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  • 00:03

    BAM!!! Mr. Tarrou. How are you doing today? I am going to be talking about imaginary numbers.
    BAM!!! Mr. Tarrou. How are you doing today? I am going to be talking about imaginary numbers.

  • 00:09

    Like they hardly even exist but yet we still need them in mathematics. Now imaginary numbers
    Like they hardly even exist but yet we still need them in mathematics. Now imaginary numbers

  • 00:16

    really in a textbook are called Complex Numbers. They have two parts, they have a real part
    really in a textbook are called Complex Numbers. They have two parts, they have a real part

  • 00:23

    and an imaginary part...or a plus b times i. Now what do I mean by imaginary. What does
    and an imaginary part...or a plus b times i. Now what do I mean by imaginary. What does

  • 00:28

    that even mean, or come up in math? Maybe if you have started solving parabolas, or
    that even mean, or come up in math? Maybe if you have started solving parabolas, or

  • 00:33

    doing the quadratic formula, you might have noticed that sometimes you don't get an answer.
    doing the quadratic formula, you might have noticed that sometimes you don't get an answer.

  • 00:39

    You try and do the square root and you end up with a negative number inside your radical.
    You try and do the square root and you end up with a negative number inside your radical.

  • 00:43

    That is one example where imaginary numbers come in to play. The square root of negative
    That is one example where imaginary numbers come in to play. The square root of negative

  • 00:47

    1 is equal to "i" Now there are some things in math that we teach as a rule that I can
    1 is equal to "i" Now there are some things in math that we teach as a rule that I can

  • 00:53

    do a pretty good job of teaching my students like why that works or how. This is one is
    do a pretty good job of teaching my students like why that works or how. This is one is

  • 00:58

    a little bit tough for that, so we are just going to let this letter "i" stand for the
    a little bit tough for that, so we are just going to let this letter "i" stand for the

  • 01:04

    square root of negative one. i for imaginary! Why is it imaginary? Because if you square
    square root of negative one. i for imaginary! Why is it imaginary? Because if you square

  • 01:11

    root negative one, or you try to come up with a square root, there are not two numbers that
    root negative one, or you try to come up with a square root, there are not two numbers that

  • 01:16

    you multiply by themselves to get you negative one. The square root of 4 is 2. Why? Because
    you multiply by themselves to get you negative one. The square root of 4 is 2. Why? Because

  • 01:21

    2 times 2 is 4. But what two real numbers can you multiply together to get negative
    2 times 2 is 4. But what two real numbers can you multiply together to get negative

  • 01:26

    one, they don't exist. Negative one times negative one is positive one. Maybe it is
    one, they don't exist. Negative one times negative one is positive one. Maybe it is

  • 01:31

    the square root of negative one is positive one. Well one times one is still positive
    the square root of negative one is positive one. Well one times one is still positive

  • 01:35

    one. You cannot take an even root of a negative number. So we must give it this imaginary
    one. You cannot take an even root of a negative number. So we must give it this imaginary

  • 01:41

    value if i. Well one fact that we are going to use quite a bit today is about i squared.
    value if i. Well one fact that we are going to use quite a bit today is about i squared.

  • 01:52

    Now i squared. i squared is just as it says, i times i or the square root of negative one
    Now i squared. i squared is just as it says, i times i or the square root of negative one

  • 02:01

    squared. What happens when you square a square root, don't they cancel out? They do so those
    squared. What happens when you square a square root, don't they cancel out? They do so those

  • 02:08

    are going to cancel out and you get that i squared is equal to negative one. So i is
    are going to cancel out and you get that i squared is equal to negative one. So i is

  • 02:13

    the square root of negative one, but then i squared is negative one. This is how we
    the square root of negative one, but then i squared is negative one. This is how we

  • 02:20

    can use these imaginary numbers in math calculations because as we manipulate it more and more
    can use these imaginary numbers in math calculations because as we manipulate it more and more

  • 02:24

    sometimes we are able to cancel out that square root that we have and actually get a real
    sometimes we are able to cancel out that square root that we have and actually get a real

  • 02:31

    number back that we can use in a later calculation. Ok, i to the third is i times i squared. We
    number back that we can use in a later calculation. Ok, i to the third is i times i squared. We

  • 02:41

    just said that i squared is what, i squared is negative one...so i to the third is negative
    just said that i squared is what, i squared is negative one...so i to the third is negative

  • 02:49

    i. i to the forth is i squared times i squared which is negative one times negative one which
    i. i to the forth is i squared times i squared which is negative one times negative one which

  • 03:01

    is positive one. Now I am giving you these four values or powers of i because in your
    is positive one. Now I am giving you these four values or powers of i because in your

  • 03:08

    book you may be using those for some questions. Now I am not going to use the third and forth
    book you may be using those for some questions. Now I am not going to use the third and forth

  • 03:13

    powers for what I am about to run through for my students in PreCalculus...and this
    powers for what I am about to run through for my students in PreCalculus...and this

  • 03:18

    also applies to an algebra 2 class. We are going to be focusing on i is the square root
    also applies to an algebra 2 class. We are going to be focusing on i is the square root

  • 03:22

    of negative one and i squared then becomes negative one, not the square root but just
    of negative one and i squared then becomes negative one, not the square root but just

  • 03:28

    negative one. We are not doing this in my math class, but in case you are... if you
    negative one. We are not doing this in my math class, but in case you are... if you

  • 03:33

    are trying to graph or draw where these imaginary numbers are on the x y axis. Well we don't
    are trying to graph or draw where these imaginary numbers are on the x y axis. Well we don't

  • 03:40

    really have an x y axis if we are talking about imaginary numbers. No, we have a real
    really have an x y axis if we are talking about imaginary numbers. No, we have a real

  • 03:44

    axis and an imaginary axis. Now you notice that I took out the x and put in real which
    axis and an imaginary axis. Now you notice that I took out the x and put in real which

  • 03:56

    is the first number. I took out y and put in imaginary which is the coefficient of the
    is the first number. I took out y and put in imaginary which is the coefficient of the

  • 04:01

    second term. So when you go to graph these, it is going to seem like are graphing a regular
    second term. So when you go to graph these, it is going to seem like are graphing a regular

  • 04:06

    point. 2 + 3i is over two and up three. -1 -4i is left and down, so negative one minus
    point. 2 + 3i is over two and up three. -1 -4i is left and down, so negative one minus

  • 04:17

    four i is there. This is just 6i, where is the real part? Right there! No left right
    four i is there. This is just 6i, where is the real part? Right there! No left right

  • 04:28

    movement, just go up six... 1,2,3,4,5,6 So there is a super quick example of how you
    movement, just go up six... 1,2,3,4,5,6 So there is a super quick example of how you

  • 04:33

    can graph these three complex values. Now let's get into what my kids are doing which
    can graph these three complex values. Now let's get into what my kids are doing which

  • 04:40

    is just some basic algebraic manipulation that involve the value of i. Let me make sure
    is just some basic algebraic manipulation that involve the value of i. Let me make sure

  • 04:51

    I have some time left. Ten minutes. Ok, let's run through them. I am going to go as quickly
    I have some time left. Ten minutes. Ok, let's run through them. I am going to go as quickly

  • 05:00

    as I can because I want to get a lot of examples done in the next ten minutes. We have got
    as I can because I want to get a lot of examples done in the next ten minutes. We have got

  • 05:05

    3 plus 2i minus 4 plus 5i. I am just giving you some binomials, two complex numbers, and
    3 plus 2i minus 4 plus 5i. I am just giving you some binomials, two complex numbers, and

  • 05:19

    asked you to basically simplify. You are going to do this just like you did your generic
    asked you to basically simplify. You are going to do this just like you did your generic

  • 05:24

    little combining like terms and getting rid of parenthesis problems. Distribute to get
    little combining like terms and getting rid of parenthesis problems. Distribute to get

  • 05:31

    the negative through the parenthesis and look for like terms. Three and negative four is negative one. What about two and negative
    the negative through the parenthesis and look for like terms. Three and negative four is negative one. What about two and negative

  • 05:44

    five? Well two minus five is negative three. Real and imaginary. So this looks very similar
    five? Well two minus five is negative three. Real and imaginary. So this looks very similar

  • 05:51

    to basic algebra skills you have been using for a long time. Let's take a look at this
    to basic algebra skills you have been using for a long time. Let's take a look at this

  • 05:56

    example. Negative three plus seven i squared. OHHHH how many times do I see kids mess this
    example. Negative three plus seven i squared. OHHHH how many times do I see kids mess this

  • 06:03

    up. This is a binomial squared. It is not negative three squared and seven squared.
    up. This is a binomial squared. It is not negative three squared and seven squared.

  • 06:07

    no no no no... It is negative 3 plus 7i squared is negative 3 plus 7i times negative 3 plus
    no no no no... It is negative 3 plus 7i squared is negative 3 plus 7i times negative 3 plus

  • 06:18

    7i. Please take the time and save your teacher a lot of grief and actually distribute those
    7i. Please take the time and save your teacher a lot of grief and actually distribute those

  • 06:25

    together so you have all the terms, you don't miss one. Negative 3 times negative 3 is 9.
    together so you have all the terms, you don't miss one. Negative 3 times negative 3 is 9.

  • 06:30

    Negative 3 and 7 is negative 21. And going to the next set of lines we have negative
    Negative 3 and 7 is negative 21. And going to the next set of lines we have negative

  • 06:37

    21 again...i And then positive 49 i squared. Here we start, this is the only place where
    21 again...i And then positive 49 i squared. Here we start, this is the only place where

  • 06:52

    there is a difference when you are working with complex numbers compared to just your
    there is a difference when you are working with complex numbers compared to just your

  • 06:57

    regular algebra. Instead of this being just like an x squared, it is i squared. We just
    regular algebra. Instead of this being just like an x squared, it is i squared. We just

  • 07:02

    talked about...oops I erased it... We just talked about how i squared was equal to negative
    talked about...oops I erased it... We just talked about how i squared was equal to negative

  • 07:08

    one. So this i squared comes out and in its place goes negative one. Now we are going
    one. So this i squared comes out and in its place goes negative one. Now we are going

  • 07:18

    to combine like terms. We have 9 and this is going to be negative 49, so negative 49
    to combine like terms. We have 9 and this is going to be negative 49, so negative 49

  • 07:25

    plus 9 is negative 40. The two like terms with i's, now they are not i squares so the
    plus 9 is negative 40. The two like terms with i's, now they are not i squares so the

  • 07:32

    i's are going to stay there, negative 21 and negative 21 is negative 42 i. Now for proper
    i's are going to stay there, negative 21 and negative 21 is negative 42 i. Now for proper

  • 07:39

    form I need to write this in the other way because the real should come before the imaginary.
    form I need to write this in the other way because the real should come before the imaginary.

  • 07:44

    So negative 40 minus 42 i. BAM!!! Alright. Moving on. Did I wake you up? Let's take a
    So negative 40 minus 42 i. BAM!!! Alright. Moving on. Did I wake you up? Let's take a

  • 07:54

    look at this. The square root of negative 81 times the square root of negative 32. You
    look at this. The square root of negative 81 times the square root of negative 32. You

  • 08:03

    cannot work with radicals that have negative numbers inside of them. If you just multiply
    cannot work with radicals that have negative numbers inside of them. If you just multiply

  • 08:09

    these together and get a positive whatever, I can't do that in my head, you are going
    these together and get a positive whatever, I can't do that in my head, you are going

  • 08:14

    to get a number that is much too big...excuse me, not too big, you will have a sign error.
    to get a number that is much too big...excuse me, not too big, you will have a sign error.

  • 08:18

    Your answer...you will believe your answer should be positive but you will see in a minute
    Your answer...you will believe your answer should be positive but you will see in a minute

  • 08:21

    it is actually supposed to be negative. If you have negatives inside your square root
    it is actually supposed to be negative. If you have negatives inside your square root

  • 08:27

    symbols, you need to pull those out as i's before you start. Now 81, the square root
    symbols, you need to pull those out as i's before you start. Now 81, the square root

  • 08:31

    of 81 is what? It is 9. And what is the square root of negative one? Because the 81 is negative.
    of 81 is what? It is 9. And what is the square root of negative one? Because the 81 is negative.

  • 08:39

    The square root of negative one is i. Times.... Well what is going on over here? I am going
    The square root of negative one is i. Times.... Well what is going on over here? I am going

  • 08:46

    to break this up a bit more just to make sure you know what I am doing. I have got negative
    to break this up a bit more just to make sure you know what I am doing. I have got negative

  • 08:52

    one, so the square root of negative one. And 32 won't square root but there is a perfect
    one, so the square root of negative one. And 32 won't square root but there is a perfect

  • 08:57

    square in 32. I am going to reduce this. There is a 16 times 2. So the negative one I am
    square in 32. I am going to reduce this. There is a 16 times 2. So the negative one I am

  • 09:05

    going to leave separate. Square root of 16 and the square root of 2. The square root
    going to leave separate. Square root of 16 and the square root of 2. The square root

  • 09:10

    of negative one is i and the square root of 16 is 4. So we are going to have 9i times
    of negative one is i and the square root of 16 is 4. So we are going to have 9i times

  • 09:16

    4i times the square root of 2. 9 times 4 is 36 i squared and the square root of 2. Now
    4i times the square root of 2. 9 times 4 is 36 i squared and the square root of 2. Now

  • 09:27

    if I had left this the way it was and just multiplied, my answer would be positive 36
    if I had left this the way it was and just multiplied, my answer would be positive 36

  • 09:33

    square root of 2. But if I take the i out, like I am supposed to, i squared is what?
    square root of 2. But if I take the i out, like I am supposed to, i squared is what?

  • 09:39

    Yeah it is. Negative one. So negative 36 square root of 2. BAM!!! Alright. Let's take a look
    Yeah it is. Negative one. So negative 36 square root of 2. BAM!!! Alright. Let's take a look

  • 09:49

    at another example. Let's take a look at 4 - 3i over 2-5i. Simplify this. Well it looks
    at another example. Let's take a look at 4 - 3i over 2-5i. Simplify this. Well it looks

  • 10:07

    pretty simple already. If this were an x instead of an i we would be done. But i is a square
    pretty simple already. If this were an x instead of an i we would be done. But i is a square

  • 10:14

    root...right? i is the square root of negative one. You cannot leave a radical in the denominator
    root...right? i is the square root of negative one. You cannot leave a radical in the denominator

  • 10:21

    if you are writing your final answer. So this, though it does not look like it, needs to
    if you are writing your final answer. So this, though it does not look like it, needs to

  • 10:25

    be rationalized. We need to multiply by something that will get my square root of negative one,
    be rationalized. We need to multiply by something that will get my square root of negative one,

  • 10:32

    my i, out of the denominator. You do that when you have binomials by multiplying by
    my i, out of the denominator. You do that when you have binomials by multiplying by

  • 10:37

    what is called a conjugate. That means that you simply change the sign, the middle sign,
    what is called a conjugate. That means that you simply change the sign, the middle sign,

  • 10:46

    and multiply that to the top and bottom of the fraction. So we are going to get 4 times
    and multiply that to the top and bottom of the fraction. So we are going to get 4 times

  • 10:55

    2 is 8. 4 times 5 is positive 20i. Negative 3 and 2 is negative 6i. Negative 3 and 5 is
    2 is 8. 4 times 5 is positive 20i. Negative 3 and 2 is negative 6i. Negative 3 and 5 is

  • 11:06

    negative 15 i squared. All over 2 times 2 is 4. 2 times positive 5 is 10i. Negative
    negative 15 i squared. All over 2 times 2 is 4. 2 times positive 5 is 10i. Negative

  • 11:16

    5 and 2 is negative 10i. You might not want to show this work because you know the middle
    5 and 2 is negative 10i. You might not want to show this work because you know the middle

  • 11:20

    term is going to cancel out, but I am teaching so I have to. Negative 5 and positive 5 is
    term is going to cancel out, but I am teaching so I have to. Negative 5 and positive 5 is

  • 11:26

    negative 25 i squared. So we have got the positive and negative 10 canceling out. Because
    negative 25 i squared. So we have got the positive and negative 10 canceling out. Because

  • 11:35

    of space, I am not going to write a new line, but what is i squared? Negative 1. What is
    of space, I am not going to write a new line, but what is i squared? Negative 1. What is

  • 11:42

    i squared...negative one. So we have 8 and that is positive 15 right...so 8 plus 15 is
    i squared...negative one. So we have 8 and that is positive 15 right...so 8 plus 15 is

  • 11:51

    23. What do we have here? We have 20 minus 6 is 14 i. All over 4 minus...not minus...that
    23. What do we have here? We have 20 minus 6 is 14 i. All over 4 minus...not minus...that

  • 12:04

    is plus, so 4 and positive 25 is 29. Now this is not done. Couples numbers have a real and
    is plus, so 4 and positive 25 is 29. Now this is not done. Couples numbers have a real and

  • 12:13

    an imaginary part. I am simply going to split this apart and give each fraction it's own
    an imaginary part. I am simply going to split this apart and give each fraction it's own

  • 12:18

    denominator of 29. BAM!!! Got it done. I have 2.5 minutes. Let's see if we can try and solve
    denominator of 29. BAM!!! Got it done. I have 2.5 minutes. Let's see if we can try and solve

  • 12:36

    a quadratic formula...shall we...or quadratic equation. Last example. Let's see if we can
    a quadratic formula...shall we...or quadratic equation. Last example. Let's see if we can

  • 12:48

    solve this equation. Don't forget when you solve an equation your are trying to find
    solve this equation. Don't forget when you solve an equation your are trying to find

  • 13:00

    out where that equation crosses the x axis. We are trying to find out when y equals 0,
    out where that equation crosses the x axis. We are trying to find out when y equals 0,

  • 13:05

    where is this graph crossing the x axis. Well this is a parabola. There is only one squared
    where is this graph crossing the x axis. Well this is a parabola. There is only one squared

  • 13:11

    term. And it might factor, but I want to show you the use of the quadratic formula which
    term. And it might factor, but I want to show you the use of the quadratic formula which

  • 13:15

    is negative b plus or minus the square root of b squared minus 4ac all over 2a. When you
    is negative b plus or minus the square root of b squared minus 4ac all over 2a. When you

  • 13:25

    have an equation solved for 0 and your exponents count down, this is a, this is b, and this
    have an equation solved for 0 and your exponents count down, this is a, this is b, and this

  • 13:32

    is c. So let's plug it in and see what happens. The opposite of b plus or minus the square
    is c. So let's plug it in and see what happens. The opposite of b plus or minus the square

  • 13:40

    root of b squared minus 4 times a times c all over 2 times a. Now you might notice every
    root of b squared minus 4 times a times c all over 2 times a. Now you might notice every

  • 13:55

    time I do substitution I use parenthesis. That will save you your butt when it comes
    time I do substitution I use parenthesis. That will save you your butt when it comes

  • 14:00

    to not making sign errors later on. Especially if you are using your TI-83 and TI-84's and
    to not making sign errors later on. Especially if you are using your TI-83 and TI-84's and

  • 14:03

    any kind of scientific calculator that allows you to type in the syntax. You want to be
    any kind of scientific calculator that allows you to type in the syntax. You want to be

  • 14:10

    careful with your signs of course and make sure that your answers are right, parenthesis
    careful with your signs of course and make sure that your answers are right, parenthesis

  • 14:14

    will help that happen. So we have negative 12 plus or minus the square root of 144, 4
    will help that happen. So we have negative 12 plus or minus the square root of 144, 4

  • 14:22

    times 2 is 8 and 8 times 20 is negative 160. All of this over 2 times 2 which is 4. This
    times 2 is 8 and 8 times 20 is negative 160. All of this over 2 times 2 which is 4. This

  • 14:30

    comes out to be negative 12 plus and minus the square root of negative 16 all over 4.
    comes out to be negative 12 plus and minus the square root of negative 16 all over 4.

  • 14:38

    I am running out of space. Look I am trying to square root a negative 16. The square root
    I am running out of space. Look I am trying to square root a negative 16. The square root

  • 14:45

    of 16, I can do it is 4. But the negative comes out to be imaginary so we get negative
    of 16, I can do it is 4. But the negative comes out to be imaginary so we get negative

  • 14:51

    twelve plus or minus 4i over 4.
    twelve plus or minus 4i over 4.

All idiom
I am
//

idiom

Modeled on the phrase "I am woman, hear me roar" from the 1972 song “I am Woman” by Helen Reddy. Either used genuinely as a phrase of empowerment for some person, or else used humorously or sarcastically to deride or poke fun at someone.

Complex (imaginary) Numbers Part 1

105,549 views

Video Language:

  • English

Caption Language:

  • English (en)

Accent:

  • English (US)

Speech Time:

99%
  • 14:51 / 15:00

Speech Rate:

  • 148 wpm - Conversational

Category:

  • Education

Intro:

BAM!!! Mr. Tarrou. How are you doing today? I am going to be talking about imaginary numbers.
Like they hardly even exist but yet we still need them in mathematics. Now imaginary numbers
really in a textbook are called Complex Numbers. They have two parts, they have a real part
and an imaginary part...or a plus b times i. Now what do I mean by imaginary. What does
that even mean, or come up in math? Maybe if you have started solving parabolas, or
doing the quadratic formula, you might have noticed that sometimes you don't get an answer.
You try and do the square root and you end up with a negative number inside your radical.
That is one example where imaginary numbers come in to play. The square root of negative
1 is equal to "i" Now there are some things in math that we teach as a rule that I can
do a pretty good job of teaching my students like why that works or how. This is one is
a little bit tough for that, so we are just going to let this letter "i" stand for the
square root of negative one. i for imaginary! Why is it imaginary? Because if you square
root negative one, or you try to come up with a square root, there are not two numbers that
you multiply by themselves to get you negative one. The square root of 4 is 2. Why? Because
2 times 2 is 4. But what two real numbers can you multiply together to get negative
one, they don't exist. Negative one times negative one is positive one. Maybe it is
the square root of negative one is positive one. Well one times one is still positive
one. You cannot take an even root of a negative number. So we must give it this imaginary
value if i. Well one fact that we are going to use quite a bit today is about i squared.
Now i squared. i squared is just as it says, i times i or the square root of negative one

Video Vocabulary

/sälv/

verb

find answer to, explanation for, or means of effectively dealing with.

/stärt/

verb

begin or be reckoned from particular point in time or space.

/ˌkalkyəˈlāSH(ə)n/

noun other

mathematical determination. An educated guess about something.

/iɡˈzampəl/

noun verb

thing characteristic of its kind. be illustrated or exemplified.

/ˈpäzədiv/

adjective noun

Having the charge produced by electrons. positive quality.

/ˈməltəˌplī/

verb

add number to itself specified number of times.

/ˈneɡədiv/

adjective exclamation noun verb

characterized by absence of distinguishing features. no. Reply to a question or statement that means 'no'. refuse to accept.

/ˈsəmˌtīmz/

adverb

Only at certain times; occasionally.

/iˈmajəˌnerē/

adjective

existing only in imagination.

/təˈɡeT͟Hər/

adjective adverb

self-confident, level-headed, or well organized. In the same place; not far in a family or group.

/ˈak(t)SH(o͞o)əlē/

adverb

as truth or facts.

/T͟Həmˈselvz/

pronoun

used to refer to group of people or things previously mentioned.

/bēˈkəz/

conjunction

for reason that.

/kwäˈdradik/

adjective noun

Polynomial of the second degree. quadratic equation.

/ˈteks(t)ˌbo͝ok/

adjective noun

conforming or corresponding to established standard or type. book used as standard work for study of particular subject.