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Math Help - what is i^3?

  1. #1
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    what is i^3?

    if i^2=-1, does that make i^3=1, im not really sure?
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  2. #2
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    i^3 = i^{2+1} = i^2\cdot i = (-1)\cdot i = -i
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  3. #3
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    @josipive
    why are you allowed to do the 3rd step (-1).i,
    while I am writing I might know the answer, is it because the root of i is -1?
    sorry got confused becasue you wrote (-1).i instead of i.(-1)
    my bad
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  4. #4
    Member Miss's Avatar
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    Quote Originally Posted by MarNie View Post
    @josipive
    why are you allowed to do the 3rd step (-1).i,
    while I am writing I might know the answer, is it because the root of i is -1?
    sorry got confused becasue you wrote (-1).i instead of i.(-1)
    my bad
    1- i^2=-1.
    2- -i=i-.
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  5. #5
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    Quote Originally Posted by hyzlemon View Post
    if i^2=-1, does that make i^3=1, im not really sure?
    Hi hyzlemon,

    i^2=-1 makes i^2(i^2)=-1(-1)=1

    All odd powers of i are {\pm}\ i
    All even powers of i are {\pm}1

    i=\sqrt{-1}

    i^2=\sqrt{-1}\sqrt{-1}=-1

    i^3=i^2i=ii^2=(-1)i=i(-1)=-i

    i^4=i^2i^2=(-1)(-1)=1\ or\ i^3i=ii^3=i(-i)=-i^2=-(-1)=1

    i^5=i^4i=(1)i=i

    Hence all higher powers continue on in this cycle i,\ i^2,\ i^3,\ i^4,\ i^5=i,\ i^6=i^2.\ i^7=i^3,\ i^8=i^4,\ i^9=i....
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  6. #6
    A riddle wrapped in an enigma
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    I have found it useful when teaching algebra students about the powers of i, is just jot down the following:

    i^0=1

    i^1=i

    i^2=-1

    i^3=-i

    Now when encountering any power of i, divide the exponent by 4. Using a little modular arithmetic, we arrive at:

    [1] If the remainder is 0, then the answer is 1.

    [2] If the remainder is 1, then the answer is i.

    [3] If the remainder is 2, then the answer is -1.

    [4] If the remainder is 3, then the answer is -i.

    Example: i^{111}=-i because 111 \div 4 = 27 \: {\text{Remainder 3}}
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