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# what does -i equals to

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what does -i equals to?

Oct 2, 2014

#2
+95369
+13

$$\\-i=-\sqrt{-1}\\ or\\ -i=-1*\sqrt{-1}\\ -i=i^2*i\\ -i=i^3\\ or\\ -i=(-1)^3*i\\ -i=(i^2)^3*i\\ -i=i^6*i\\ -i=i^7\\ continuing with this pattern I think\\ -i=i^{4n+3}\qquad Where n \in Z  and  n\ge 0$$

Now it has got me thinking more.

what if n was a negative integer?  Would that work?

$$\\i^{-4n+3}\\ =i^{-4n}*i^3\\ =\frac{1}{i^{4n}}*i^3\\ =\frac{1}{(i^{4})^n}*i^3\\ =\frac{1}{(1)^n}*i^3\\ =\frac{1}{1}*i^3\\ =i^3\\ =i^2*i\\ =-i\\\\ so\\\\ -i=i^{4n+3} \qquad  where  n \in Z\\ n can be any integer, it does not have to be positive$$$. Oct 3, 2014 ### 9+0 Answers #1 +8263 +8 $$-i=$$ $$-{\mathtt{1}}{i}$$ My guess is $$-{\mathtt{1}}{i}$$. Oct 3, 2014 #2 +95369 +13 Best Answer $$\\-i=-\sqrt{-1}\\ or\\ -i=-1*\sqrt{-1}\\ -i=i^2*i\\ -i=i^3\\ or\\ -i=(-1)^3*i\\ -i=(i^2)^3*i\\ -i=i^6*i\\ -i=i^7\\ continuing with this pattern I think\\ -i=i^{4n+3}\qquad Where n \in Z and n\ge 0$$ Now it has got me thinking more. what if n was a negative integer? Would that work? $$\\i^{-4n+3}\\ =i^{-4n}*i^3\\ =\frac{1}{i^{4n}}*i^3\\ =\frac{1}{(i^{4})^n}*i^3\\ =\frac{1}{(1)^n}*i^3\\ =\frac{1}{1}*i^3\\ =i^3\\ =i^2*i\\ =-i\\\\ so\\\\ -i=i^{4n+3} \qquad where n \in Z\\ n can be any integer, it does not have to be positive$$$

Melody Oct 3, 2014
#3
+8263
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WOW! OUR SUPERSTAR EXPLAINED THE ANSWER, AND MY ANSWER IS CORRECT TOO!!!

WOW MELODY! YOU ARE A SUPERMODEL!!!!!

Oct 3, 2014
#4
+95369
+3

Yes I think that we are both right.

Oct 3, 2014
#5
+8263
0

YAY!

Oct 3, 2014
#6
+94619
+3

Very nice proof, Melody !!!

Oct 3, 2014
#7
+8263
0

when melody has the chance to explain an aswer, she DEEPLY EXPLAINS IT.

Oct 3, 2014
#8
+95369
+3

Thanks Chris and dragon.

I was just playing.  It is good to do that sometimes.

Oct 3, 2014
#9
+8263
0

playing? i dont think so. you did a great answer, so that is no joke! are you one of those super genious people on Earth???

Oct 3, 2014