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Thread: 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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    $\displaystyle i^3 = i^{2+1} = i^2\cdot i = (-1)\cdot i = -i$
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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
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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- $\displaystyle i^2=-1$.
    2- $\displaystyle -i=i-$.
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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,

    $\displaystyle i^2=-1$ makes $\displaystyle i^2(i^2)=-1(-1)=1$

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

    $\displaystyle i=\sqrt{-1}$

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

    $\displaystyle i^3=i^2i=ii^2=(-1)i=i(-1)=-i$

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

    $\displaystyle i^5=i^4i=(1)i=i$

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

    $\displaystyle i^0=1$

    $\displaystyle i^1=i$

    $\displaystyle i^2=-1$

    $\displaystyle i^3=-i$

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