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# Imaginary and complex numbers

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Why does raised to an exponent change when using even numbers? You'd expect that a negative number with an even exponent to always come out positive. Can someone explain the pattern:

i^2= -1, i^4+ 1, i^6= 1, i^8= -1, i^10= 1, and so on. Why does it bounce from negative to positve?

Guest Nov 18, 2014

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By definition

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

Remember this is not a real number it is an imaginary number!

squares and square roots cancle each other out so

$$\\i^2=(\sqrt{-1})^2 = -1\\\\ i^4=i^2\times i^2=-1\times-1=+1\\\\ i^6=i^4\times i^2=1\times-1=-1\\\\ i^8=i^6\times i^2=-1\times-1=+1\\\\ i^{10}=i^8\times i^2=+1\times-1=-1\\\\$$

Melody  Nov 19, 2014
Sort:

#1
+92221
+10

By definition

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

Remember this is not a real number it is an imaginary number!

squares and square roots cancle each other out so

$$\\i^2=(\sqrt{-1})^2 = -1\\\\ i^4=i^2\times i^2=-1\times-1=+1\\\\ i^6=i^4\times i^2=1\times-1=-1\\\\ i^8=i^6\times i^2=-1\times-1=+1\\\\ i^{10}=i^8\times i^2=+1\times-1=-1\\\\$$