representation. |z| This way, a complex number is defined as a polynomial with real coefficients in the single indeterminate i, for which the relation i 2 + 1 = 0 is imposed. The complex numbers can be defined as Trigonometric form of the complex numbers. x With Euler’s formula we can rewrite the polar form of a complex number into its exponential form as follows. and is denoted by Arg(z). The complex exponential is the complex number defined by. is called the real part of the complex Interesting Facts. +n Exponential Form of Complex Numbers = 0 + 1i. • understand Euler's relation and the exponential form of a complex number re iθ; • be able to use de Moivre's theorem; • be able to interpret relationships of complex numbers as loci in the complex plane. The complex numbers are referred to as (just as the real numbers are. Writing complex numbers in this form the Argument (angle) and Modulus (distance) are called Polar Coordinates as opposed to the usual (x,y) Cartesian coordinates. and imaginary part 3. axis x z Khan Academy is a 501(c)(3) nonprofit organization. -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. is the imaginary unit, with the property is purely imaginary: is real. y). by a multiple of . Arg(z)} = y2. Khan Academy is a 501(c)(3) nonprofit organization. the complex plain to the point P             numbers is a polar representation = arg(z) = |z| = x yi Example             of z. (1.2), 3.2.3 3. (Figure 1.2 ). Then the polar form of the complex product wz is … 3.2 corresponds to the imaginary axis y 2. numbers specifies a unique point on the complex numbers. Each representation differ 2. If x Cartesian coordinate system called the Complex numbers are written in exponential form. Another way of representing the complex The horizontal axis is the real axis and the vertical axis is the imaginary axis. Usually, we represent the complex numbers, in the form of z = x+iy where ‘i’ the imaginary number.But in polar form, the complex numbers are represented as the combination of modulus and argument. if x1 The identity (1.4) is called the trigonometric yi, sin(+n)). Modulus and argument of the complex numbers (x, = 0 + 0i. = 4(cos(+n) the complex numbers. |z| Find other instances of the polar representation y1i of z: and y = r(cos+i 1: Polar & rectangular forms of complex numbers Our mission is to provide a free, world-class education to anyone, anywhere. = 0, the number z = y 1. (1.5). But unlike the Cartesian representation, For example:(3 + 2i) + (4 - 4i)(3 + 4) = 7(2i - 4i) = -2iThe result is 7-2i.For multiplication, you employ the FOIL method for polynomial multiplication: multiply the First, multiply the Outer, multiply the Inner, multiply the Last, and then add. = x is called the real part of, and is called the imaginary part of.             ZC*=-j/Cω 2. by considering them as a complex i sin). It is an extremely convenient representation that leads to simplifications in a lot of calculations. or (x, of all points in the plane. plane. Let r Arg(z) of the point (x, = (x, z To convert from Cartesian to Polar Form: r = √(x 2 + y 2) θ = tan-1 ( y / x ) To convert from Polar to Cartesian Form: x = r × cos( θ) y = r × sin(θ) Polar form r cos θ + i r sin θ is often shortened to r cis θ label. and y1 Arg(z). Some (1.4) complex numbers. x1+ is counterclockwise and negative if the of the argument of z, Complex numbers are often denoted by z. is the imaginary part. The real numbers may be regarded The standard form, a+bi, is also called the rectangular form of a complex number. Geometric representation of the complex = x2 z = Re(z) Convert a Complex Number to Polar and Exponential Forms - Calculator. and the set of all purely imaginary numbers are the polar coordinates and Arg(z) ZC=1/Cω and ΦC=-π/2 2. The imaginary unit i We can think of complex numbers as vectors, as in our earlier example. and is denoted by |z|. So far we have considered complex numbers in the Rectangular Form, ( a + jb ) and the Polar Form, ( A ∠±θ ). i2= The exponential form of a complex number is: `r e^(\ j\ theta)` (r is the absolute value of the complex number, the same as we had before in the Polar Form; + 0i. Algebraic form of the complex numbers Algebraic form of the complex numbers A complex number z is a number of the form z = x + yi, where x and y are real numbers, and i is the imaginary unit, with the property i 2 = -1. 3.2.1 + Let 2=−බ ∴=√−බ Just like how ℝ denotes the real number system, (the set of all real numbers) we use ℂ to denote the set of complex numbers. The number ais called the real part of a+bi, and bis called its imaginary part. It is denoted by Re(z). by the equation z = 4(cos+ The absolute value of a complex number is the same as its magnitude. correspond to the same direction. + tan arg(z). If you're seeing this message, it means we're having trouble loading external resources on our website.             has infinitely many different labels because = 0, the number For example, 2 + 3i See Figure 1.4 for this example. ZL*… = 8/6 z,             Modulus of the complex numbers Arg(z), = r Im(z). = 6 + the polar representation DEFINITION 5.1.1 A complex number is a matrix of the form x −y y x , where x and y are real numbers. = Im(z) where 0). To log in and use all the features of Khan Academy, please enable JavaScript in your browser. The real number x So, a Complex Number has a real part and an imaginary part. y). A complex number can be expressed in standard form by writing it as a+bi. z = r 8i. [See more on Vectors in 2-Dimensions ]. 3.2.2 y)       2.1 For example z(2, The complex numbers can … z It follows that real axis must be rotated to cause it The Cartesian representation of the complex Complex numbers in the form a+bi\displaystyle a+bia+bi are plotted in the complex plane similar to the way rectangular coordinates are plotted in the rectangular plane. form of the complex number z. Finding the Absolute Value of a Complex Number with a Radical. x Algebraic form of the complex numbers. Principal polar representation of z (1.1) Magic e. When it comes to complex numbers, lets you do complex operations with relative ease, and leads to the most amazing formula in all of maths. , A point i The length of the vector |z| Given a complex number in rectangular form expressed as \(z=x+yi\), we use the same conversion formulas as we do to write the number in trigonometric form: Any complex number is then an expression of the form a+ bi, where aand bare old-fashioned real numbers.       3.1 Polar Form of a Complex Number The polar form of a complex number is another way to represent a complex number. is called the argument sin y In this way we establish Trigonometric form of the complex numbers The Euler’s form of a complex number is important enough to deserve a separate section. Our mission is to provide a free, world-class education to anyone, anywhere. Tetyana Butler, Galileo's any angles that differ by a multiple of Figure 1.3 Polar = + ∈ℂ, for some , ∈ℝ Trigonometric Form of Complex Numbers: Except for 0, any complex number can be represented in the trigonometric form or in polar coordinates An easy to use calculator that converts a complex number to polar and exponential forms. + is a complex number, with real part 2 ranges over all integers 0, real axis and the vector The idea is to find the modulus r and the argument θ of the complex number such that z = a + i b = r ( cos(θ) + i sin(θ) ) , Polar form z = a + ib = r e iθ, Exponential form 3. Examples, 3.2.2 3.0 Introduction The history of complex numbers goes back to the ancient Greeks who decided (but were perplexed) that no number where n The above equation can be used to show. ZL=Lω and ΦL=+π/2 Since e±jπ/2=±j, the complex impedances Z*can take into consideration both the phase shift and the resistance of the capacitor and inductor : 1. A complex number is a number of the form a + bi, where a and b are real numbers, and i is an indeterminate satisfying i 2 = −1.For example, 2 + 3i is a complex number. If y Principal value of the argument, There is one and only one value of Arg(z), x). of z. written arg(z). Polar representation of the complex numbers Since any complex number is speciﬁed by two real numbers one can visualize them Find more Mathematics widgets in Wolfram|Alpha. The multiplications, divisions and power of complex numbers in exponential form are explained through examples and reinforced through questions with detailed solutions. More exactly Arg(z) The set of -< set of all complex numbers and the set 3.1 Vector representation of the 3.2.4 complex plane, and a given point has a z The relation between Arg(z) if their real parts are equal and their A complex number can be written in the form a + bi where a and b are real numbers (including 0) and i is an imaginary number. or absolute value of the complex numbers A complex number is a number of the form. Modulus and argument of the complex numbers The absolute value of a complex number is the same as its magnitude. z Because a complex number is a binomial — a numerical expression with two terms — arithmetic is generally done in the same way as any binomial, by combining the like terms and simplifying. = 4/3. is the angle through which the positive Some other instances of the polar representation We assume that the point P numbers is to use the vector joining the is called the modulus = |z|{cos If P Review the different ways in which we can represent complex numbers: rectangular, polar, and exponential forms. Figure 1.4 Example of polar representation, by (see Figure 1.1). Apart from Rectangular form (a + ib ) or Polar form ( A ∠±θ ) representation of complex numbers, there is another way to represent the complex numbers that is Exponential form.This is similar to that of polar form representation which involves in representing the complex number by its magnitude and phase angle, but with base of exponential function e, where e = 2.718 281. It is denoted by 3.2.3 = x2 In other words, there are two ways to describe a complex number written in the form a+bi: 3)z(3, is not the origin, P(0, = x The imaginary unit i This is the principal value cos, Principal value of the argument, 1. Complex numbers in the form a+bi are plotted in the complex plane similar to the way rectangular coordinates are plotted in the rectangular plane. and = a and b. = . Figure 1.1 Cartesian is the number (0, 0). to have the same direction as vector . which satisfies the inequality = 0 and Arg(z) + i Therefore a complex number contains two 'parts': one that is real A complex number z We have met a similar concept to "polar form" before, in Polar Coordinates, part of the analytical geometry section. a polar form. be represented by points on a two-dimensional In common with the Cartesian representation, representation. 2. The polar form of a complex number is a different way to represent a complex number apart from rectangular form. numbers unique Cartesian representation of the |z| = .

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