SOLUTION: find the ff. quantities absolute value of z-1 squared. squared is outside the absolute value

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Question 1178085: find the ff. quantities
absolute value of z-1 squared. squared is outside the absolute value
Found 2 solutions by MathLover1, ikleyn:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

%28abs%28+z-1%29%29%5E2
Apply absolute rule : %28abs%28+x%29%29%5E2=+x%5E2 assuming x+elementR
%28abs%28+z-1%29%29%5E2=%28+z-1%29%5E2

Answer by ikleyn(52806) About Me  (Show Source):
You can put this solution on YOUR website!
.

            The  "solution"  by  @MathLover1 is   A B S O L U T E L Y   W R O N G.

            So I came to bring a correct solution. 


We are given a complex number  z;  Let  z%5Bc%5D be the complex conjugate to  z  complex number. 


Then  |z-1|^2 = %28z-1%29%2A%28z%5Bc%5D-1%29 = z%2Az%5Bc%5D - %28z%2Bz%5Bc%5D%29 + 1 = |z|^2 - 2*Re(z) + 1.    ANSWER


Here  Re(z)  is the real part of the complex number z.

Solved.

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On complex numbers, see introductory lessons
    - Complex numbers and arithmetical operations on them
    - Complex plane
    - Addition and subtraction of complex numbers in complex plane
    - Multiplication and division of complex numbers in complex plane

    - Solved problems on taking roots of complex numbers
    - Solved problems on arithmetic operations on complex numbers
    - Solved problem on taking square root of complex number
in this site.

Also, you have this free of charge online textbook in ALGEBRA-II in this site
    - ALGEBRA-II - YOUR ONLINE TEXTBOOK.

The referred lessons are the part of this online textbook under the topic "Complex numbers".


Save the link to this textbook together with its description

Free of charge online textbook in ALGEBRA-II
https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson

into your archive and use when it is needed.