Complex Numbers I INTRODUCTION Complex Numbers, in mathematics, the sum of a real number and an imaginary number.

Complex Numbers I INTRODUCTION Complex Numbers, in mathematics, the sum of a real number and an imaginary number. An imaginary number is a multiple of i, where i is the square root of -1. Complex numbers can be expressed in the form a + bi, where a and b are real numbers. They have the algebraic structure of a field in mathematics. In engineering and physics, complex numbers are used extensively to describe electric circuits and electromagnetic waves (see Electromagnetic Radiation). The number i appears explicitly in the Schrödinger wave equation (see Schrödinger, Erwin), which is fundamental to the quantum theory of the atom. Complex analysis, which combines complex numbers with ideas from calculus, has been widely applied to subjects as different as the theory of numbers and the design of airplane wings. II HISTORY Historically, complex numbers arose in the search for solutions to equations such as x2=-1. Because there is no real number x for which the square is -1, early mathematicians believed this equation had no solution. However, by the middle of the 16th century, Italian mathematician Gerolamo Cardano and his contemporaries were experimenting with solutions to equations that involved the square roots of negative numbers. Cardano suggested that the real number 40 could be expressed as Swiss mathematician Leonhard Euler introduced the modern symbol i for in 1777 and expressed the famous relationship epi =-1 which connects four of the fundamental numbers of mathematics. For his doctoral dissertation in 1799, German mathematician Carl Friedrich Gauss proved the fundamental theorem of algebra, which states that every polynomial with complex coefficients has a complex root. The study of complex functions was continued by French mathematician Augustin Louis Cauchy, who in 1825 generalized the real definite integral of calculus to functions of a complex variable. III PROPERTIES For a complex number a + bi, a is called the real part and b is called the imaginary part. Thus, the complex number -2 + 3i has the real part -2 and the imaginary part 3. Addition of complex numbers is performed by adding the real and imaginary parts separately. To add 1 + 4i and 2 - 2i, for example, add the real parts 1 and 2 and then the imaginary parts 4 and -2 to obtain the complex number 3 + 2i. The general rule for addition is (a+ bi) + (c+di) = (a+ c) + (b +d )iMultiplication of complex numbers is based on the premise that i×i=-1 and the assumption that multiplication distributes over addition. This gives the rule (a+bi) × (c+di) = ( ac-bd) + (ad+bc)i For example, (1 + 4i) × (2 - 2i) = 10 + 6i If z=a+bi is any complex number, then, by definition, the complex conjugate of z is and the absolute value, or modulus, of z is For example, the complex conjugate of 1 + 4i is 1 - 4i, and the modulus of 1 + 4i is A basic relationship connecting absolute value and complex conjugate is IV THE COMPLEX PLANE Figure 2: Complex Plane in Polar Coordinates This graph illustrates the multiplication of two complex numbers by using vectors in the complex plane with polar coordinates. The product of vectors z and w is a vector whose length is the product of the lengths of z and w, and whose angle with the x-axis is the sum of the angles that z and w make with the x-axis. © Microsoft Corporation. All Rights Reserved. In the same way that real numbers can be thought of as points on a line, complex numbers can be thought of as points in a plane. The number a+bi is identified with the point in the plane with x coordinate a and y coordinate b. The points 1 + 4i and 2 - 2i are plotted in Figure 1 and correspond to the points (1,4) and (2,-2). In 1806 Swiss bookkeeper Jean Robert Argand was one of the first people to express complex numbers geometrically as points in the plane. For this reason, Figure 1 is sometimes referred to as an Argand diagram. If a complex number in the plane is thought of as a vector joining the origin to that point, then addition of complex numbers corresponds to standard vector addition. Figure 1 shows the complex number 3 + 2i obtained by adding the vectors 1 + 4i and 2 - 2i. Figure 1: Complex Plane in Cartesian Coordinates This graph illustrates the addition of two complex numbers by using vectors in the complex plane with cartesian coordinates. The parallelogram shows that the sum of 1 + 4i and 2 - 2i is 3 + 2i. © Microsoft Corporation. All Rights Reserved. Since points in the plane can be written in terms of the polar coordinates r and ? (see Coordinate System), every complex number z can be written in the form z= r (cos ? +i sin ? )Here, r is the modulus, or distance to the origin, and ? is the argument of z or the angle that z makes with the x axis. If z=r (cos ? +i sin ? ) and w=s (cos ? +i sin ? ) are two complex numbers in polar form, then their product in polar form is given by zw=rs (cos (? +? ) +i sin (? +? )) This has a simple geometric interpretation that is illustrated in Figure 2. V SOLUTIONS TO POLYNOMIALS There are many polynomial equations that have no real solutions, such as x2+ 1 = 0However, if x is allowed to be complex, the equation has the solutions x=±i, where i and - i are roots of the polynomial x2+ 1. The equation x2- 2x+ 2 = 0has the solutions x= l ±i. In his fundamental theory of algebra, Gauss showed that every nontrivial (having at least one nonzero root) polynomial with complex coefficients must have at least one complex root. From this it follows that every complex polynomial of degree n must have exactly n roots, although some roots may be the same. Consequently, every complex polynomial of degree n can be written as a product of exactly n linear, or first-degree, factors. Contributed By: William James Ralph Microsoft ® Encarta ® 2009. © 1993-2008 Microsoft Corporation. All rights reserved.