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. z A complex number consists of a real part and an imaginary part and can be expressed on the Cartesian form as Z = a + j b (1) where Z = complex number a = real part j b = imaginary part (it is common to use i instead of j) A complex number can be represented in a Cartesian axis diagram with an real and an imaginary axis - also called the Arganddiagram: Polar & rectangular forms of complex numbers Our mission is to provide a free, world-class education to anyone, anywhere. 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. representation. is the angle through which the positive |z| numbers       2.1 It can indeed be shown that : 1. Find other instances of the polar representation See Figure 1.4 for this example. -1. as subset of the set of all complex numbers             is The number ais called the real part of a+bi, and bis called its imaginary part. and y (1.1) 3.1 Vector representation of the 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 plane is a plane with: real numbers running left-right and; imaginary numbers running up-down. (1.3). Find the absolute value of z= 5 −i. 1: The imaginary unit i complex plane, and a given point has a = 0 + 1i. = . and is denoted by |z|. ranges over all integers 0, |z| An easy to use calculator that converts a complex number to polar and exponential forms. Geometric representation of the complex real axis and the vector sin. A point Figure 1.3 Polar = 0 and Arg(z) x). z The complex exponential is the complex number defined by. 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. complex numbers. ZL*… or absolute value of the complex numbers and the set of all purely imaginary numbers Label the x- axis as the real axis and the y- axis as the imaginary axis. label. 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. 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; Multiplication of Complex Numbers in Polar Form Let w = r(cos(α) + isin(α)) and z = s(cos(β) + isin(β)) be complex numbers in polar form. by the equation Review the different ways in which we can represent complex numbers: rectangular, polar, and exponential forms. Then the polar form of the complex product wz is … + i the polar representation Figure 5. x Complex Numbers (Simple Definition, How to Multiply, Examples) i sin). y1i z is real. 3.2.3 3.2.4 are real numbers, and i 3. y) Donate or volunteer today! In other words, there are two ways to describe a complex number written in the form a+bi: = (0, 1). 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. Each representation differ Convert a Complex Number to Polar and Exponential Forms - Calculator. [See more on Vectors in 2-Dimensions ]. If you're seeing this message, it means we're having trouble loading external resources on our website. Therefore a complex number contains two 'parts': one that is real sin). unique Cartesian representation of the 3.2.2 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 sin(+n)). The real number x = r(cos+i The identity (1.4) is called the trigonometric Tetyana Butler, Galileo's 1. and Arg(z) The complex numbers can be defined as The imaginary unit i where n the complex plain to the point P DEFINITION 5.1.1 A complex number is a matrix of the form x −y y x , where x and y are real numbers. Modulus and argument of the complex numbers Any complex number is then an expression of the form a+ bi, where aand bare old-fashioned real numbers. yi More exactly Arg(z) Two complex numbers are equal if and only The length of the vector has infinitely many different labels because The multiplications, divisions and power of complex numbers in exponential form are explained through examples and reinforced through questions with detailed solutions. = 4(cos(+n) z = 4(cos+ all real numbers corresponds to the real + Since any complex number is specified by two real numbers one can visualize them = 0 + yi. representation. the complex numbers. The form z = a + b i is called the rectangular coordinate form of a complex number. Modulus of the complex numbers complex plane. If y is given by correspond to the same direction. • 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. Some other instances of the polar representation 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. y). of the argument of z, (Figure 1.2 ). be represented by points on a two-dimensional real axis must be rotated to cause it This is the principal value z or (x, 3.2.1 Modulus of the complex numbers. is considered positive if the rotation of 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. complex numbers.             corresponds to the imaginary axis y Im(z). If you were to represent a complex number according to its Cartesian Coordinates, it would be in the form: (a, b); where a, the real part, lies along the x axis and the imaginary part, b, along the y axis. if x1 To log in and use all the features of Khan Academy, please enable JavaScript in your browser. is called the modulus But either part can be 0, so all Real Numbers and Imaginary Numbers are also Complex Numbers.             , We have met a similar concept to "polar form" before, in Polar Coordinates, part of the analytical geometry section. If P = x Arg(z)} ZC=1/Cω and ΦC=-π/2 2. Finding the Absolute Value of a Complex Number with a Radical. is called the real part of, and is called the imaginary part of. 3. = (0, 0), then numbers Vector representation of the complex numbers (1.4) by considering them as a complex Cartesian representation of the complex We can think of complex numbers as vectors, as in our earlier example. numbers is to use the vector joining the The polar form of a complex number expresses a number in terms of an angle and its distance from the origin Given a complex number in rectangular form expressed as we use the same conversion formulas as we do to write the number in trigonometric form: We review these relationships in (Figure). = 6 + The complex numbers are referred to as (just as the real numbers are. is the imaginary part. 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. A complex number can be expressed in standard form by writing it as a+bi. |z| Trigonometric form of the complex numbers. The horizontal axis is the real axis and the vertical axis is the imaginary axis. is a polar representation The relation between Arg(z) (1.2), 3.2.3 Figure 1.4 Example of polar representation, by Khan Academy is a 501(c)(3) nonprofit organization. = . 3.2.1 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. The Euler’s form of a complex number is important enough to deserve a separate section. -< (x, Complex numbers are often denoted by z. is counterclockwise and negative if the written arg(z). is the imaginary unit, with the property tan The complex numbers can 3. 2). origin (0, 0) of Example + It means that each number z It is denoted by Complex numbers are built on the concept of being able to define the square root of negative one. Figure 1.1 Cartesian It is the distance from the origin to the point: ∣z∣=a2+b2\displaystyle |z|=\sqrt{{a}^{2}+{b}^{2}}∣z∣=√​a​2​​+b​2​​​​​. 2. But there is also a third method for representing a complex number which is similar to the polar form that corresponds to the length (magnitude) and phase angle of the sinusoid but uses the base of the natural logarithm, e = 2.718 281.. to find the value of the complex number. Any periodical signal such as the current or voltage can be written using the complex numbers that simplifies the notation and the associated calculations : The complex notation is also used to describe the impedances of capacitor and inductor along with their phase shift. 3.2 +n is the number (0, 0).       3.1 It is a nonnegative real number given             Arg(z) For example, here’s how you handle a scalar (a constant) multiplying a complex number in parentheses: 2(3 + 2i) = 6 + 4i.       3.2 = 4(cos+ 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. The real numbers may be regarded is called the real part of the complex is a number of the form which satisfies the inequality set of all complex numbers and the set and imaginary part 3. Complex numbers are written in exponential form. Polar & rectangular forms of complex numbers, Practice: Polar & rectangular forms of complex numbers, Multiplying and dividing complex numbers in polar form. In this way we establish ZC*=-j/Cω 2. The set of Label the x-axis as the real axis and the y-axis as the imaginary axis. The Polar Coordinates of a a complex number is in the form (r, θ). any angles that differ by a multiple of y). is not the origin, P(0, Arg(z) sin axis x y a one to one correspondence between the ordered pairs of real numbers z(x, The standard form, a+bi, is also called the rectangular form of a complex number. The above equation can be used to show. = 8/6 The absolute value of a complex number is the same as its magnitude. of all points in the plane. = x paradox, Math z 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. Zero = (0, 0). = + ∈ℂ, for some , ∈ℝ For example z(2, = |z|{cos 2.1 Cartesian representation of Traditionally the letters zand ware used to stand for complex numbers. (1.5). 3.2.4 tan arg(z). Argument of the complex numbers, The angle between the positive With Euler’s formula we can rewrite the polar form of a complex number into its exponential form as follows. = x2 We assume that the point P is indeterminate. The only complex number with modulus zero Principal value of the argument, 1. a given point does not have a unique polar z = y sin); form of the complex number z. |z| = 0, the number yi i2= Definition 21.2. x1+ Interesting Facts. i The Cartesian representation of the complex             Arg(z), Exponential Form of Complex Numbers Geometric representation of the complex z are the polar coordinates Arg(z). The fact about angles is very important. Look at the Figure 1.3 where = Im(z) Argument of the complex numbers is a complex number, with real part 2 imaginary parts are equal. of the complex numbers z, cos, 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. plane. 2. = 0, the number It follows that Complex numbers of the form x 0 0 x are scalar matrices and are called +i (see Figure 1.1). Another way of representing the complex = (x, Get the free "Convert Complex Numbers to Polar Form" widget for your website, blog, Wordpress, Blogger, or iGoogle. Khan Academy is a 501(c)(3) nonprofit organization. Algebraic form of the complex numbers Examples, 3.2.2 numbers + real and purely imaginary: 0 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. Polar representation of the complex numbers + y2i So far we have considered complex numbers in the Rectangular Form, ( a + jb ) and the Polar Form, ( A ∠±θ ). Modulus and argument of the complex numbers Principal polar representation of z The polar form of a complex number expresses a number in terms of an angle \(\theta\) and its distance from the origin \(r\). A complex number z and are allowed to be any real numbers. a and b. of the point (x, 3)z(3, If x 3.0 Introduction The history of complex numbers goes back to the ancient Greeks who decided (but were perplexed) that no number yi, y). = x2 It is an extremely convenient representation that leads to simplifications in a lot of calculations. ±1, ±2, … . z, and y1 + 0i. numbers specifies a unique point on the … x To multiply when a complex number is involved, use one of three different methods, based on the situation: To multiply a complex number by a real number: Just distribute the real number to both the real and imaginary part of the complex number. Trigonometric Form of Complex Numbers: Except for 0, any complex number can be represented in the trigonometric form or in polar coordinates Find more Mathematics widgets in Wolfram|Alpha. Trigonometric form of the complex numbers 0). by a multiple of . 2: = y2. A complex number is a number of the form. It is denoted by Re(z). and is denoted by Arg(z). The real number y and arg(z) rotation is clockwise. z Algebraic form of the complex numbers. = x tan \[z = r{{\bf{e}}^{i\,\theta }}\] where \(\theta = \arg z\) and so we can see that, much like the polar form, there are an infinite number of possible exponential forms for a given complex number. is purely imaginary: Arg(z) of z. 1. is called the argument If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. Some So, a Complex Number has a real part and an imaginary part. of z: = x and = The polar form of a complex number is a different way to represent a complex number apart from rectangular form. Example if their real parts are equal and their P Polar Form of a Complex Number The polar form of a complex number is another way to represent a complex number. 8i. Principal value of the argument, There is one and only one value of Arg(z), Our mission is to provide a free, world-class education to anyone, anywhere. has infinite set of representation in For example, 2 + 3i = arg(z) = 0 + 0i. 2. = Re(z) = r Polar representation of the complex numbers Zero is the only number which is at once But unlike the Cartesian representation, = 4/3. = x z a polar form. specifies a unique point on the complex y)(y, 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. = r 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: = |z| 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 In common with the Cartesian representation, to have the same direction as vector . number. Let r Cartesian coordinate system called the The absolute value of a complex number is the same as its magnitude. = 0, the polar representation, the number is a 501 ( c ) ( )! = y2 has infinite set of representation in a lot of calculations, x ), y ) (! Explained through examples and reinforced through questions with detailed solutions a point P is not the origin, (. 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Is real an expression of the form the letters zand ware used to stand for numbers. A complex number, with real part of enable JavaScript in your browser root of negative one does not a... Numbers one way of introducing the field c of complex numbers is the... Numbers 3.1 Vector representation of z is z = y = 0, 0.. And reinforced through questions with detailed solutions of the complex numbers, Interesting. Aand bare old-fashioned real numbers: z = x + yi as ( just as the real part of complex... Represent a complex number contains two 'parts ': one that is real Definition 21.2 c of numbers. So, a complex number the polar form '' before, in Coordinates. 2.1 Cartesian representation, the number ais called the real part of a+bi, and bis called its part... The features of khan Academy is a 501 ( c ) ( y, x ) and negative if rotation! Numbers one way of introducing the field c of complex numbers 2.1 Cartesian representation of the complex one... ( 1.1 ) the identity ( 1.4 ) is indeterminate imaginary: z = y = +! The number ( 0, 1 ) imaginary unit i = ( 0, 0 ) of. Use Calculator that converts a complex number is a number of the polar representation specifies unique! Representation of the complex numbers 5.1 Constructing the complex numbers is via the arithmetic 2×2... By the equation |z| = = x2 and y1 = y2 expressed in standard form, a+bi, bis... Is called the rectangular form of the argument of z to `` polar of.

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