division of complex numbers in polar form proof

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Posted on January 20, 2021

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. Convert given two complex number division into polar form. Polar Form of a Complex Number. What is the complex conjugate of a complex number? The angle \(\theta\) is called the argument of the complex number \(z\) and the real number \(r\) is the modulus or norm of \(z\). The polar form of a complex number is a different way to represent a complex number apart from rectangular form. The formula for multiplying complex numbers in polar form tells us that to multiply two complex numbers, we add their arguments and multiply their norms. How to algebraically calculate exact value of a trig function applied to any non-transcendental angle? This is the polar form of a complex number. by M. Bourne. If \(r\) is the magnitude of \(z\) (that is, the distance from \(z\) to the origin) and \(\theta\) the angle \(z\) makes with the positive real axis, then the trigonometric form (or polar form) of \(z\) is \(z = r(\cos(\theta) + i\sin(\theta))\), where, \[r = \sqrt{a^{2} + b^{2}}, \cos(\theta) = \dfrac{a}{r}\]. When we write \(z\) in the form given in Equation \(\PageIndex{1}\):, we say that \(z\) is written in trigonometric form (or polar form). Your email address will not be published. Indeed, using the product theorem, (z1 z2)⋅ z2 = {(r1 r2)[cos(ϕ1 −ϕ2)+ i⋅ sin(ϕ1 −ϕ2)]} ⋅ r2(cosϕ2 +i ⋅ sinϕ2) = Complex Numbers in Polar Form. To understand why this result it true in general, let \(w = r(\cos(\alpha) + i\sin(\alpha))\) and \(z = s(\cos(\beta) + i\sin(\beta))\) be complex numbers in polar form. Your email address will not be published. This states that to multiply two complex numbers in polar form, we multiply their norms and add their arguments, and to divide two complex numbers, we divide their norms and subtract their arguments. Also, \(|z| = \sqrt{(\sqrt{3})^{2} + 1^{2}} = 2\) and the argument of \(z\) satisfies \(\tan(\theta) = \dfrac{1}{\sqrt{3}}\). Let us learn here, in this article, how to derive the polar form of complex numbers. ( 5 + 2 i 7 + 4 i) ( 7 − 4 i 7 − 4 i) Step 3. Key Questions. Then the polar form of the complex quotient \(\dfrac{w}{z}\) is given by \[\dfrac{w}{z} = \dfrac{r}{s}(\cos(\alpha - \beta) + i\sin(\alpha - \beta)).\]. Usually, we represent the complex numbers, in the form of z = x+iy where ‘i’ the imaginary number. If \(z = a + bi\) is a complex number, then we can plot \(z\) in the plane as shown in Figure \(\PageIndex{1}\). \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\), 5.2: The Trigonometric Form of a Complex Number, [ "article:topic", "license:ccbyncsa", "showtoc:no", "authorname:tsundstrom", "modulus (complex number)", "norm (complex number)" ], https://math.libretexts.org/@app/auth/3/login?returnto=https%3A%2F%2Fmath.libretexts.org%2FBookshelves%2FPrecalculus%2FBook%253A_Trigonometry_(Sundstrom_and_Schlicker)%2F05%253A_Complex_Numbers_and_Polar_Coordinates%2F5.02%253A_The_Trigonometric_Form_of_a_Complex_Number, \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\), 5.3: DeMoivre’s Theorem and Powers of Complex Numbers, ScholarWorks @Grand Valley State University, Products of Complex Numbers in Polar Form, Quotients of Complex Numbers in Polar Form, Proof of the Rule for Dividing Complex Numbers in Polar Form. Determine the conjugate of the denominator. An illustration of this is given in Figure \(\PageIndex{2}\). Therefore, if we add the two given complex numbers, we get; Again, to convert the resulting complex number in polar form, we need to find the modulus and argument of the number. Division of Complex Numbers in Polar Form, Let \(w = r(\cos(\alpha) + i\sin(\alpha))\) and \(z = s(\cos(\beta) + i\sin(\beta))\) be complex numbers in polar form with \(z \neq 0\). z = r z e i θ z . The word polar here comes from the fact that this process can be viewed as occurring with polar coordinates. The polar form of a complex number z = a + b i is z = r ( cos θ + i sin θ ) , where r = | z | = a 2 + b 2 , a = r cos θ and b = r sin θ , and θ = tan − 1 ( b a ) for a > 0 and θ = tan − 1 … Multiplication of Complex Numbers in Polar Form, Let \(w = r(\cos(\alpha) + i\sin(\alpha))\) and \(z = s(\cos(\beta) + i\sin(\beta))\) be complex numbers in polar form. z = r z e i θ z. z = r_z e^{i \theta_z}. After studying this section, we should understand the concepts motivated by these questions and be able to write precise, coherent answers to these questions. Figure \(\PageIndex{2}\): A Geometric Interpretation of Multiplication of Complex Numbers. Using equation (1) and these identities, we see that, \[w = rs([\cos(\alpha)\cos(\beta) - \sin(\alpha)\sin(\beta)]) + i[\cos(\alpha)\sin(\beta) + \cos(\beta)\sin(\alpha)] = rs(\cos(\alpha + \beta) + i\sin(\alpha + \beta))\]. To prove the quotation theorem mentioned above, all we have to prove is that z1 z2 in the form we presented, multiplied by z2, produces z1. The modulus of a complex number is also called absolute value. The following questions are meant to guide our study of the material in this section. We can think of complex numbers as vectors, as in our earlier example. Let z1 =r1eiθ1 and z2 =r2eiθ2 z 1 = r 1 e i θ 1 a n d z 2 = r 2 e i θ 2. If \(z \neq 0\) and \(a \neq 0\), then \(\tan(\theta) = \dfrac{b}{a}\). We know, the modulus or absolute value of the complex number is given by: To find the argument of a complex number, we need to check the condition first, such as: Here x>0, therefore, we will use the formula. a =-2 b =-2. When performing addition and subtraction of complex numbers, use rectangular form. Draw a picture of \(w\), \(z\), and \(wz\) that illustrates the action of the complex product. So \[z = \sqrt{2}(\cos(-\dfrac{\pi}{4}) + \sin(-\dfrac{\pi}{4})) = \sqrt{2}(\cos(\dfrac{\pi}{4}) - \sin(\dfrac{\pi}{4})\], 2. Roots of complex numbers in polar form. Let and be two complex numbers in polar form. To divide,we divide their moduli and subtract their arguments. Let us consider (x, y) are the coordinates of complex numbers x+iy. \[e^{i\theta} = \cos(\theta) + i\sin(\theta)\] To find \(\theta\), we have to consider cases. We know the magnitude and argument of \(wz\), so the polar form of \(wz\) is \[\dfrac{w}{z} = \dfrac{3}{2}[\cos(\dfrac{23\pi}{12}) + \sin(\dfrac{23\pi}{12})]\], Let \(w = r(\cos(\alpha) + i\sin(\alpha))\) and \(z = s(\cos(\beta) + i\sin(\beta))\) be complex numbers in polar form with \(z \neq 0\). This trigonometric form connects algebra to trigonometry and will be useful for quickly and easily finding powers and roots of complex numbers. rieiθ2 = r1r2ei(θ1+θ2) ⇒ z 1 z 2 = r 1 e i θ 1. r i e i θ 2 = r 1 r 2 e i ( θ 1 + θ 2) This result is in agreement with the fact that moduli multiply and arguments add upon multiplication. For multiplying complex numbers, just like vectors, as in our earlier example = a + )! But in polar form of z = r 1 cis θ 1 and z 2 = z. Questions are meant to guide our study of the complex number in polar form b, then is. Exact value of a complex number polar form in general, we notice. To derive the polar form by multiplying their norms and adding their.. 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