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A conjugate example (click to view in the calculator): In physics and electrical engineering, a complex conjugate is often denoted as z*. For example, the following two numbers are complex conjugates: The complex conjugate of a number is found by changing the sign of the imaginary part. Two complex numbers x + yi and n + mi are equal if and only if x = n and y = m. The quantity of i is treated as a constant and whenever an i² is encountered, it is replaced by –1. You will find more information about complex number visualization, phasors and conversion from polar to rectangular and vice versa in our Phasor Conversion Calculator.Ĭomplex numbers follow the same algebra rules as ordinary numbers. In this representation, the terms “amplitude” and “phase” are used instead of the terms “modulus” (“magnitude”) and “argument”.Ī complex number representing a sinusoidal function with amplitude A, angular frequency ω and initial phase θ is called a phasor (from phase vector). In physics and electrical engineering, the polar representation of complex numbers is widely used for the representation of sinusoidal voltages and currents. To obtain the rectangular coordinates from the polar ones, we use the following formula:Įuler’s formula establishes the relationship between the trigonometric functions and the complex exponential function for any real number φ:Įuler’s formula allows representing a sinusoid as a complex exponential function, which is convenient in many fields. The magnitude r and the argument φ together represent complex numbers in the polar form because their combination specifies a unique position of the point representing the complex number on the polar plane. The argument φ is determined using the two-argument arctangent arctan2( y,x) function: The magnitude of a complex number z = x + iy is given by the following: This representation uses the magnitude (modulus) r of a vector starting at the origin and ending in the complex point z, and the angle φ between this vector and the positive real axis measured in a clockwise direction.
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Polar complex planeĪ complex number z = x + jy = r ∠φ is represented as a point and a vector in the complex planeĪ complex number z can also be represented in polar notation, which uses another type of the complex plane in the polar coordinate system.
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We can see that the real number line is the same as the real (horizontal) axis of the complex plane because the imaginary part of real numbers is zero. The horizontal axis of the complex plane corresponds to the real part of the complex number and the vertical axis corresponds to the imaginary part. Just as all real numbers can be thought of as points on a number line, a complex number z, which is identified with an ordered pair of real numbers (Re( z), Im( z)), can be represented by a point in a two-dimensional space called the complex plane. The mathematical notation of complex numbers uses two operators for separating a complex number into its real and imaginary parts: Re( z) and Im( z). Representation of Complex Numbers Cartesian complex plane Usually, the imaginary part is reduced to a real number multiplied by the square root of minus one. The real part is a real number and the imaginary part is an imaginary number, which is the square root of a negative number. In other words, i is the square root of minus one (√–1). The symbol i or j in electrical engineering (electrical engineers think differently from the rest of the world!) is called the imaginary unit and is defined by the equation i² = –1. A complex number is a number in the form of a sum of a real part and an imaginary part a + bi.