Math for Non-Geeks/Exercises: Complex numbers

oben Real and imaginary part Determining imaginary and real part Determine the real part and the imaginary part of the following complex numbers: There is So and . There is So and . There is So and . Plane of complex numbers Sketch the following sets on the plane of complex numbers: The set describes all complex numbers which fulfil the condition . In the complex plane, this corresponds exactly to the numbers , whose distance from the number is at most . So describes a closed disk centered at with radius . The boundary of the circle is included in the set: center|400px|The set M1 For all elements of the set , should apply. So their distance from the origin should be at least , but at most . Thus, the set describes a closed annulus between the circles with radii and , including both boundary circles: center|400px|The set M2 The set describes a closed disk centered at the origin with radius : center|350px|First part of the set M3 The second part of the set describes all complex numbers whose real part is larger than their imaginary part. On the bisector of the first and third quadrant ("diagonal line") we have all numbers whose real and imaginary part are equal. To the right of this line, there are all complex numbers with a real part greater than the imaginary part: center|350px|Second part of the set M3 The intersection of both sets yields: center|350px|Intersection of the sets With this you can sketch the set . Note that one of the boundary lines is only dashed. That means, the numbers directly on the diagonal line through the origin do not belong to the set , since the real part of each element must be strictly bigger than its imaginary part: center|400px|Menge M3 Polar representation Transform to polar representation Compute the complex numbers and convert them into polar representation: The absolute value is . Since and there is . Thus . This formula appears frequently, so you might want to learn it by heart. We have the absolute value . With and the number lies in the fourth quadrant. The formula for this is . So there is . You get the polar representation of by first complexly conjugating the polar representation of and then multiplying it by . There is: Under complex conjugation, is replaced by in the exponent in the polar representation. Thus . By multiplying it out, we get . So . With and the number lies in the second quadrant. For the angle there is . This corresponds in degrees to an angle of . Altogether we get the result . First, we determine the polar representation of the three factors of : Then, there is First we determine the absolute value and the angle of . There is and . Now we can determine the polar representation of : unten