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Cyclic Property of Imaginary Number and Negative Imaginary Number

Imaginary number i have its cycle property. It repeat a pattern for every 4 multiplication cycles.

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i^0 = +1
i^1 = +i
i^2 = -1
i^3 = -i
i^4 = +1
i^5 = +i

Now the problem is, what if it starts from negative i.

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(-i)^0 = +1
(-i)^1 = -i
(-i)^2 = -1
(-i)^3 = +i
(-i)^4 = +1
(-i)^5 = -i

Now that is interesting. Comparing the result to positive i, result of negative is reversing some of the signs. But what is the pattern?

Think spacially. Not literally. Here is a clock, made by all possible product from the mulitiplication of imaginary number i. And you start from the top.

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         *start
        +1 
       /   \
    -i      +i
       \   / 
        -1

Go clockwise, you end up landing on the position of +i. Go anti-clockwise, you end up landing on the position of -i.

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         *start
        +1 
       /   \
    -i      +i *step1
       \   / 
        -1

Whether you go clockwise or anti-clockwise, the action of "go" is in fact multiplying the current state by +i or -i.

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     clockwise = multiplying current state by +i
anti-clockwise = multiplying current state by -i

You go clockwise, then go anti-clockwise. You land on the same position. The action of clockwise and anti-clockwise cancel out each other.

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         *step2
         *start
        +1 
       /   \
    -i      +i *step1
       \   / 
        -1

+i * -i = 1 (identity)

So why the degenerats writing math textbook can never tell you how to do chain multiplation of both +i and -i mixing up together is that they don't even understand the structure.

What if +i * -i * +i * +i * -i * +i * +i * -i * -i ?

You just need to count.

5 clockwise and 4 anticlowise = +5 -4 = 1 = 1 step toward clockwise = +i

What if the result exceed 4?

Since imaginary number repeat at a cycle of 4, you do a modula 4 to it.

999999 step toward clockwise = 999999 mod 4 step toward clockwise = 3 step toward clockwise

-999999 step toward clockwise = -999999 mod 4 step toward clockwise = 1 step toward clockwise

Done.

Last update: 2023-06-28
Created: 2023-06-28