Question 1073485
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<pre>
Let z = a + bi be our complex number under the question.
We need to find its real and imaginary parts "a" and "b".

As you, probably, know  |z| = {{{sqrt(a^2 + b^2)}}}.

So, the given equation becomes

a + bi - {{{sqrt(a^2 + b^2)}}} = 8 + 4i.


It actually deploys in two independent equations for the real part and the imaginary part separately:

a - {{{sqrt(a^2 + b^2)}}} = 8     (1)     and
b            = 4.    (2)

Equation (1) is equivalent to

a - 8 = {{{sqrt(a^2 + b^2)}}},

{{{(a-8)^2}}} = {{{a^2 + b^2}}},

{{{a^2 - 16a + 64}}} = {{{a^2 + 4^2}}},    (I substituted b= 4)

-16a + 64 = 16  --->  -16a = 16-64  --->  -16a = -48  ---> a = 3.


So, the logic leads us to the solution  a = 3, b = 4, i.e. z = 3 + 4i.


Now CHECK this solution: 3 + 4i - {{{sqrt(3^2 + 4^2)}}} = 3 + 4i - 5 = -2 + 4i.


It doesn't check !!!


The conclusion is: The given equation HAS NO solutions.
</pre>

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When I got this result, &nbsp;I was very surprised: &nbsp;it is for the first time in my life I see the equation for complex numbers, 
which has no solution.


But reviewing the problem and the solution, &nbsp;I got the understanding WHY it happened in this case.


The equation of the form  &nbsp;z - |z| = u + iw  &nbsp;CAN NOT have a solution if the real part "u" of the complex number 
in the right hand side is positive.


Actually, &nbsp;it is OBVIOUS !


Now I am calm. &nbsp;Because I know the reason.
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See my lessons on complex numbers in this site 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/complex/Complex-numbers-and-arithmetical-operations.lesson>Complex numbers and arithmetic operations on them</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/complex/Complex-plane.lesson>Complex plane</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/complex/Addition-and-subtraction-of-complex-numbers-in-complex-plane.lesson>Addition and subtraction of complex numbers in complex plane</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/complex/Multiplication-and-division-of-complex-numbers-in-complex-plane-.lesson>Multiplication and division of complex numbers in complex plane</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/complex/Raising-a-complex-number-to-an-integer-power.lesson>Raising a complex number to an integer power</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/complex/How-to-take-a-root-of-a-complex-number.lesson>How to take a root of a complex number</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/complex/Solution-of-the-quadratic-equation-with-real-coefficients-on-complex-domain.lesson>Solution of the quadratic equation with real coefficients on complex domain</A>


&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/complex/Solved-problems-on-taking-roots-of-complex-numbers.lesson>Solved problems on taking roots of complex numbers</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/complex/Solved-problems-on-arithmetic-operations-on-complex-numbers.lesson>Solved problems on arithmetic operations on complex numbers</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/complex/Solved-problem-on-taking-square-roots-of-complex-numbers.lesson>Solved problem on taking square root of complex number</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/complex/Miscellaneous-problems-on-complex-numbers.lesson>Miscellaneous problems on complex numbers</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/complex/Advanced-problem-in-complex-numbers.lesson>Advanced problem on complex numbers</A>



Also, &nbsp;you have this free of charge online textbook in ALGEBRA-II in this site

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-II - YOUR ONLINE TEXTBOOK</A>.


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