Question 756000
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*[tex \LARGE i^0\ =\ 1] by definition of *[tex \LARGE x^0]


*[tex \LARGE i^1\ =\ i] by definition of *[tex \LARGE x^1]


*[tex \LARGE i^2\ = -1] by definition of *[tex \LARGE i]


*[tex \LARGE i^3\ =\ i^2\,\cdot\,i^1\ =\ -i] by laws of exponents and substitution from above.


*[tex \LARGE i^4\ =\ i^2\,\cdot\,i^2\ =\ -1\,\cdot\,-1\ = 1\ =\ i^0]


and so on.


In general *[tex \LARGE i^n\ =\ i^{n\,mod\,4]


where *[tex \LARGE n\ mod\ m] is the modulo function that returns the remainder of the integer division of *[tex \LARGE n] by *[tex \LARGE m]


John
*[tex \LARGE e^{i\pi}\ +\ 1\ =\ 0]
<font face="Math1" size="+2">Egw to Beta kai to Sigma</font>
My calculator said it, I believe it, that settles it
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