eight-fold power symmetry

[AntonioLao]

Using complex numbers it can be demonstrated that power symmetry exists. A complex numbers z is given by z=a+bi where i is the imaginary unit and a and b are real numbers. Using simple multiplications show that the product of any complex number and its conjugate is always real. However, the power of a complex number can be real, imaginary or complex depending on the degree of its power. And where and when the absolute values of a and b are equal, |a|=|b|, the 4th power is always negative real while the 8th power is always positive real.

[Profpat]

I'm a little confused Antonio. Could you explain why the 4th power is negative and the 8th would be positive?

[AntonioLao]

why the 4th power is negative and the 8th would be positive

These can be easily demonstrated if you got a graphing calculator that do complex numbers multiplications. Please multiply (1+i)(1-i)=real but (1+i)^4=(1-i)^4=negative while (1+i)^8=(1-i)^8=positive real.

[Profpat]

I think I'll take your word on it Antonio. You are a very wise member.

[AntonioLao]

I'll take your word

But that is not necessary. You can extend to the 100th power and found that it is also negative but to the 200th, 400th and 600th are all positive real then after that the calculator returns an overflow error.