ME50 ADVANCED ENGINEERING MATHEMATICS

1 Complex Numbers √ A complex number is an ordered pair (x, y) of real numbers x and y. For example, (−2.1, 3.5), (π, 2), (0, 0) are complex numbers. Let z = (x, y) be a complex number. The real part of z, denoted by Re z, is the real number x. The imaginary part of z, denoted by Im z, is the real number y. Re z = x Im z = y Two complex numbers z1 = (a1, b1) and z2 = (a2, b2) are equal, written z1 = z2 or (a1, b1) = (a2, b2) if and only if a1 = a2 and b1 = b2. For example, if (x, 2) = (3, c) then x = 3 and c = 2. Since a complex number is denoted by an ordered pair (x, y) of real numbers x and y, then we may view the complex number (x, y) as the point with abscissa x and ordinate y. The complex plane consists of all the points that represent the complex numbers. For example, let us indicate the following complex numbers in the complex plane: z1 = (−3, −2), z2 = (0, 1), z3 = (4, 2), z4 = (5, −1)

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z3

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z2

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z1

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z4

The complex plane showing four complex numbers z1, z2, z3, and z4 1.1 Operations on Complex Numbers

Some binary operations on complex numbers are addition, multiplication, and division. They are deﬁned as follows: Let z1 = (a1, b1) and z2 = (a2, b2). Then 1. Addition. z1 + z2 = (a1 + a2, b1 + b2). For example, (2, −3) + (−1, 2) = (1, −1). 2. Multiplication. z1z2 = (a1a2−b1b2, a1b2+a2b1). For example, (2, −3)(−1, 2) = ((2)(−1)−(−3)(2), (2)(2)+ (−3)(−1)) = (4, 7) 3. Division. If z2 = (0, 0), then z1 = z2 For example, (2, −3) = (−1, 2) 2

a1a2 + b1b2 −a1b2 + a2b1 , a2 + b2 a2 + b2 2 2 2 2 8 1 − ,− 5 5

Remark. The complex numbers of the form (x, 0) are actually the real numbers x in the following sense: 1. (a1, 0) + (a2, 0) = (a1 + a2, 0), which corresponds to the sum a1 + a2. 2. (a1, 0)(a2, 0) = (a1a2, 0), which corresponds to the product a1a2 3. (a1, 0) = (a2, 0) a1 a2 , 0

, which corresponds to the quotient

a1 a2 .

1.2 Scalar Multiple

A complex number z = (x, y) may be multiplied by a real number c and the result is cz = (cx, cy) For example, if z = (2, −3), then 5z = (10, −15). The additive inverse or negative of a complex number z = (x, y), denoted by −z, is deﬁned by −z = (−1)z. For example, if z = (2, −3), then −z = (−1)z = (−2, 3). Remark. We may deﬁne subtraction denoted by z1 − z2 in terms of addition and negative as follows: z1 − z2 = z1 + (−1)z2. For example, (1, 2) − (2, −2) = (1, 2) + (−2, 2) = (−1, 4).

1.3 Conjugate

The conjugate of a complex number z = (x, y), denoted by the symbol z, is the complex number (x, −y). For example, if z = (5, −2), then z = (5, 2). Note that if we plot z and z on the complex plane, then 3

these two points are reﬂections of each other with respect to the x axis. . . .. ... . . ..... . . . . . . . ...... . . . . . . . . . . . . . . . . . . . . . . . . . . . ...... . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . .. ..................................................................................................................................................................................................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....