Question 1201677
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<pre>

The number 2-2i is located in 4th quadrant, QIV.


Its modulus is  {{{sqrt(2^2+2^2)}}} = {{{sqrt(4+4)}}} = {{{sqrt(8)}}} = {{{2*sqrt(2)}}}.


Its argument is -45°  or  360°-45° = 315°.   

(You may use any of these two values, -45° or 315°, they both represent the correct arguments).



So, the polar form of a complex number 2-2i is

    2 - 2i = {{{2*sqrt(2)*(cos(-45^o) + i*sin(-45^o))}}}.



It is the same as to say that the polar form of a complex number 2-2i is

    2 - 2i = {{{2*sqrt(2)*(cos(315^o) + i*sin(315^o))}}}.



In "cis"-form,  it is the same as

    2 - 2i = {{{2*sqrt(2)*cis(-45^o)}}} = {{{2*sqrt(2)*cis(315^o)}}}.



The modulus is {{{2*sqrt(2)}}};  the argument  is -45° or 315°  or any other number of degrees 
which differs by a multiple of 360° from -45° or from 315°.


To make the argument unique, sometimes the conception of a PRINCIPAL ARGUMENT is used and the principal argument is taken: 

    it is the value of an argument in the range [0°,360°),

    i.e. 315° in this problem.
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On complex numbers, &nbsp;see introductory lessons

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/complex/Complex-numbers-and-arithmetical-operations.lesson>Complex numbers and arithmetical 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=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/Solving-polynomial--equations-in-complex-domain.lesson>Solving polynomial equations in complex domain</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>

in this site.


Also, &nbsp;you have this free of charge online textbook on &nbsp;ALGEBRA-II &nbsp;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 &nbsp;"<U>Complex numbers</U>".



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.