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Hello! I was doing some alcumus, and I came upon a review problem: (A problem that you can't skip), and it was about conjugates in intermediate algebra. I have fun doing alcumus, and I  recommend playing it, but I came across this problem. This does not require much work, but I never quite learned about conjugates so I need your help :D

If |z| = 1, express $\overline{z}$ as a simplified fraction in terms of z.

 

Thanks in advance!

 Apr 23, 2021
 #1
avatar+2266 
+1

Hello :))

 

I'm not quite sure how to do it either, but I'll try my best to help. 

A conjugate is like an inverse, a+ bi conjugate is a - bi. 

The absolute value of a complex number is sqrt(a^2 + b^2), a + bi. 

So we have a complex number (a+bi) where sqrt(a^2 + b^2) = 1. 

 

 

z = a + bi

sqrt(a^2 + b^2) = 1

a^2 + b^2 = 1

 

We're looking for a - bi in terms of z. 

(a-bi)(a+bi) = a^2 + b^2

But we know a^2 + b^2 = 1. 

(a-bi)(a+bi) = 1

We also know that a + bi = z. 

(a-bi)(z) = 1

a-bi = 1/z

 

Our answer is 1/z hopefully

 

I hope this helped. :))

 

=^._.^=

 Apr 23, 2021
 #2
avatar+115 
+2

Thanks! It was correct! I had the concept of the conjugate from complex numbers, but I didn't know much about the variables itself!

OofPirate  Apr 23, 2021
 #3
avatar+2266 
+1

ohh yayyy

I'm quite happy. 

I've strugling with this topic for a month in alcumus

complex numbers are very hard :((

 

=^._.^=

catmg  Apr 23, 2021

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