to understand . I have 16 buckets of rocks or | |
| 08:42 | whatever . I have 16 pencils . That's easy to | |
| 08:44 | understand . Positive numbers are easy , but at some | |
| 08:46 | point in history somebody must have thought , hey , | |
| 08:49 | the idea of a negative number is useful too . | |
| 08:52 | It sounds crazy to you because you're so used to | |
| 08:54 | dealing with negative numbers . But there was a time | |
| 08:56 | when nobody knew what the negative number was , Somebody | |
| 08:58 | had to invent the idea of a negative number . | |
| 09:01 | Why in order to solve an equation like this , | |
| 09:04 | when I subtract something bigger from something smaller than I | |
| 09:08 | can't use a positive number , I have to introduce | |
| 09:10 | the idea of kind of owing somebody . I always | |
| 09:12 | say negative numbers is like if I owe you something | |
| 09:15 | , so if I get an answer of X is | |
| 09:16 | equal to negative one , it doesn't mean I have | |
| 09:18 | $1 in my bank . It means I'm book keeping | |
| 09:22 | that . I actually owe you negative , I owe | |
| 09:24 | you a dollar . So I don't have that dollar | |
| 09:26 | , I actually owe it to somebody else . That's | |
| 09:28 | the idea of what a negative number is . But | |
| 09:30 | that number was not something that we're born understanding . | |
| 09:33 | You have to be taught that . And it's a | |
| 09:35 | useful idea to solve an equation like this , which | |
| 09:38 | is a different kind of equation , right ? It | |
| 09:39 | requires negative numbers . Uh as an answer or as | |
| 09:43 | a uh as we need to invent this in order | |
| 09:46 | to solve this kind of equation . So let's continue | |
| 09:48 | on , let's take another kind of equation . A | |
| 09:50 | little bit down the way in history , somebody tried | |
| 09:52 | to solve an equation like this , X plus 1/4 | |
| 09:54 | is equal to one half . You all know how | |
| 09:56 | to solve this . No problem . I'll just subtract | |
| 09:58 | the 1/4 from both sides . What am I gonna | |
| 10:00 | get ? I'm gonna say that X is equal to | |
| 10:02 | Well , if I take one half minus the fourth | |
| 10:04 | , of course you can do the common denominator but | |
| 10:06 | you all know one half minus 1/4 is equal to | |
| 10:09 | 1/4 . Right ? Again , this is a real | |
| 10:12 | number . I can have 1/4 of a sandwich , | |
| 10:14 | I can have 1/4 of a candy bar . Of | |
| 10:18 | course it's a real number and uh it's a positive | |
| 10:22 | number . Also , I'm not writing down every little | |
| 10:24 | attribute of all these numbers is a positive number , | |
| 10:26 | just like this too is however it's different than the | |
| 10:28 | others because we call this rational a rational number , | |
| 10:34 | remember is a is a number that can be written | |
| 10:36 | as a fraction any fraction . So 1/4 of course | |
| 10:40 | is a rational number , but I didn't write it | |
| 10:42 | down for these two . These two are also rational | |
| 10:44 | numbers because two can be written as to over one | |
| 10:46 | , that's a fraction negative , one can be written | |
| 10:49 | as negative 1/1 . So that's a fraction . So | |
| 10:51 | let's take a little bit of an aside to talk | |
| 10:53 | about these rational these rational they give you some examples | |
| 10:56 | some more rational numbers . The number 10 is a | |
| 10:58 | rational number because I can write it as 10/1 . | |
| 11:01 | The number 3/7 is a rational number because it's a | |
| 11:03 | fraction . Of course The number of negative 2/3 , | |
| 11:06 | that's a rational number because that obviously is a fraction | |
| 11:08 | . It happens to be negative as well . That's | |
| 11:10 | fine . It's still rational . Let's take something crazy | |
| 11:13 | . 1028 1028 , divided by 37 . That's a | |
| 11:17 | fraction because it's a crazy numerator and a crazy denominator | |
| 11:21 | . Let's keep on going because we can make up | |
| 11:23 | a few more and there's a point to this . | |
| 11:24 | I promise . Let's take a look at 0.75 you | |
| 11:27 | might say . Well that's not that's not a a | |
| 11:31 | fraction . However , you know , you can write | |
| 11:32 | this as 3/4 . That's a fraction . So this | |
| 11:34 | decimal is disguised . It's irrational number two . It | |
| 11:38 | turns out that other decimals are rational for instance , | |
| 11:40 | 2.125 . Any decimal that terminates terminates any industrial that | |
| 11:48 | terminates like this . That can also always be written | |
| 11:51 | as a fraction . So this is a rational number | |
| 11:53 | two , I can go find a fraction That's equal | |
| 11:55 | to this . Right ? Let's take our last crazy | |
| 11:57 | one . What about two negative two point uh , | |
| 12:00 | that's not an equal sign there . Sorry about that | |
| 12:02 | . Negative 2.1414 But I'm gonna put a repeating bar | |
| 12:06 | over the last 14 So this is two point 14141414141414 | |
| 12:12 | goes on forever . As long as you have a | |
| 12:15 | repeating pattern in your decimal , you may have learned | |
| 12:17 | from previous lessons in algebra or another math classes . | |
| 12:20 | As long as it's some sort of repeating pattern then | |
| 12:23 | it's always rational . I can find a fraction that | |
| 12:26 | this is that equal to this even though the decimal | |
| 12:28 | goes on to infinity . As long as it is | |
| 12:31 | repeating like this , I can call it . I | |
| 12:33 | can find a fraction that's equal to that . All | |
| 12:36 | right , so I'm gonna write down this is a | |
| 12:38 | repeating special assistant decimal . Let's put the G here | |
| 12:45 | , decimal . Yeah . Why am I going through | |
| 12:49 | all this stuff ? Because all of these , every | |
| 12:51 | one of these things , pretty much every number that | |
| 12:53 | you've really been exposed to , up until now has | |
| 12:55 | been a real number and it's been a rational number | |
| 12:57 | . So these are all rational , Right . But | |
| 13:02 | the point is is that throughout history first we didn't | |
| 13:05 | need to know about fractions at some point in time | |
| 13:07 | there was some person that decided that it was useful | |
| 13:10 | to know what a half of a loaf of bread | |
| 13:11 | was and represented as one half right ? Or three | |
| 13:15 | quarters of a dollar is worth something . So somebody | |
| 13:17 | had to invent fractions . But these these different types | |
| 13:20 | of numbers that we've used have all been invented to | |
| 13:23 | solve different kinds of equations . This one requires real | |
| 13:25 | positive numbers . This one requires real negative numbers . | |
| 13:28 | This one requires real rational numbers . And then once | |
| 13:31 | we figure that out we realize these are also rational | |
| 13:33 | numbers to but then a long time ago , probably | |
| 13:38 | the ancient Greeks , I'm not totally sure but probably | |
| 13:40 | the ancient Greeks um figured out that some numbers do | |
| 13:44 | not terminate like this , the decimals don't stop . | |
| 13:46 | And also if the decimals go on and on forever | |
| 13:49 | , there's some special numbers that don't have any repeating | |
| 13:51 | pattern , like this was 141414 on forever . There | |
| 13:54 | are some numbers that have infinite decimals but they don't | |
| 13:58 | repeat in any pattern that we know about . And | |
| 14:00 | those are useful to solve certain kind of equations as | |
| 14:03 | well . So the famous equation , the area of | |
| 14:05 | a circle is pi times r squared , you know | |
| 14:09 | that equation . Of course you do . Everybody learns | |
| 14:11 | that equation back in fifth or sixth grade . Right | |
| 14:13 | ? But in order to solve this equation we need | |
| 14:15 | to invent a new kind of number . Right ? | |
| 14:17 | So we need what we need pie right ? Which | |
| 14:21 | can be approximately written us 3.14159 dot dot dot dot | |
| 14:26 | dot It goes on and on forever . But the | |
| 14:28 | pattern of the decimal , this 14159 It never repeats | |
| 14:32 | . We have supercomputers calculating pi out to 100,000 million | |
| 14:37 | trillion decimal places . There is no pattern to pie | |
| 14:40 | . It's called an irrational number because the rational numbers | |
| 14:44 | can be written as fractions because they have decimals that | |
| 14:46 | either stop or they go on forever forever in a | |
| 14:49 | pattern . But this is a new kind of number | |
| 14:51 | that we've never needed before . But we need this | |
| 14:53 | number to solve this kind of equation . The area | |
| 14:56 | of a circle requires an irrational number . So this | |
| 14:59 | is called this is real . Of course this is | |
| 15:02 | a real number , right ? I can hold it | |
| 15:04 | in my hand . Of course it's a positive number | |
| 15:06 | . I'm gonna put pos for that . But it's | |
| 15:08 | a new kind of number called an irrational . And | |
| 15:12 | that means I cannot write that as a as a | |
| 15:14 | fraction . There is no way to write pi as | |
| 15:16 | a fraction , you may think well , I should | |
| 15:19 | say there's no way to write Pie is a fraction | |
| 15:22 | of whole numbers . You can't write it as 22/7 | |
| 15:25 | . Some people say Pie is 22/7 , that's just | |
| 15:27 | an approximation . It's not the true value of pi | |
| 15:30 | , right , But this is not the only irrational | |
| 15:32 | number that we've discovered . We have tons of irrational | |
| 15:35 | numbers , we have . Let's talk about another equation | |
| 15:37 | . what about X squared is equal to two ? | |
| 15:39 | This is another kind of equation . How do I | |
| 15:40 | solve this equation ? I'm gonna take the square root | |
| 15:42 | of both sides , right ? So that means I'm | |
| 15:44 | gonna say that X is equal to plus or minus | |
| 15:46 | the square root of two , We've learned about that | |
| 15:48 | . And if you take the square root of two | |
| 15:50 | and you stick it in your calculator and you look | |
| 15:51 | at the decimals , there is no repeating pattern to | |
| 15:55 | the square root of two . It's an infinite number | |
| 15:57 | of decimals that numbers that go beyond the decimal , | |
| 15:59 | but there is no pattern just like Pie has no | |
| 16:01 | pattern . It is a irrational number . We cannot | |
| 16:04 | write the square root of two as any kind of | |
| 16:06 | fraction , so plus square root of two or minus | |
| 16:09 | square root of two . Uh Let's see what we | |
| 16:12 | can just say . The square root of two is | |
| 16:13 | real positive in irrational . I'll put IR for that | |
| 16:19 | and negative square root of two is real books . | |
| 16:24 | Uh Instead of positive , it's negative and it's also | |
| 16:27 | irrational because you cannot write it as a decimal . | |
| 16:30 | All right . And it just in case you're curious | |
| 16:32 | , the square root of two is something like 1.412135 | |
| 16:38 | dot dot dot dot . It does not repeat . | |
| 16:40 | It goes on and on forever and ever . We | |
| 16:42 | can calculate them now . Why am I going through | |
| 16:44 | this trip down memory lane ? I think you can | |
| 16:45 | probably figure it out right . The reason is because | |
| 16:48 | all of these different kinds of numbers that we have | |
| 16:50 | in math were invented at some point in time to | |
| 16:52 | solve certain kinds of equations that we needed to solve | |
| 16:55 | . The real positive numbers were used and we needed | |
| 16:57 | to solve that equation . The real negative numbers , | |
| 17:00 | we had to invent a negative number to solve this | |
| 17:02 | kind of equation . We needed to understand the idea | |
| 17:04 | of a fraction in order to solve equations with fractions | |
| 17:06 | to call them rational numbers . And then we had | |
| 17:08 | to have irrational numbers to solve certain kinds of equations | |
| 17:11 | that we have outlined here , decimals that go forever | |
| 17:14 | . That you cannot write the number as any kind | |
| 17:16 | of a fraction . And now were taken to the | |
| 17:18 | very famous I the imaginary number . We invent the | |
| 17:22 | concept of I because it is useful to solve certain | |
| 17:25 | equations that require I as a solution that requires the | |
| 17:28 | square root of negative numbers . That is the reason | |
| 17:31 | why we have I because we have to have it | |
| 17:33 | to solve certain kinds of equations . And these equations | |
| 17:36 | that we're gonna learn algebra , we're gonna solve tons | |
| 17:37 | of equations that have eyes and answer right ? You're | |
| 17:40 | going to get them you're gonna circle , then you're | |
| 17:41 | gonna say great . But the truth is we use | |
| 17:43 | I probably more than almost anything else in in math | |
| 17:47 | and higher level learning to solve all kinds of things | |
| 17:50 | . Quantum mechanics , advanced physics and chemistry , all | |
| 17:54 | almost always the solution to very common equations in advanced | |
| 17:58 | math . Actually use the concept of i in the | |
| 18:00 | solutions and I'm going to talk more about that at | |
| 18:03 | the very end , we're going to give you a | |
| 18:04 | concrete example of that . But the idea is we | |
| 18:06 | invent this concept divide to solve equations . What kind | |
| 18:09 | of equations do we need to solve ? Right , | |
| 18:12 | let's solve this equation . Very simple . I'm just | |
| 18:15 | gonna put it to you like this . What if | |
| 18:16 | you have the equation X squared is equal to negative | |
| 18:18 | one . How do you solve that ? Well , | |
| 18:21 | in the previous , you know lectures we would have | |
| 18:23 | just taken the square root of both sides . That's | |
| 18:25 | exactly what we're gonna do . We're gonna say that | |
| 18:28 | X is equal to plus or minus the square root | |
| 18:30 | of the right hand side , negative one . But | |
| 18:31 | then we say , well we don't know how to | |
| 18:33 | take the square root of negative one . There is | |
| 18:35 | no real number that actually can be multiplied by itself | |
| 18:38 | to get me negatives one . So up until now | |
| 18:40 | we just stopped and we said , well we can't | |
| 18:41 | do that , it's undefined , but in higher math | |
| 18:43 | , which is where you're at right now , we | |
| 18:45 | don't say it's undefined , we say there's a new | |
| 18:47 | kind of number to solve this equation , just like | |
| 18:50 | we needed those new kinds of numbers to solve those | |
| 18:52 | equations and this number is called the imaginary number , | |
| 18:55 | so we don't stop there , right ? Uh We | |
| 18:59 | go further and we say uh let's take another equation | |
| 19:02 | . X squared is equal to negative four . We | |
| 19:04 | do the same thing , we're gonna take the square | |
| 19:05 | to both sides . We say X is equal to | |
| 19:07 | plus or minus the square root of negative floor . | |
| 19:10 | In the past we would have stopped there , we | |
| 19:12 | would have said , well we don't have any answers | |
| 19:14 | , we're gonna stop , it's undefined . But now | |
| 19:15 | we go further , we say that really X is | |
| 19:18 | equal to plus or minus the square root of negative | |
| 19:20 | one , which we're defining to be eye . So | |
| 19:23 | we say plus or minus . I that's the answer | |
| 19:25 | to that equation . Right over here , we're gonna | |
| 19:29 | do a lot more problems here . What we have | |
| 19:31 | here is the square root of negative for you can | |
| 19:32 | kind of think of , we know how to take | |
| 19:34 | the square root of four . That's just too right | |
| 19:37 | , but the square root of the negative ones still | |
| 19:39 | in there . When we take the square root of | |
| 19:41 | that it's I . So it's really too I and | |
| 19:43 | we still have the plus or minus . So you | |
| 19:46 | see the punch line is when you take the square | |
| 19:48 | root of negative numbers , you take them ignoring the | |
| 19:51 | negative sign , you just pretend the negative signs not | |
| 19:53 | there , you just write the number down just like | |
| 19:55 | always . But then you tack an eye on there | |
| 19:58 | because the negative inside of it produces an imaginary number | |
| 20:01 | . So this is not a real number , you | |
| 20:03 | cannot have two I potatoes , you cannot have six | |
| 20:07 | I strawberries . It's an imaginary number . It doesn't | |
| 20:10 | exist in a tangible form that you can touch . | |
| 20:12 | However , as I've mentioned before , if I have | |
| 20:14 | four I potatoes over here is a solution of some | |
| 20:16 | equation . I may have to multiply it with some | |
| 20:20 | other number of potatoes over here when I can multiply | |
| 20:22 | those those imaginary numbers together , I get a real | |
| 20:26 | number back because when you square I you always get | |
| 20:30 | a real number back , you get negative one . | |
| 20:32 | That's why imaginary numbers are useful . We say they | |
| 20:35 | don't exist initially , but now we say , hey | |
| 20:37 | , we're gonna define an imaginary number , we're gonna | |
| 20:38 | keep track of them . But then sometimes they'll combine | |
| 20:41 | and give us real numbers back anyway . All right | |
| 20:44 | now what I wanna do is solve some very simple | |
| 20:46 | additional problems to give you practice actually taking the square | |
| 20:49 | root of negative numbers and then at the end of | |
| 20:51 | the lesson , I'm gonna do a little bit more | |
| 20:53 | into the philosophy of why imaginary numbers are useful for | |
| 20:57 | modern day stuff . So stick with me to the | |
| 20:59 | end and we'll get there , let's solve some additional | |
| 21:02 | problems . Let's go over here , let's say we | |
| 21:05 | have the square root of negative nine . As I | |
| 21:10 | said , you just ignore the negative . You pretend | |
| 21:12 | that it's the squared of nine . You already know | |
| 21:14 | what that is . That is uh three , right | |
| 21:18 | ? And you know what the square root of negative | |
| 21:19 | one is ? That's just gonna be I so let | |
| 21:21 | me just take it a little bit more step by | |
| 21:23 | step . If you take the squared of the negative | |
| 21:26 | inside , kind of pull it out and take the | |
| 21:28 | square root , you're going to get the eye of | |
| 21:30 | the negative one , but you'll still have this squared | |
| 21:32 | of nine left over . So this is equal to | |
| 21:35 | the eye that I comes about because we're taking the | |
| 21:37 | square root of the negative one in there , poof | |
| 21:39 | ! It turns into an eye , but the square | |
| 21:41 | root of nine is still left over , which you | |
| 21:42 | all know is three . So you can write it | |
| 21:44 | as I times three if you want . But the | |
| 21:46 | way we write imaginary numbers is we always put the | |
| 21:48 | number first . So we say that the squared of | |
| 21:50 | negative nine is three . I . So what you | |
| 21:53 | do is you take the square of the number and | |
| 21:55 | then any time you have a negative there you just | |
| 21:57 | have to tack an eye at the end . That's | |
| 21:58 | all you do . It's extremely , extremely easy to | |
| 22:01 | work with imaginary numbers . So let's take some more | |
| 22:04 | rapid fire example square to 49 . You know what | |
| 22:06 | the square to 49 is it seven ? But you | |
| 22:08 | have that negative so you have to take the square | |
| 22:10 | to that too . So you get seven I that's | |
| 22:12 | the final answer of this problem . This is the | |
| 22:14 | final answer of this problem . What if I have | |
| 22:17 | the square root of negative 36 ? I think you | |
| 22:19 | can see the pattern here . We know that six | |
| 22:21 | times six is 36 . And the square root of | |
| 22:23 | negative one there is the eye . So that's not | |
| 22:26 | a real number . This is an imaginary number six | |
| 22:28 | times the base a number that we call . I | |
| 22:31 | What about negative 100 ? We're gonna take the square | |
| 22:34 | root of this . Well we know how to take | |
| 22:35 | the square root of 100 , that's 10 . And | |
| 22:37 | the square root of the negative there means we have | |
| 22:40 | to put an eye there . So that's a 10 | |
| 22:42 | . I right now we learned up until now we've | |
| 22:46 | learned how to take the square root of much more | |
| 22:48 | complicated things in these simple things . We've learned how | |
| 22:50 | to take the square root of things like 24 . | |
| 22:52 | But what if you have the squared of negative 24 | |
| 22:55 | ? So what you're gonna do is you're gonna completely | |
| 22:57 | ignore the negative side in there . You're gonna come | |
| 22:59 | over to the side of your paper and you're gonna | |
| 23:01 | write that 24 down and build a factor tree . | |
| 23:03 | Like always you're going to ignore the negative . So | |
| 23:06 | for 24 , you know that eight times three is | |
| 23:08 | 24 , but you know that eight is um two | |
| 23:12 | times four and you know that four is two times | |
| 23:14 | two . So you've got your tree built , you're | |
| 23:16 | looking for square roots . So that's a pair . | |
| 23:18 | I still have a two and 23 left over right | |
| 23:21 | ? So what you have is when you write the | |
| 23:23 | answer here , the square root of negative 24 you | |
| 23:26 | just pretend that the negative completely gone . And so | |
| 23:30 | if I if there was no negative there at all | |
| 23:32 | , I would take a single to out , I | |
| 23:34 | would have a square root of two . Times three | |
| 23:37 | is six , right ? I would have that leftover | |
| 23:39 | . Let me write it explicitly on the inside . | |
| 23:41 | It would be two times three leftover . However , | |
| 23:44 | because I took the square to that negative , I | |
| 23:46 | still have an eye here . We don't usually write | |
| 23:48 | the eye after the radical . I mean it's all | |
| 23:50 | multiply . It doesn't really matter if you put the | |
| 23:52 | eye here , we put it in front of the | |
| 23:53 | radical to keep it away because if you put the | |
| 23:56 | eye at the end , you might accidentally leave it | |
| 23:59 | under the radical and you don't want that right . | |
| 24:01 | Eyes do not , imaginary numbers do not live under | |
| 24:04 | the radical in general . So too , I times | |
| 24:06 | the square root of two times three . So it's | |
| 24:07 | two times I times square root of six . This | |
| 24:10 | is the answer to this problem . So whereas before | |
| 24:14 | We don't know what the square root of this negative | |
| 24:16 | 24 is here . You can see all we're doing | |
| 24:18 | is we're writing the square root as if it were | |
| 24:19 | positive under the radical . But then attacking an eye | |
| 24:22 | on because it's an imaginary number . So working with | |
| 24:24 | these imaginary numbers is exceedingly easy . Right ? What | |
| 24:28 | if you have the square root of -5 ? Well | |
| 24:30 | , we can't simplify the negative five . There's no | |
| 24:33 | factor tree that's going to help us . But we | |
| 24:35 | can take the square root of this negative one in | |
| 24:37 | there . So that comes out as an eye . | |
| 24:39 | But what's left over is what's underneath this . The | |
| 24:41 | square defies left over just like we have the left | |
| 24:44 | over six here . In our answer , we have | |
| 24:45 | to leave it as a square root there as well | |
| 24:47 | . So at the end of the day , you | |
| 24:48 | should never ever have negative numbers under a radical . | |
| 24:51 | If the only thing you do is just pull the | |
| 24:53 | negative one out as an eye , then that's fine | |
| 24:56 | . I can't do anything else with the five . | |
| 24:57 | Leave it there under that radical . What about the | |
| 25:00 | squared of negative 12 right same sort of thing . | |
| 25:03 | You pretend this is a positive number . You go | |
| 25:05 | over here or I guess you could go underneath if | |
| 25:07 | you want . But just ignore this . When you | |
| 25:09 | build your factor tree , ignore it . You can | |
| 25:11 | say six times two and six is three times to | |
| 25:14 | you're looking for pairs . So there's your pair right | |
| 25:16 | there . So if there were no nothing , no | |
| 25:19 | negative number there , the two would come out and | |
| 25:22 | then underneath you would have a square root of three | |
| 25:24 | which is left over . But you still have the | |
| 25:26 | square of this negative one , which means an eye | |
| 25:28 | comes out . So it's two times I times the | |
| 25:30 | square root of three . Now on this page , | |
| 25:33 | what I have done is I have tried to show | |
| 25:38 | you a basic idea about how to take square roots | |
| 25:41 | of negative numbers . So you just do the factor | |
| 25:43 | tree as usual . And if you have a negative | |
| 25:45 | under there you just tack an eye on extremely simple | |
| 25:48 | . Alright . But we also learned another property of | |
| 25:52 | rat of this imaginary number because it's defined this way | |
| 25:56 | the I square the square . The times itself is | |
| 26:00 | not imaginary anymore . It's equal to a negative one | |
| 26:02 | which is a real number . So any time in | |
| 26:05 | an equation you see an eye squared . Then in | |
| 26:08 | the very next step you just substitute negative one . | |
| 26:11 | You don't have to think about it , you don't | |
| 26:12 | have to drive it or calculate it anytime you see | |
| 26:15 | I squared , you just write down negative one . | |
| 26:17 | So because of that we can do some simple problems | |
| 26:20 | here . So let's just get some practice . The | |
| 26:21 | simplest one is what is I squared ? You just | |
| 26:24 | replace it with negative one . You don't have to | |
| 26:25 | think about it . You have to prove it . | |
| 26:27 | What if I have two times I squared ? Well | |
| 26:30 | the two is totally normal but the I squared is | |
| 26:33 | negative one . So you just substitute for I squared | |
| 26:35 | negative one and the answer to this is negative two | |
| 26:38 | . So you see what I mean , how you | |
| 26:39 | have to imaginary numbers multiplied together . But at the | |
| 26:42 | end of the day you don't get an imaginary number | |
| 26:44 | back . You get a real number back . Right | |
| 26:47 | . Same thing we got here . What if we | |
| 26:49 | have three ? I wrapped up inside of the parentheses | |
| 26:52 | and that whole thing squared will be treated as we | |
| 26:55 | do anything . You pretend this is kind of like | |
| 26:56 | a variable . The exponent will apply to the three | |
| 26:59 | into the eye separately . So you'll have three square | |
| 27:02 | times I squared . The same rules of algebra apply | |
| 27:04 | to complex number two , imaginary numbers three squared is | |
| 27:07 | nine and you're still gonna have this I squared . | |
| 27:10 | But anytime you see an eye square you substitute negative | |
| 27:13 | one . Anytime you see I scared you just put | |
| 27:15 | negative one there , so it's negative nine . This | |
| 27:17 | is the final answer . So when we have three | |
| 27:20 | I quantity squared , we get a real number back | |
| 27:22 | which is negative nine . Now it turns out that | |
| 27:25 | these imaginary numbers if you have them in a fraction | |
| 27:28 | , they can cancel . Just like variables can cancel | |
| 27:31 | . So for instance , four times I squared in | |
| 27:33 | the numerator with eye on the bottom right ? So | |
| 27:37 | I'm gonna rewrite it just so I don't clutter up | |
| 27:39 | my problem statement . So this is for I squared | |
| 27:41 | over I . So if these were variables , if | |
| 27:43 | this was like four X squared over X , what | |
| 27:45 | you would do is you would say well I'm gonna | |
| 27:46 | cancel this one and I'm gonna cancel with this one | |
| 27:49 | . I'm gonna leave an explosion of one left behind | |
| 27:51 | . Basically I'm dividing away one of them and the | |
| 27:53 | same thing is true of imaginary numbers . So I | |
| 27:56 | had an imaginary on the bottom and imaginary squared on | |
| 27:59 | the top . All I'm left with is the four | |
| 28:01 | and the eye on the top to the first power | |
| 28:04 | , it's basically gone divided away in the bottom . | |
| 28:06 | So I get an answer of four times I okay | |
| 28:10 | . What if I have something more complicated ? What | |
| 28:12 | about five times I . Quantity Q . But on | |
| 28:16 | the bottom I also have and I I don't handle | |
| 28:19 | that well I still have this numerator three is going | |
| 28:23 | to apply to the five as an exponent . It's | |
| 28:25 | also going to go in and apply to this I | |
| 28:27 | right so you have five cubed times I . Cube | |
| 28:31 | . And on the bottom you still have this I | |
| 28:33 | right but notice that I have the same variable just | |
| 28:35 | like it's not a variable , it's an imaginary number | |
| 28:38 | but you can treat it as a variable that can | |
| 28:39 | cancel with this . Leaving me too behind . So | |
| 28:43 | what I have left with is five cube five times | |
| 28:45 | five times five is going to be 125 . But | |
| 28:49 | what's left over is I squared but we now know | |
| 28:53 | that I square . It is very easy to remember | |
| 28:55 | any time you see it you just replace it with | |
| 28:56 | a negative one . So what you have is negative | |
| 28:59 | 125 . So this whole expression which looks very complicated | |
| 29:03 | with imaginary numbers everywhere actually is not imaginary in the | |
| 29:07 | end , this is one of the things I've been | |
| 29:09 | trying to harp on to tell you is that oftentimes | |
| 29:12 | when you're working with imaginary numbers , you'll get real | |
| 29:14 | numbers as your answers . The last one I'm gonna | |
| 29:17 | do for you is a really interesting one . What | |
| 29:21 | if you have I cubed some , some students will | |
| 29:24 | look at that and say I have no idea . | |
| 29:25 | You didn't tell me what I cube was equal to | |
| 29:27 | until you remember that I cube can be written as | |
| 29:30 | I times I squared . Why ? Because you can | |
| 29:34 | treat it as a variable . I can add these | |
| 29:36 | exponents of course it's equal to this and I know | |
| 29:39 | that this I squared is negative one . So really | |
| 29:41 | it's I times negative one which means it's gonna be | |
| 29:44 | negative . I so the imaginary number I can be | |
| 29:47 | positive , it can be negative , it can have | |
| 29:49 | fractions involved , decimals involved , it's all fine . | |
| 29:52 | It's just a placeholder for the square root of negative | |
| 29:55 | one . Anytime I see an I squared anywhere I | |
| 29:58 | have to replace it with the real number which is | |
| 30:00 | negative one . Now I've talked at great length or | |
| 30:04 | at least in the introduction to try to give you | |
| 30:06 | some motivation why imaginary numbers are so useful . What | |
| 30:10 | usually happens is you study imaginary numbers in algebra and | |
| 30:13 | then in pre calculus you do more with it and | |
| 30:15 | then in calculus you do even more with it . | |
| 30:17 | And then when you get in really advanced math beyond | |
| 30:19 | calculus you take advanced math classes where we solve special | |
| 30:23 | equations . Doesn't matter what they're called . They're called | |
| 30:25 | differential equations but that's that's the name of it . | |
| 30:27 | But anyway their equations and you deal with imaginary numbers | |
| 30:30 | every day and then finally at the end you understand | |
| 30:33 | why they're so useful and how common they are and | |
| 30:36 | how real that they are in the solutions of our | |
| 30:38 | mathematics right ? But I want a short circuit that | |
| 30:41 | because I don't want you to wait six years to | |
| 30:43 | figure after seven or eight years to figure out why | |
| 30:44 | they're useful . I'm gonna give you a quick 10 | |
| 30:46 | minutes or five minutes explaining that to you . Okay | |
| 30:49 | so I've talked about in general the idea that I | |
| 30:52 | can have an imaginary number like I have I strawberries | |
| 30:56 | and I can have another imaginary number in my other | |
| 30:59 | hand another I strawberries . So individually this is an | |
| 31:01 | imaginary number and this is also separately an imaginary number | |
| 31:05 | . But when I multiply them together they form I | |
| 31:08 | squared , which I know is equal to negative one | |
| 31:10 | . So this is one of the main reasons why | |
| 31:12 | they're so useful because separately they're not real numbers . | |
| 31:15 | But when I combine them together , that can give | |
| 31:17 | me a real number back , which is a tangible | |
| 31:19 | thing . This is something I can touch in the | |
| 31:21 | real world . I mean , yeah , it's negative | |
| 31:23 | , but it's still a number that exists in our | |
| 31:25 | everyday , you know , life so we can get | |
| 31:27 | as a result a real result . Now , this | |
| 31:32 | isn't too exciting . The I times I and all | |
| 31:34 | that . It's not too exciting . But there's one | |
| 31:36 | example I'm gonna give you that's way beyond the scope | |
| 31:38 | of an algebra class . But I want to give | |
| 31:39 | it to you because it is so powerful for you | |
| 31:42 | to understand how how common imaginary numbers are in real | |
| 31:46 | world problems . I want to give it to you | |
| 31:47 | . I don't want you to wait until your second | |
| 31:49 | year of college to see it . All right , | |
| 31:51 | I want to introduce something to you that we're gonna | |
| 31:53 | get to later in this math class , in this | |
| 31:55 | algebra class , I want to talk to you about | |
| 31:57 | something called the sign of some number . Right ? | |
| 32:01 | So the sign , we're gonna get to this in | |
| 32:03 | algebra later on in our in our set of lectures | |
| 32:05 | , I'm gonna explain this to you in a future | |
| 32:07 | lesson . So this is not something crazy that you're | |
| 32:10 | never ever going to see . This is something uh | |
| 32:12 | quite common that that you that you learn . So | |
| 32:15 | the sign is a special function . It's a very | |
| 32:18 | special function . Sine of X . This is the | |
| 32:20 | X . Axis , and this is the sine of | |
| 32:22 | X . It's very special because it has this wavy | |
| 32:25 | shape to it , so it goes like up and | |
| 32:27 | then it goes down and it goes up and then | |
| 32:28 | it goes down and on the other side it continues | |
| 32:31 | the other direction it goes down and up and down | |
| 32:33 | so on . So you can see it's this general | |
| 32:35 | wave shape right ? That you that you recognize ? | |
| 32:39 | Okay , why is this important to study this thing | |
| 32:41 | called a sign ? Why is it so important to | |
| 32:43 | study something that has a wavy shape ? Well , | |
| 32:45 | I'll try to explain it to you . You know | |
| 32:47 | , light we get into advanced physics much much later | |
| 32:51 | , but light one theory of light is that it | |
| 32:53 | behaves like a wave , right ? What kind of | |
| 32:56 | wave here it is . This is the kind of | |
| 32:58 | wave that a light behave . The lightwave actually is | |
| 33:01 | . So later on in your education , when you | |
| 33:03 | write down , what does a lightwave look like or | |
| 33:05 | what does a lightwave behave like ? You're going to | |
| 33:08 | write down a sign ? Or you might write down | |
| 33:10 | its cousin which is called the co sign . I'm | |
| 33:12 | not gonna get into that because it's the same shape | |
| 33:14 | . A CO sign is the same shape as a | |
| 33:16 | sign . It's a wavy shape , a very special | |
| 33:18 | shape that's extremely common in nature . All uh lightwaves | |
| 33:23 | , r sine waves , right ? Which means x | |
| 33:25 | rays , gamma rays , infrared , visible light . | |
| 33:29 | The light that goes into your eyes , microwaves that | |
| 33:31 | you heat your food , radio waves , which your | |
| 33:33 | your cell phone use is in your your your walkie | |
| 33:36 | talkies to talk to people . All of those things | |
| 33:38 | are sine waves . Right ? So I'm I'm building | |
| 33:41 | it up for you because of the following thing . | |
| 33:45 | This sine wave is numbers that are plotted on the | |
| 33:48 | Y axis here as a function of this is just | |
| 33:51 | like any other function and just goes up and down | |
| 33:53 | and up and down and up and down . It | |
| 33:55 | can be proven in advanced math classes . It's can | |
| 34:01 | be shown , I'm not going to show it here | |
| 34:04 | because it's way beyond the scope of this . But | |
| 34:06 | it can be shown that this thing that we call | |
| 34:09 | the sign can be written like this equals E . | |
| 34:14 | To the I times X minus E to the negative | |
| 34:18 | , I , times X . All divided by two | |
| 34:22 | times I . This is incredibly important for later mathematics | |
| 34:27 | . It's not important for algebra . I don't care | |
| 34:29 | if you remember this , I don't care if you | |
| 34:31 | toss it in the in the in the trash can | |
| 34:32 | and just forget about it . That's fine . When | |
| 34:34 | I'm trying to show you though , is notice what | |
| 34:37 | you have here . E is just a number we're | |
| 34:39 | gonna talk about . E actually not too much longer | |
| 34:41 | from now . That's a number . It's a number | |
| 34:43 | like pi it's an irrational number . That's very common | |
| 34:45 | in nature . It's not equal to pi it's equal | |
| 34:47 | to about 2.7 and some some uh infinite number of | |
| 34:51 | decimal places after that . But it's just a number | |
| 34:53 | . So this E is not a variable , it's | |
| 34:55 | a number , it's some number . To the to | |
| 34:57 | the what ? To the power of I . X | |
| 34:59 | . So to an imaginary power . So this is | |
| 35:02 | an imaginary power , this is an imaginary power to | |
| 35:05 | but it just has a negative sign in front . | |
| 35:07 | So if I take this guy to this imaginary power | |
| 35:10 | and then I subtract E raised to the exact same | |
| 35:13 | thing but to the negative a copy of itself . | |
| 35:15 | But to the negative imaginary power if I take the | |
| 35:18 | answer to that and I divide by two times I | |
| 35:20 | which is another imaginary number . I'm putting these imaginary | |
| 35:24 | numbers in a pot and I'm stirring them around in | |
| 35:26 | this function . What you get out is not an | |
| 35:30 | imaginary function , it's a real number . So imaginary | |
| 35:34 | numbers go in here , imaginary numbers go in here | |
| 35:37 | , imaginary numbers go in there . What you get | |
| 35:38 | out is not an imaginary number . You get a | |
| 35:40 | real number back and that real number is a plot | |
| 35:43 | . I can plot this real number as a function | |
| 35:46 | of whatever I stick in here . See this is | |
| 35:48 | exit goes in here . Well you would put it | |
| 35:49 | in here , put it in here , calculated all | |
| 35:51 | get the thing but you don't get an imaginary number | |
| 35:54 | back . Why ? It's for the same basic reason | |
| 35:56 | that we don't get an imaginary number back here . | |
| 35:59 | I mean it's beyond the scope . I don't want | |
| 36:00 | to prove it . But basically you can see how | |
| 36:02 | you can combine imaginary numbers and get real numbers back | |
| 36:06 | . Well I'm combining imaginary numbers in a much more | |
| 36:08 | complicated way and you're just gonna have to take my | |
| 36:11 | word for it that you don't get an imaginary number | |
| 36:13 | back . So why am I doing all of this | |
| 36:15 | ? Because when I first learned what an imaginary number | |
| 36:18 | was , I thought they were useless , I thought | |
| 36:20 | they would just I would just circle some answers for | |
| 36:22 | the teacher and be done . But it turns out | |
| 36:24 | that almost every equation that we solve in real life | |
| 36:28 | looks like some kind of wave and if it looks | |
| 36:30 | like some kind of wave with a sine wave then | |
| 36:33 | it can be written as imaginary numbers . And so | |
| 36:35 | a lot of times you'll get an equation , you'll | |
| 36:37 | solve it , you'll get some imaginary numbers in the | |
| 36:39 | answer . I might have to take that and multiply | |
| 36:42 | by another equation which has some other imaginary numbers when | |
| 36:45 | they get all mixed together . Sometimes what pops out | |
| 36:47 | is not an imaginary number at all , it's a | |
| 36:49 | real number and that's why it's so useful . Another | |
| 36:53 | example would be quantum mechanics way future physics , you'll | |
| 36:56 | study that long time from now . All the solutions | |
| 36:58 | of quantum occasions , they all but most of the | |
| 37:00 | solutions of quantum mechanics are all waves . They all | |
| 37:04 | involve these signs are they involve these exponents with I | |
| 37:08 | in there and you mix them around and oftentimes you | |
| 37:11 | get real numbers back , right , That is why | |
| 37:13 | imaginary numbers are so useful . We don't know how | |
| 37:16 | to take the square root of negative one with a | |
| 37:18 | real answer , but as a placeholder we say we | |
| 37:21 | just say it's equal to I will keep it as | |
| 37:23 | a placeholder and we start combining it with other things | |
| 37:25 | and oftentimes pop poof , the imaginary numbers go away | |
| 37:28 | and we have a real answer back . So it's | |
| 37:30 | very , very tangible . When you get some more | |
| 37:32 | advanced math , a lot of your answers will be | |
| 37:34 | imaginary , you'll get very used to it , they | |
| 37:36 | won't seem crazy or weird , and then often they | |
| 37:38 | combine back into real numbers anyway , which we can | |
| 37:41 | measure in the real world . That is why imaginary | |
| 37:44 | numbers are useful . So make sure you understand this | |
| 37:46 | and then follow me on the next few lessons . | |
| 37:48 | We're gonna learn how to work with imaginary numbers , | |
| 37:51 | a little more , a little more detail than we've | |
| 37:54 | done here . We get a lot of practice by | |
| 37:55 | solving problems . |
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