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Lecture 1 Complex Numbers Deﬁnitions. Just as R is the set of real numbers, C is the set of complex numbers.Ifz is a complex number, z is of the form z = x+ iy ∈ C, for some x,y ∈ R. e.g. The union of the set of all imaginary numbers and the set of all real numbers is the set of complex numbers. DEFINITION 5.1.1 A complex number is a matrix of the form x −y y x , where x and y are real numbers. Complex Numbers and the Complex Exponential 1. COMPLEX NUMBERS 5.1 Constructing the complex numbers One way of introducing the ﬁeld C of complex numbers is via the arithmetic of 2×2 matrices. Complex numbers of the form x 0 0 x are scalar matrices and are called 1–2 WWLChen : Introduction to Complex Analysis Note the special case a =1and b =0. Figure 1: Complex numbers can be displayed on the complex plane. Suppose that z = x+iy, where x,y ∈ R. The real number x is called the real part of z, and denoted by x = Rez.The real number y is called the imaginary part of z, and denoted by y = Imz.The set C = {z = x+iy: x,y ∈ R} is called the set of all complex numbers. Since complex numbers are composed from two real numbers, it is appropriate to think of them graph-ically in a plane. Let i2 = −1. A complex number z1 = a + bi may be displayed as an ordered pair: (a,b), with the “real axis” the usual x-axis and the “imaginary axis” the usual y-axis. Introduction to Complex Numbers: YouTube Workbook 6 Contents 6 Polar exponential form 41 6.1 Video 21: Polar exponential form of a complex number 41 6.2 Revision Video 22: Intro to complex numbers + basic operations 43 6.3 Revision Video 23: Complex numbers and calculations 44 6.4 Video 24: Powers of complex numbers via polar forms 45 (Note: and both can be 0.) 3 + 4i is a complex number. Introduction to COMPLEX NUMBERS 1 BUSHRA KANWAL Imaginary Numbers Consider x2 = … The horizontal axis representing the real axis, the vertical representing the imaginary axis. z= a+ ib a= Re(z) b= Im(z) = argz r = jz j= p a2 + b2 Figure 1: The complex number z= a+ ib. Complex numbers The equation x2 + 1 = 0 has no solutions, because for any real number xthe square x 2is nonnegative, and so x + 1 can never be less than 1.In spite of this it turns out to be very useful to assume that there is a number ifor which one has 1What is a complex number? For instance, d3y dt3 +6 d2y dt2 +5 dy dt = 0 Introduction to Complex Numbers Adding, Subtracting, Multiplying And Dividing Complex Numbers SPI 3103.2.1 Describe any number in the complex number system. Introduction to Complex Numbers. View complex numbers 1.pdf from BUSINESS E 1875 at Riphah International University Islamabad Main Campus. Introduction. 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