By Titu Andreescu

It is most unlikely to visualize smooth arithmetic with out complicated numbers. the second one version of *Complex Numbers from A to … Z* introduces the reader to this interesting topic that from the time of L. Euler has develop into probably the most applied principles in mathematics.

The exposition concentrates on key thoughts after which effortless effects touching on those numbers. The reader learns how advanced numbers can be utilized to unravel algebraic equations and to appreciate the geometric interpretation of advanced numbers and the operations related to them.

The theoretical elements of the e-book are augmented with wealthy workouts and difficulties at a number of degrees of trouble. Many new difficulties and options were extra during this moment version. a unique characteristic of the booklet is the final bankruptcy a variety of remarkable Olympiad and different vital mathematical contest difficulties solved by means of applying the tools already presented.

The ebook displays the original event of the authors. It distills an unlimited mathematical literature so much of that is unknown to the western public and captures the essence of an considerable challenge tradition. the objective viewers contains undergraduate highschool scholars and their teacher's mathematical contestants (such as these education for Olympiads or the W. L. Putnam Mathematical festival) and their coaches in addition to an individual drawn to crucial mathematics.

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**Extra resources for Complex Numbers from A to ... Z**

**Example text**

5. 5 Problems 1. Find the polar coordinates for the following points, given their Cartesian coordinates: √ (a) M1 (−3, 3); (b) M2 (−4 3, −4); (c) M3 (0, −5); (d) M4 (−2, −1); (e) M5 (4, −2). 2. Find the Cartesian coordinates for the following points, given their polar coordinates: π 3 ; (b) P2 4, 2π − arcsin (a) P1 2, ; (c) P3 (2, π); 3 5 π 3π ). (d) P4 (3, −π); (e) P5 (1, ); (f) P6 (4, 2 2 3. Express arg(z) and arg(−z) in terms of arg(z). 4. Find the geometric images for the complex numbers z in each of the following cases: (a) |z| = 2; (b) |z + i| ≥ 2; (c) |z − i| ≤ 3; 5π 3π π (d) π < arg z < ; (e) arg z ≥ ; (f) arg z < ; 4 2 2 π π π (g) arg(−z) ∈ , ; (h) |z + 1 + i| < 3 and 0 < arg z < .

1 Geometric Interpretation of a Complex Number We have deﬁned a complex number z = (x, y) = x + yi to be an ordered pair of real numbers (x, y) ∈ R × R, so it is natural to let a complex number z = x + yi correspond to a point M (x, y) in the plane R × R. For a formal introduction, let us consider P to be the set of points of a given plane Π equipped with a coordinate system xOy. Consider the bijective function ϕ : C → P, ϕ(z) = M (x, y). Definition. The point M (x, y) is called the geometric image of the complex number z = x + yi .

Zn is real. 33. Let z1 , z2 , z3 be distinct complex numbers such that |z1 | = |z2 | = |z3 | > 0. If z1 + z2 z3 , z2 + z1 z3 , and z3 + z1 z2 are real numbers, prove that z1 z2 z3 = 1. 34. Let x1 and x2 be the roots of the equation x2 − x + 1 = 0. Compute the following: + x2000 ; (b) x1999 + x1999 ; (c) xn1 + xn2 , for n ∈ N. (a) x2000 1 2 1 2 35. Factorize (in linear polynomials) the following polynomials: (a) x4 + 16; (b) x3 − 27; (c) x3 + 8; (d) x4 + x2 + 1. 36. Find all quadratic equations with real coeﬃcients that have one of the following roots: (a) (2 + i)(3 − i); (b) 5+i ; (c) i51 + 2i80 + 3i45 + 4i38 .