Lesson Solving non-standard equations in complex numbers
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<H2>Solving non-standard equations in complex numbers</H2> <H3>Problem 1</H3>Find all complex numbers z such that {{{z^2}}} is equal to the complex conjugate of z. <B>Solution</B> <pre> It is better and much easier to analyse and to solve this problem in polar trigonometric form. If z^2 = z complex conjugate, then, firstly, the modulus of z is equal to 1. In other words, z lies on the unit circle in a coordinate plane. Next, if the argument of z is polar angle {{{alpha}}}, then the polar angle of z^2 is {{{2*alpha}}}, while the polar angle of (z conjugate) is {{{-alpha}}} or {{{2pi - alpha}}}. So, we get the equation for the polar angle <U>Case 1</U>. {{{2*alpha}}} = -{{{alpha}}}, which implies {{{3alpha}}} = 0, or {{{alpha}}} = 0. Then the solution is z = 1. OR <U>Case 2</U>. {{{2*alpha}}} = {{{2pi- alpha}}}, which implies {{{3alpha}}} = {{{2pi}}}. Hence, {{{alpha}}} = {{{2pi/3}}}. <U>ANSWER</U>. There are TWO solutions. One solution is z = 1. The other solution is z = {{{cos(2pi/3) + i*sin(2pi/3)}}} = {{{(-1/2) + i*(sqrt(3)/2)}}}. </pre> <H3>Problem 2</H3>Find the solution set to equation {{{iz^2}}} - {{{i^3*abs(2z^2)}}} = {{{3i*(conj(z))^2}}} in complex numbers. <B>Solution</B> <pre> Our starting equation is {{{iz^2}}} - {{{i^3*abs(2z^2)}}} = {{{3i*(conj(z))^2}}}. (1) Since {{{i^2}}} = -1, this equation is the same as {{{iz^2}}} + {{{i*abs(2z^2)}}} = {{{3i*(conj(z))^2}}}. We can write it in this equivalent form {{{i*abs(2z^2)}}} = {{{3i*(conj(z))^2}}} - {{{iz^2}}}. We can cancel common factor " i " in both sides {{{abs(2z^2)}}} = {{{3*(conj(z))^2}}} - {{{z^2}}}. (2) Left side is a real number --- hence, right side must be a real number, too. Ok. Now let z = a+bi. Then from (2), right side of (2) is {{{3*(a-bi)^2}}} - {{{(a+bi)^2}}} = 3*(a^2 - 2abi - b^2) - (a^2 +2abi - b^2) = = (3a^2 - 3b^2- a^2 + b^2) + (-6abi - 2abi) = (2a^2 - 2b^2) - 8abi. (3) Since the right side of (2) is a real number, it implies from (3) that ab = 0, i.e. EITHER a= 0 OR b= 0 (or BOTH). So, below we analyze two cases. (a) b= 0. It means that "z" is, actually, a real number. It is easy to check, that then equation (2) is valid for any value of real number z, which means that any real number is the solution for the original equation. (b) a= 0, b=/= 0. It means that "z" is, actually, a pure imaginary number. Then in (2), the left side is positive real number, while the right side (3) is negative real number, which creates a CONTRADICTORY. +-----------------------------------------------------------------------------+ | So, the {{{highlight(ANSWER)}}} to the problem is THIS: | | the solution set to the given equation is the set of all real numbers. | +-----------------------------------------------------------------------------+ </pre> My other lessons on complex numbers in this site are - <A HREF=http://www.algebra.com/algebra/homework/complex/Complex-numbers-and-arithmetical-operations.lesson>Complex numbers and arithmetic operations on them</A> - <A HREF=http://www.algebra.com/algebra/homework/complex/Complex-plane.lesson>Complex plane</A> - <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> - <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> - <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> - <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> - <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> - <A HREF=http://www.algebra.com/algebra/homework/complex/How-to-take-a-square-root-of-a-complex-number.lesson>How to take a square root of a complex number</A> - <A HREF=http://www.algebra.com/algebra/homework/complex/Solution-of-the-quadratic-equation-with-complex-coefficients-on-complex-domain.lesson>Solution of the quadratic equation with complex coefficients on complex domain</A> - <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> - <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> - <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> - <A HREF=https://www.algebra.com/algebra/homework/complex/Solving-polynomial--equations-in-complex-domain.lesson>Solving polynomial equations in complex domain</A> - <A HREF=https://www.algebra.com/algebra/homework/complex/Miscellaneous-problems-on-complex-numbers.lesson>Miscellaneous problems on complex numbers</A> - <A HREF=https://www.algebra.com/algebra/homework/complex/Advanced-problem-in-complex-numbers.lesson>Advanced problem on complex numbers</A> - <A HREF=https://www.algebra.com/algebra/homework/complex/Solved-problems-on-de%27Moivre-formula.lesson>Solved problems on de'Moivre formula</A> - <A HREF=https://www.algebra.com/algebra/homework/complex/Proving-identities-using-complex-numbers.lesson>Proving identities using complex numbers</A> - <A HREF=https://www.algebra.com/tutors/18-Calculating-1sin%281%B0%29%2B2sin%282%B0%29%2B3sin%283%B0%29%2B-%2B180sin%28180%B0%29.lesson>Calculating the sum 1*sin(1°) + 2*sin(2°) + 3*sin(3°) + . . . + 180*sin(180°)</A> - <A HREF=https://www.algebra.com/algebra/homework/complex/An-equation-in-complex-numbers-which-HAS-NO-a-solution.lesson>A curious example of an equation in complex numbers which HAS NO a solution</A> - <A HREF=https://www.algebra.com/algebra/homework/complex/Upper-level-problem-on-complex-numbers.lesson>Upper level problem on complex numbers</A> - <A HREF=https://www.algebra.com/algebra/homework/complex/Determine-locus-of-points-using-complex-numbers.lesson>Determine locus of points using complex numbers</A> - <A HREF=https://www.algebra.com/algebra/homework/complex/Joke-problems-on-complex-numbers.lesson>Joke problems on complex numbers</A> - <A HREF=https://www.algebra.com/algebra/homework/complex/Review-of-lessons-on-complex-numbers.lesson>OVERVIEW of lessons on complex numbers</A> Use this file/link <A HREF=https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-II - YOUR ONLINE TEXTBOOK</A> to navigate over all topics and lessons of the online textbook ALGEBRA-II.
