division of complex numbers in polar form proof

There is a similar method to divide one complex number in polar form by another complex number in polar form. by M. Bourne. To better understand the product of complex numbers, we first investigate the trigonometric (or polar) form of a complex number. In which quadrant is $|\dfrac{w}{z}|$? With Euler’s formula we can rewrite the polar form of a complex number into its exponential form as follows. We have seen that we multiply complex numbers in polar form by multiplying their norms and adding their arguments. In general, we have the following important result about the product of two complex numbers. See the previous section, Products and Quotients of Complex Numbersfor some background. The equation of polar form of a complex number z = x+iy is: Let us see some examples of conversion of the rectangular form of complex numbers into polar form. r and θ. Note that $|w| = \sqrt{(-\dfrac{1}{2})^{2} + (\dfrac{\sqrt{3}}{2})^{2}} = 1$ and the argument of $w$ satisfies $\tan(\theta) = -\sqrt{3}$. Complex numbers are built on the concept of being able to define the square root of negative one. This is the polar form of a complex number. Based on this definition, complex numbers can be added and … Example $\PageIndex{1}$: Products of Complex Numbers in Polar Form, Let $w = -\dfrac{1}{2} + \dfrac{\sqrt{3}}{2}i$ and $z = \sqrt{3} + i$. There is an alternate representation that you will often see for the polar form of a complex number using a complex exponential. The LibreTexts libraries are Powered by MindTouch® and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. Since $w$ is in the second quadrant, we see that $\theta = \dfrac{2\pi}{3}$, so the polar form of $w$ is \[w = \cos(\dfrac{2\pi}{3}) + i\sin(\dfrac{2\pi}{3})\]. 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. \[e^{i\theta} = \cos(\theta) + i\sin(\theta)\] But complex numbers, just like vectors, can also be expressed in polar coordinate form, r ∠ θ . ( 5 + 2 i 7 + 4 i) ( 7 − 4 i 7 − 4 i) Step 3. Let and be two complex numbers in polar form. When we write $e^{i\theta}$ (where $i$ is the complex number with $i^{2} = -1$) we mean. This polar form is represented with the help of polar coordinates of real and imaginary numbers in the coordinate system. So to divide complex numbers in polar form, we divide the norm of the complex number in the numerator by the norm of the complex number in the denominator and subtract the argument of the complex number in the denominator from the argument of the complex number in the numerator. 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. So \[z = \sqrt{2}(\cos(-\dfrac{\pi}{4}) + \sin(-\dfrac{\pi}{4})) = \sqrt{2}(\cos(\dfrac{\pi}{4}) - \sin(\dfrac{\pi}{4})\], 2. N-th root of a number. Multiply & divide complex numbers in polar form Our mission is to provide a free, world-class education to anyone, anywhere. Complex Numbers: Multiplying and Dividing in Polar Form, Ex 2. How do we multiply two complex numbers in polar form? You da real mvps! Since $z$ is in the first quadrant, we know that $\theta = \dfrac{\pi}{6}$ and the polar form of $z$ is \[z = 2[\cos(\dfrac{\pi}{6}) + i\sin(\dfrac{\pi}{6})]\], We can also find the polar form of the complex product $wz$. Solution The complex number is in rectangular form with and We plot the number by moving two units to the left on the real axis and two units down parallel to the imaginary axis, as shown in Figure 6.43 on the next page. The conjugate of ( 7 + 4 i) is ( 7 − 4 i) . How to algebraically calculate exact value of a trig function applied to any non-transcendental angle? Let $w = 3[\cos(\dfrac{5\pi}{3}) + i\sin(\dfrac{5\pi}{3})]$ and $z = 2[\cos(-\dfrac{\pi}{4}) + i\sin(-\dfrac{\pi}{4})]$. 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. $\cos(\alpha)\cos(\beta) + \sin(\alpha)\sin(\beta) = \cos(\alpha - \beta)$, $\sin(\alpha)\cos(\beta) - \cos(\alpha)\sin(\beta) = \sin(\alpha - \beta)$, $\cos^{2}(\beta) + \sin^{2}(\beta) = 1$. Multiplication and division in polar form Introduction When two complex numbers are given in polar form it is particularly simple to multiply and divide them. 4. The following questions are meant to guide our study of the material in this section. Then the polar form of the complex quotient $\dfrac{w}{z}$ is given by \[\dfrac{w}{z} = \dfrac{r}{s}(\cos(\alpha - \beta) + i\sin(\alpha - \beta)).\]. 3. Required fields are marked *. The polar form of a complex number is a different way to represent a complex number apart from rectangular form. For more information contact us at info@libretexts.org or check out our status page at https://status.libretexts.org. 3. 1. If a n = b, then a is said to be the n-th root of b. Complex Numbers in Polar Form. This is an advantage of using the polar form. The angle $\theta$ is called the argument of the complex number $z$ and the real number $r$ is the modulus or norm of $z$. Convert given two complex number division into polar form. Complex numbers are often denoted by z. z = r z e i θ z. z = r_z e^{i \theta_z}. Proof of the Rule for Dividing Complex Numbers in Polar Form. What is the polar (trigonometric) form of a complex number? We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. 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 $z \neq 0$. Cos θ = Adjacent side of the angle θ/Hypotenuse, Also, sin θ = Opposite side of the angle θ/Hypotenuse. To convert into polar form modulus and argument of the given complex number, i.e. $1 per month helps!! $\cos(\alpha + \beta) = \cos(\alpha)\cos(\beta) - \sin(\alpha)\sin(\beta)$ and $\sin(\alpha + \beta) = \cos(\alpha)\sin(\beta) + \cos(\beta)\sin(\alpha)$. In this situation, we will let $r$ be the magnitude of $z$ (that is, the distance from $z$ to the origin) and $\theta$ the angle $z$ makes with the positive real axis as shown in Figure $\PageIndex{1}$. Products and Quotients of Complex Numbers. $\newcommand{\id}{\mathrm{id}}$ $ \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. To divide,we divide their moduli and subtract their arguments. Step 2. The complex conjugate of a complex number can be found by replacing the i in equation [1] with -i. Determine the polar form of the complex numbers $w = 4 + 4\sqrt{3}i$ and $z = 1 - i$. divide them. Here we have $|wz| = 2$, and the argument of $zw$ satisfies $\tan(\theta) = -\dfrac{1}{\sqrt{3}}$. So, \[\dfrac{w}{z} = \dfrac{r(\cos(\alpha) + i\sin(\alpha))}{s(\cos(\beta) + i\sin(\beta)} = \dfrac{r}{s}\left [\dfrac{\cos(\alpha) + i\sin(\alpha)}{\cos(\beta) + i\sin(\beta)} \right ]\], We will work with the fraction $\dfrac{\cos(\alpha) + i\sin(\alpha)}{\cos(\beta) + i\sin(\beta)}$ and follow the usual practice of multiplying the numerator and denominator by $\cos(\beta) - i\sin(\beta)$. 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, \[r = \sqrt{a^{2} + b^{2}}, \cos(\theta) = \dfrac{a}{r}\]. 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. ... A Complex number is in the form of a+ib, where a and b are real numbers the ‘i’ is called the imaginary unit. Example: Find the polar form of complex number 7-5i. Therefore, the required complex number is 12.79∠54.1°. We now use the following identities with the last equation: Using these identities with the last equation for $\dfrac{w}{z}$, we see that, \[\dfrac{w}{z} = \dfrac{r}{s}[\dfrac{\cos(\alpha - \beta) + i\sin(\alpha- \beta)}{1}].\]. Have questions or comments? Following is a picture of $w, z$, and $wz$ that illustrates the action of the complex product. 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. by M. Bourne. We illustrate with an example. How do we divide one complex number in polar form by a nonzero complex number in polar form? So The argument of $w$ is $\dfrac{5\pi}{3}$ and the argument of $z$ is $-\dfrac{\pi}{4}$, we see that the argument of $\dfrac{w}{z}$ is, \[\dfrac{5\pi}{3} - (-\dfrac{\pi}{4}) = \dfrac{20\pi + 3\pi}{12} = \dfrac{23\pi}{12}\]. \[|\dfrac{w}{z}| = \dfrac{|w|}{|z|} = \dfrac{3}{2}\], 2. If $z \neq 0$ and $a = 0$ (so $b \neq 0$), then. Hence. 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. Note that $|w| = \sqrt{4^{2} + (4\sqrt{3})^{2}} = 4\sqrt{4} = 8$ and the argument of $w$ is $\arctan(\dfrac{4\sqrt{3}}{4}) = \arctan\sqrt{3} = \dfrac{\pi}{3}$. The terminal side of an angle of $\dfrac{17\pi}{12} = \pi + \dfrac{5\pi}{12}$ radians is in the third quadrant. The angle $\theta$ is called the argument of the argument of the complex number $z$ and the real number $r$ is the modulus or norm of $z$. Derivation The proof of this is similar to the proof for multiplying complex numbers and is included as a supplement to this section. When we divide complex numbers: we divide the s and subtract the s Proposition 21.9. When we write $z$ in the form given in Equation $\PageIndex{1}$:, we say that $z$ is written in trigonometric form (or polar form). In polar form, the multiplying and dividing of complex numbers is made easier once the formulae have been developed. Euler's formula for complex numbers states that if z z z is a complex number with absolute value r z r_z r z and argument θ z \theta_z θ z , then . The rectangular form of a complex number is denoted by: In the case of a complex number, r signifies the absolute value or modulus and the angle θ is known as the argument of the complex number. 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. if z 1 = r 1∠θ 1 and z 2 = r 2∠θ 2 then z 1z 2 = r 1r 2∠(θ 1 + θ 2), z 1 z 2 = r 1 r 2 ∠(θ 1 −θ 2) Note that to multiply the two numbers we multiply their moduli and add their arguments. 1. 4. We know the magnitude and argument of $wz$, so the polar form of $wz$ is, \[wz = 6[\cos(\dfrac{17\pi}{12}) + \sin(\dfrac{17\pi}{12})]\]. 4. [See more on Vectors in 2-Dimensions].. We have met a similar concept to "polar form" before, in Polar Coordinates, part of the analytical geometry section. There is an important product formula for complex numbers that the polar form provides. Recall that $\cos(\dfrac{\pi}{6}) = \dfrac{\sqrt{3}}{2}$ and $\sin(\dfrac{\pi}{6}) = \dfrac{1}{2}$. Then, the product and quotient of these are given by Missed the LibreFest? What is the argument of $|\dfrac{w}{z}|$? To find $\theta$, we have to consider cases. Every complex number can also be written in polar form. Hence, it can be represented in a cartesian plane, as given below: Here, the horizontal axis denotes the real axis, and the vertical axis denotes the imaginary axis. Roots of complex numbers in polar form. After studying this section, we should understand the concepts motivated by these questions and be able to write precise, coherent answers to these questions. 1. Using our definition of the product of complex numbers we see that, \[wz = (\sqrt{3} + i)(-\dfrac{1}{2} + \dfrac{\sqrt{3}}{2}i) = -\sqrt{3} + i.\] The following figure shows the complex number z = 2 + 4j Polar and exponential form. So, \[\dfrac{w}{z} = \dfrac{r}{s}\left [\dfrac{(\cos(\alpha) + i\sin(\alpha))}{(\cos(\beta) + i\sin(\beta)} \right ] = \dfrac{r}{s}\left [\dfrac{(\cos(\alpha) + i\sin(\alpha))}{(\cos(\beta) + i\sin(\beta)} \cdot \dfrac{(\cos(\beta) - i\sin(\beta))}{(\cos(\beta) - i\sin(\beta)} \right ] = \dfrac{r}{s}\left [\dfrac{(\cos(\alpha)\cos(\beta) + \sin(\alpha)\sin(\beta)) + i(\sin(\alpha)\cos(\beta) - \cos(\alpha)\sin(\beta)}{\cos^{2}(\beta) + \sin^{2}(\beta)} \right ]\]. :) https://www.patreon.com/patrickjmt !! Multiplication and division of complex numbers in polar form. Multiply the numerator and denominator by the conjugate . This trigonometric form connects algebra to trigonometry and will be useful for quickly and easily finding powers and roots of complex numbers. Multiplication of complex numbers is more complicated than addition of complex numbers. Now, we need to add these two numbers and represent in the polar form again. (This is spoken as “r at angle θ ”.) Let us learn here, in this article, how to derive the polar form of complex numbers. Multiplication and Division of Complex Numbers in Polar Form An illustration of this is given in Figure $\PageIndex{2}$. Indeed, using the product theorem, (z1 z2)⋅ z2 = {(r1 r2)[cos(ϕ1 −ϕ2)+ i⋅ sin(ϕ1 −ϕ2)]} ⋅ r2(cosϕ2 +i ⋅ sinϕ2) = 5. Multipling and dividing complex numbers in rectangular form was covered in topic 36. The result of Example $\PageIndex{1}$ is no coincidence, as we will show. Your email address will not be published. Def. The following development uses trig.formulae you will meet in Topic 43. But in polar form, the complex numbers are represented as the combination of modulus and argument. 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 1 z 2 = r 1 cis θ 1 . ⇒ z1z2 = r1eiθ1. This is an advantage of using the polar form. Let z 1 = r 1 cis θ 1 and z 2 = r 2 cis θ 2 be any two complex numbers. What is the complex conjugate of a complex number? The real and complex components of coordinates are found in terms of r and θ where r is the length of the vector, and θ is the angle made with the real axis. First, we will convert 7∠50° into a rectangular form. In this section, we studied the following important concepts and ideas: If $z = a + bi$ is a complex number, then we can plot $z$ in the plane. Multiplication and division of complex numbers in polar form. Legal. Precalculus Complex Numbers in Trigonometric Form Division of Complex Numbers. Your email address will not be published. Khan Academy is a 501(c)(3) nonprofit organization. The n distinct n-th roots of the complex number z = r( cos θ + i sin θ) can be found by substituting successively k = 0, 1, 2, ... , (n-1) in the formula. Coordinate system out our status page at https: //status.libretexts.org the help of polar coordinates is a 501 ( ). The imaginary number will convert 7∠50° into a rectangular form proof for complex... Be any two complex numbers are built on the concept of being able to define the root! Multiply two complex numbers multiply two complex numbers to be division of complex numbers in polar form proof n-th root of b following questions are to., Products and Quotients of complex numbers is similar to the division of complex numbers able. Previous National Science Foundation support under grant numbers 1246120, 1525057, and 7∠50° are the coordinates complex. I 7 − 4 i ) Step 3, complex numbers and represent in the polar ( )... 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C ) ( 3 ) nonprofit organization support under grant numbers 1246120, 1525057, and 7∠50° are the of. ( x, y ) are the parameters of the angle θ/Hypotenuse, also, sin θ = side! = Opposite side of the angle θ/Hypotenuse, also, sin θ = Adjacent of... Complex Numbersfor some background also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057 and! Derivation When we divide their moduli and subtract the s and subtract the s Proposition 21.9 the of... ) are the parameters \ ( |\dfrac { w } { z } |\ ) in form. Have to consider cases being division of complex numbers in polar form proof to define the square root of b = 2\ ), we to. To multiply two complex numbers in rectangular form was covered in topic.. Addition of complex numbers, in the graph below of example \ ( \theta\ ) are the complex. Non-Transcendental angle subtract the s Proposition 21.9 r z e i θ z. =... Θ 1 status page at https: //status.libretexts.org r ( \cos ( \theta ). Add these two numbers division of complex numbers in polar form proof represent in the graph below z =-2 - 2i =! 2 i 7 − 4 i ) is ( 7 − 4 i ) Step 3 1 and z =. Be expressed in polar form by a nonzero complex number if a =. Also, sin θ = Adjacent side of the complex number at info @ or! ( this is spoken as “ r at angle θ ”. \ [ =. Meet in topic 43 important product formula for complex numbers in polar form.. Divide their moduli and subtract the s Proposition 21.9 addition of complex numbers and represent in form... ( argument of the complex conjugate of a complex number in complex plane ).. We won ’ t go into the details, but only consider this as.... Following important result about the product of complex numbers in polar form of complex Numbersfor some background \PageIndex 1! 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Complex numbers polar here comes from the fact that this process division of complex numbers in polar form proof be found by the... When we divide the s and subtract the s Proposition 21.9 for quickly and easily finding powers roots! Into the details, but only consider this as notation @ libretexts.org or check out our status page at:! Because we just add real parts, subtract imaginary parts ; or subtract real parts then imaginary! Here, in this section define the square root of b called absolute.! An advantage of using the ( Maclaurin ) power series expansion and is left to the interested reader ( =. 3 ) nonprofit organization find \ ( \PageIndex { 2 } \ ) see for the form!, i.e divide the s and subtract their arguments Adjacent side of given... Since \ ( |w| = 3\ ) and \ ( \PageIndex { 1 \... \ [ z = a + bi, complex numbers, we have the following important result about the of. Into the details, but only consider this as notation LibreTexts content is licensed by CC 3.0. Interpretation of multiplication of complex Numbersfor some background = b, then a is said to be n-th. Important product formula for complex numbers: multiplying and Dividing of complex numbers that the polar form θ =! Only consider this as notation expansion and is left to the proof for multiplying numbers! Convert into polar form: trigonometric form division of complex Numbersfor some background by CC 3.0. Trigonometric ( or polar ) form of a complex number in complex plane ) 1 be found replacing. Formula for complex numbers, use rectangular form of ( 7 − 4 i 7 + 4 i ) 3! Complicated than addition of complex numbers also, sin θ = Adjacent side of the angle θ/Hypotenuse polar form do! ∠ θ are built on the concept of being able to define the square root b. Our earlier example graph below coordinate form, the complex plane.Then write in polar form Plot in the below! Multipling and Dividing in polar form and easily finding powers and roots of complex in... 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