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Math Help - help on complex numbers

  1. #1
    Member Jones's Avatar
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    help on complex numbers

    Hello, i find this whole complex number thing quite difficult

    And therefore i have 2 questions.

    1
    Simplify and write on the formula a + bi
    e^(1-i)*e^(2+2i)

    Should these functions be differentiated first or can we use the product rule and the exponetial rules to "add the exponents"?


    2
    Im feeling lost when dealing with e. I don't know what im doing, I don't know what it is and where it comes from. All i know is that is got something to do with exponetial growth.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by Jones View Post
    Hello, i find this whole complex number thing quite difficult

    And therefore i have 2 questions.

    1
    Simplify and write on the formula a + bi
    e^(1-i)*e^(2+2i)

    Should these functions be differentiated first or can we use the product rule and the exponetial rules to "add the exponents"?
    Yes the laws of exponents apply so:

    e^(1-i)*e^(2+2i) = e^(1-i + 2 + 2i) = e^(3 + i)

    Now to get this into the for a+bi, we need to know that:

    e^(i theta) = cos(theta) + i sin(theta)

    for real theta.

    Now:

    e^(3+i)=e^3 e^(i) = e^3 [cos(1) + i sin(1)] = e^3 cos(1) + i e^3 sin(1).

    (Note trig function are in radians)

    2
    Im feeling lost when dealing with e. I don't know what im doing, I don't know what it is and where it comes from. All i know is that is got something to do with exponetial growth.
    e is a real number a bit like pi, and e ~= 2.718282. Like pi it is important
    in mathematics because it has special properties, one of which is that:

    d/dx (e^x) = e^x.


    RonL
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  3. #3
    Member Jones's Avatar
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    Quote Originally Posted by CaptainBlack View Post

    e is a real number a bit like pi, and e ~= 2.718282. Like pi it is important
    in mathematics because it has special properties, one of which is that:

    d/dx (e^x) = e^x.

    RonL
    How do we get e? Where does it come from?
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  4. #4
    Member Glaysher's Avatar
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    Quote Originally Posted by Jones View Post
    How do we get e? Where does it come from?
    In several ways

    e -- from Wolfram MathWorld
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  5. #5
    Member Jones's Avatar
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    One last problem.

    If we are going to write (Sqrt(3) -i) on the e^z form it should look something like this?

    e^z=e^x+iy
    The absolute value here is 2. So we get e^ln2, but how do i get the period y(obviously i don't mean that kind of period )
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  6. #6
    Grand Panjandrum
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    Quote Originally Posted by Jones View Post
    One last problem.

    If we are going to write (Sqrt(3) -i) on the e^z form it should look something like this?

    e^z=e^x+iy
    The absolute value here is 2. So we get e^ln2, but how do i get the period y(obviously i don't mean that kind of period )
    sqrt(3) - i = 2 [sqrt(3)/2 -(1/2)i] = 2 [cos(theta) + i sin(theta)]

    ..............=2 e^(i theta),

    so: sin(theta)=-1/2, and cos(theta)=sqrt(3)/2.

    Now the first of these has solutions 7 pi/6 and 11 pi/6

    (these are easier to find than you might think, first you need to know that:

    sin(30 degrees) = sin( pi/6 radian) = 1/2

    and then sketch the sine curve over a complete cycle 0 - 2pi to see
    where the sign is what you want).

    Now cos(7 pi/6) is negative which is not what we want, but cos(11pi/6)
    is negative so 11pi/6 is the angle we seek.

    so:

    sqrt(3) - i = 2 e^(i (11pi/6)).

    RonL
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  7. #7
    Member Jones's Avatar
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    Just wanted to point out, since you moved the thread, that this is still highschool math.

    And thanks for all your time and help
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