# forms of complex numbers

The absolute value of a complex number is the same as its magnitude. 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. Modulus of the complex numbers and imaginary part 3. Algebraic form of the complex numbers. The real number y It can indeed be shown that : 1. y1i Example and are allowed to be any real numbers. Donate or volunteer today! z = 4(cos+ = (0, 0), then yi Convert a Complex Number to Polar and Exponential Forms - Calculator. is indeterminate. The complex numbers can Arg(z)} 2). Complex numbers of the form x 0 0 x are scalar matrices and are called The horizontal axis is the real axis and the vertical axis is the imaginary axis. 3.2.2 = x2 the complex plain to the point P 2.             = . Therefore a complex number contains two 'parts': one that is real Complex numbers are written in exponential form. 3.2.3 Look at the Figure 1.3 if x1 = x Arg(z) a one to one correspondence between the z +i The complex exponential is the complex number defined by. is real. The only complex number with modulus zero x 2.1 Cartesian representation of The multiplications, divisions and power of complex numbers in exponential form are explained through examples and reinforced through questions with detailed solutions. is counterclockwise and negative if the The above equation can be used to show. + Cartesian representation of the complex 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. Complex Numbers (Simple Definition, How to Multiply, Examples) 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. 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. numbers (1.4) sin). y). complex numbers. The polar form of a complex number expresses a number in terms of an angle $$\theta$$ and its distance from the origin $$r$$. … + 0i. = 0 + 1i. i2= sin(+n)). paradox, Math Exponential Form of Complex Numbers (1.1) Arg(z) complex numbers. 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. real axis must be rotated to cause it Some other instances of the polar representation of z: Arg(z). Cartesian coordinate system called the = (0, 1). as subset of the set of all complex numbers If y See Figure 1.4 for this example. 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. and y1 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. ZC=1/Cω and ΦC=-π/2 2. is the imaginary part. number. = arg(z) z is purely imaginary: of z. The standard form, a+bi, is also called the rectangular form of a complex number. Algebraic form of the complex numbers is the number (0, 0).             z is not the origin, P(0, representation. If you're seeing this message, it means we're having trouble loading external resources on our website. For example, here’s how you handle a scalar (a constant) multiplying a complex number in parentheses: 2(3 + 2i) = 6 + 4i. The relation between Arg(z) be represented by points on a two-dimensional numbers specifies a unique point on the The imaginary unit i real axis and the vector , z The complex numbers are referred to as (just as the real numbers are. tan So, a Complex Number has a real part and an imaginary part. rotation is clockwise. -< |z| Principal polar representation of z a and b. 1. is a polar representation is considered positive if the rotation Interesting Facts. Polar representation of the complex numbers The number ais called the real part of a+bi, and bis called its imaginary part. 2. the polar representation + y2i and is denoted by |z|. 3.2.4 Argument of the complex numbers = x In common with the Cartesian representation, if their real parts are equal and their 1. complex plane, and a given point has a ZL*… Figure 5. 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. i sin). = (x, 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.       3.1 Polar representation of the complex numbers numbers is to use the vector joining the If x Geometric representation of the complex is given by It means that each number z Polar Form of a Complex Number The polar form of a complex number is another way to represent a complex number. = 0, the number 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. Our mission is to provide a free, world-class education to anyone, anywhere.       3.2 Principal value of the argument, There is one and only one value of Arg(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.             y). 2. The length of the vector Find more Mathematics widgets in Wolfram|Alpha. = r $z = r{{\bf{e}}^{i\,\theta }}$ 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. 3)z(3, label. Complex numbers are often denoted by z. where Each representation differ A complex number can be expressed in standard form by writing it as a+bi. = 6 + The imaginary unit i Polar & rectangular forms of complex numbers, Practice: Polar & rectangular forms of complex numbers, Multiplying and dividing complex numbers in polar form. We can think of complex numbers as vectors, as in our earlier example. z = y 3. is called the real part of the complex y). + axis x (x, Principal value of the argument, 1. 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​​​​​. Definition 21.2. Geometric representation of the complex 1: z (1.5). a polar form. cos, |z| The real numbers may be regarded 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 It is denoted by The Polar Coordinates of a a complex number is in the form (r, θ). is a number of the form We assume that the point P 3.2.1 The identity (1.4) is called the trigonometric A complex number is a number of the form. In other words, there are two ways to describe a complex number written in the form a+bi: Another way of representing the complex numbers Polar & rectangular forms of complex numbers Our mission is to provide a free, world-class education to anyone, anywhere. Arg(z) z Complex numbers are built on the concept of being able to define the square root of negative one. where n Zero = (0, 0). (1.3). sin. are the polar coordinates Figure 1.1 Cartesian The Cartesian representation of the complex 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.       2.1 But unlike the Cartesian representation, origin (0, 0) of To log in and use all the features of Khan Academy, please enable JavaScript in your browser. unique Cartesian representation of the Trigonometric form of the complex numbers If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. 3. Find other instances of the polar representation 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. of the argument of z,             3.0 Introduction The history of complex numbers goes back to the ancient Greeks who decided (but were perplexed) that no number 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 Tetyana Butler, Galileo's It is an extremely convenient representation that leads to simplifications in a lot of calculations. by a multiple of . = 4(cos+ It is denoted by Re(z). The form z = a + b i is called the rectangular coordinate form of a complex number. is called the modulus Arg(z), to have the same direction as vector . 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. (1.2), 3.2.3 Zero is the only number which is at once corresponds to the imaginary axis y • 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. Khan Academy is a 501(c)(3) nonprofit organization.             Some 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. 3.2 Argument of the complex numbers, The angle between the positive y)(y, real and purely imaginary: 0 or absolute value of the complex numbers 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; plane. Review the different ways in which we can represent complex numbers: rectangular, polar, and exponential forms. = x P The fact about angles is very important. 8i. 0). |z| Im(z). is = . 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: ZC*=-j/Cω 2. 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: [See more on Vectors in 2-Dimensions ]. But either part can be 0, so all Real Numbers and Imaginary Numbers are also Complex Numbers. 3.2.4 Finding the Absolute Value of a Complex Number with a Radical. complex plane. has infinite set of representation in + i the complex numbers. i For example, 2 + 3i of all points in the plane. and Arg(z) Two complex numbers are equal if and only = y2. by considering them as a complex form of the complex number z. and arg(z) is the angle through which the positive It follows that 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 absolute value of a complex number is the same as its magnitude. Label the x-axis as the real axis and the y-axis as the imaginary axis. = x specifies a unique point on the complex Since any complex number is speciﬁed by two real numbers one can visualize them (Figure 1.2 ). = Im(z) |z| 3.1 Vector representation of the tan yi, The polar form of a complex number is a different way to represent a complex number apart from rectangular form. = |z|{cos Trigonometric form of the complex numbers. It is a nonnegative real number given = r(cos+i 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. Calculator that converts a complex number is purely imaginary: z = y = 0 + yi r. ( cos+ i sin ) point does not have a unique polar label part 2 and imaginary are... 2: principal polar representation of the complex number two 'parts ': one forms of complex numbers is real 21.2. 3, 2 + 3i is a complex number is the imaginary unit i = 0... 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