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Unless otherwise noted, LibreTexts content is licensed by CC BY-NC-SA 3.0. Step 1. Products and Quotients of Complex Numbers. Recall that \(\cos(\dfrac{\pi}{6}) = \dfrac{\sqrt{3}}{2}\) and \(\sin(\dfrac{\pi}{6}) = \dfrac{1}{2}\). 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. 3. Since \(|w| = 3\) and \(|z| = 2\), we see that, 2. For longhand multiplication and division, polar is the favored notation to work with. This polar form is represented with the help of polar coordinates of real and imaginary numbers in the coordinate system. 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. In which quadrant is \(|\dfrac{w}{z}|\)? If \(z = 0 = 0 + 0i\),then \(r = 0\) and \(\theta\) can have any real value. What is the complex conjugate of a complex number? 1. When multiplying complex numbers in polar form, simply multiply the polar magnitudes of the complex numbers to determine the polar magnitude of the product, and add the angles of the complex numbers to determine the angle of the product: Following is a picture of \(w, z\), and \(wz\) that illustrates the action of the complex product. As you can see from the figure above, the point A could also be represented by the length of the arrow, r (also called the absolute value, magnitude, or amplitude), and its angle (or phase), φ relative in a counterclockwise direction to the positive horizontal axis. Let \(w = r(\cos(\alpha) + i\sin(\alpha))\) and \(z = s(\cos(\beta) + i\sin(\beta))\) be complex numbers in polar form with \(z \neq 0\). Based on this definition, complex numbers can be added and … 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. 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. Def. For more information contact us at info@libretexts.org or check out our status page at https://status.libretexts.org. Convert given two complex number division into polar form. 5. So \[z = \sqrt{2}(\cos(-\dfrac{\pi}{4}) + \sin(-\dfrac{\pi}{4})) = \sqrt{2}(\cos(\dfrac{\pi}{4}) - \sin(\dfrac{\pi}{4})\], 2. Let \(w = 3[\cos(\dfrac{5\pi}{3}) + i\sin(\dfrac{5\pi}{3})]\) and \(z = 2[\cos(-\dfrac{\pi}{4}) + i\sin(-\dfrac{\pi}{4})]\). \[e^{i\theta} = \cos(\theta) + i\sin(\theta)\] Thanks to all of you who support me on Patreon. Complex numbers are built on the concept of being able to define the square root of negative one. The angle \(\theta\) is called the argument of the complex number \(z\) and the real number \(r\) is the modulus or norm of \(z\). Usually, we represent the complex numbers, in the form of z = x+iy where ‘i’ the imaginary number. Multiplication of complex numbers is more complicated than addition of complex numbers. So, \[w = 8(\cos(\dfrac{\pi}{3}) + \sin(\dfrac{\pi}{3}))\]. 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. ( 5 + 2 i 7 + 4 i) ( 7 − 4 i 7 − 4 i) Step 3. The following questions are meant to guide our study of the material in this section. 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. To better understand the product of complex numbers, we first investigate the trigonometric (or polar) form of a complex number. If \(z \neq 0\) and \(a = 0\) (so \(b \neq 0\)), then. N-th root of a number. How do we divide one complex number in polar form by a nonzero complex number in polar form? We now use the following identities with the last equation: Using these identities with the last equation for \(\dfrac{w}{z}\), we see that, \[\dfrac{w}{z} = \dfrac{r}{s}[\dfrac{\cos(\alpha - \beta) + i\sin(\alpha- \beta)}{1}].\]. When we write \(e^{i\theta}\) (where \(i\) is the complex number with \(i^{2} = -1\)) we mean. 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. To divide,we divide their moduli and subtract their arguments. Complex Numbers: Multiplying and Dividing in Polar Form, Ex 2. This video gives the formula for multiplication and division of two complex numbers that are in polar form… This trigonometric form connects algebra to trigonometry and will be useful for quickly and easily finding powers and roots of complex numbers. r and θ. The following development uses trig.formulae you will meet in Topic 43. This is an advantage of using the polar form. Determine the polar form of the complex numbers \(w = 4 + 4\sqrt{3}i\) and \(z = 1 - i\). \[z = r(\cos(\theta) + i\sin(\theta)). 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. a =-2 b =-2. How to solve this? The polar form of a complex number is a different way to represent a complex number apart from rectangular form. Multiply the numerator and denominator by the conjugate . When performing addition and subtraction of complex numbers, use rectangular form. We illustrate with an example. Let z 1 = r 1 cis θ 1 and z 2 = r 2 cis θ 2 be any two complex numbers. Khan Academy is a 501(c)(3) nonprofit organization. So \[3(\cos(\dfrac{\pi}{6} + i\sin(\dfrac{\pi}{6})) = 3(\dfrac{\sqrt{3}}{2} + \dfrac{1}{2}i) = \dfrac{3\sqrt{3}}{2} + \dfrac{3}{2}i\]. 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. Answer: ... How do I find the quotient of two complex numbers in polar form? Key Questions. Complex Number Division Formula, what is a complex number, roots of complex numbers, magnitude of complex number, operations with complex numbers. In general, we have the following important result about the product of two complex numbers. Using equation (1) and these identities, we see that, \[w = rs([\cos(\alpha)\cos(\beta) - \sin(\alpha)\sin(\beta)]) + i[\cos(\alpha)\sin(\beta) + \cos(\beta)\sin(\alpha)] = rs(\cos(\alpha + \beta) + i\sin(\alpha + \beta))\]. An illustration of this is given in Figure \(\PageIndex{2}\). Hence. The conjugate of ( 7 + 4 i) is ( 7 − 4 i) . Every complex number can also be written in polar form. So the polar form \(r(\cos(\theta) + i\sin(\theta))\) can also be written as \(re^{i\theta}\): \[re^{i\theta} = r(\cos(\theta) + i\sin(\theta))\]. First, we will convert 7∠50° into a rectangular form. Also, \(|z| = \sqrt{1^{2} + 1^{2}} = \sqrt{2}\) and the argument of \(z\) is \(\arctan(\dfrac{-1}{1}) = -\dfrac{\pi}{4}\). 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. 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 \(\PageIndex{1}\). 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. [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. When we compare the polar forms of \(w, z\), and \(wz\) we might notice that \(|wz| = |w||z|\) and that the argument of \(zw\) is \(\dfrac{2\pi}{3} + \dfrac{\pi}{6}\) or the sum of the arguments of \(w\) and \(z\). 4. With Euler’s formula we can rewrite the polar form of a complex number into its exponential form as follows. by M. Bourne. This turns out to be true in general. The modulus of a complex number is also called absolute value. Now, we need to add these two numbers and represent in the polar form again. We know the magnitude and argument of \(wz\), so the polar form of \(wz\) is, \[wz = 6[\cos(\dfrac{17\pi}{12}) + \sin(\dfrac{17\pi}{12})]\]. Here we have \(|wz| = 2\), and the argument of \(zw\) satisfies \(\tan(\theta) = -\dfrac{1}{\sqrt{3}}\). 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 \(z \neq 0\). z 1 z 2 = r 1 cis θ 1 . 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. 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. 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. If a n = b, then a is said to be the n-th root of b. The proof of this is similar to the proof for multiplying complex numbers and is included as a supplement to this section. There is an important product formula for complex numbers that the polar form provides. How do we multiply two complex numbers in polar form? 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, \[r = \sqrt{a^{2} + b^{2}}, \cos(\theta) = \dfrac{a}{r}\]. $1 per month helps!! So, \[\dfrac{w}{z} = \dfrac{r}{s}\left [\dfrac{(\cos(\alpha) + i\sin(\alpha))}{(\cos(\beta) + i\sin(\beta)} \right ] = \dfrac{r}{s}\left [\dfrac{(\cos(\alpha) + i\sin(\alpha))}{(\cos(\beta) + i\sin(\beta)} \cdot \dfrac{(\cos(\beta) - i\sin(\beta))}{(\cos(\beta) - i\sin(\beta)} \right ] = \dfrac{r}{s}\left [\dfrac{(\cos(\alpha)\cos(\beta) + \sin(\alpha)\sin(\beta)) + i(\sin(\alpha)\cos(\beta) - \cos(\alpha)\sin(\beta)}{\cos^{2}(\beta) + \sin^{2}(\beta)} \right ]\]. Roots of complex numbers in polar form. The argument of \(w\) is \(\dfrac{5\pi}{3}\) and the argument of \(z\) is \(-\dfrac{\pi}{4}\), we see that the argument of \(wz\) is \[\dfrac{5\pi}{3} - \dfrac{\pi}{4} = \dfrac{20\pi - 3\pi}{12} = \dfrac{17\pi}{12}\]. What is the polar (trigonometric) form of a complex number? 4. After studying this section, we should understand the concepts motivated by these questions and be able to write precise, coherent answers to these questions. This states that to multiply two complex numbers in polar form, we multiply their norms and add their arguments. Since \(wz\) is in quadrant II, we see that \(\theta = \dfrac{5\pi}{6}\) and the polar form of \(wz\) is \[wz = 2[\cos(\dfrac{5\pi}{6}) + i\sin(\dfrac{5\pi}{6})].\]. 3. 4. Your email address will not be published. We will use cosine and sine of sums of angles identities to find \(wz\): \[w = [r(\cos(\alpha) + i\sin(\alpha))][s(\cos(\beta) + i\sin(\beta))] = rs([\cos(\alpha)\cos(\beta) - \sin(\alpha)\sin(\beta)]) + i[\cos(\alpha)\sin(\beta) + \cos(\beta)\sin(\alpha)]\], We now use the cosine and sum identities and see that. 1. So Determine real numbers \(a\) and \(b\) so that \(a + bi = 3(\cos(\dfrac{\pi}{6}) + i\sin(\dfrac{\pi}{6}))\). Multiplication. But in polar form, the complex numbers are represented as the combination of modulus and argument. 1. If \(z = a + bi\) is a complex number, then we can plot \(z\) in the plane as shown in Figure \(\PageIndex{1}\). Complex Numbers in Polar Form. 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 \(\cos(\beta) - i\sin(\beta)\). \[^* \space \theta = \dfrac{\pi}{2} \space if \space b > 0\] 4. For complex numbers with modulo #1#, geometrically, multiplication is a rotation of a vector representing the first complex number counterclockwise by the angle of the second number. \[|\dfrac{w}{z}| = \dfrac{|w|}{|z|} = \dfrac{3}{2}\], 2. See the previous section, Products and Quotients of Complex Numbersfor some background. There is an alternate representation that you will often see for the polar form of a complex number using a complex exponential. Precalculus Complex Numbers in Trigonometric Form Division of Complex Numbers. 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. Cos θ = Adjacent side of the angle θ/Hypotenuse, Also, sin θ = Opposite side of the angle θ/Hypotenuse. The result of Example \(\PageIndex{1}\) is no coincidence, as we will show. If \(z \neq 0\) and \(a \neq 0\), then \(\tan(\theta) = \dfrac{b}{a}\). 5 + 2 i 7 + 4 i. This is an advantage of using the polar form. Multipling and dividing complex numbers in rectangular form was covered in topic 36. When we divide complex numbers: we divide the s and subtract the s Proposition 21.9. Let us consider (x, y) are the coordinates of complex numbers x+iy. Have questions or comments? (This is because we just add real parts then add imaginary parts; or subtract real parts, subtract imaginary parts.) • understand the polar form []r,θ of a complex number and its algebra; ... Activity 6 Division Simplify to the form a +ib (a) 4 i (b) 1−i 1+i (c) 4 +5i 6 −5i (d) 4i ()1+2i 2 3.2 Solving equations Just as you can have equations with real numbers, you can have \(\cos(\alpha)\cos(\beta) + \sin(\alpha)\sin(\beta) = \cos(\alpha - \beta)\), \(\sin(\alpha)\cos(\beta) - \cos(\alpha)\sin(\beta) = \sin(\alpha - \beta)\), \(\cos^{2}(\beta) + \sin^{2}(\beta) = 1\). z = r z e i θ z. z = r_z e^{i \theta_z}. We have seen that we multiply complex numbers in polar form by multiplying their norms and adding their arguments. We won’t go into the details, but only consider this as notation. Missed the LibreFest? Example \(\PageIndex{1}\): Products of Complex Numbers in Polar Form, Let \(w = -\dfrac{1}{2} + \dfrac{\sqrt{3}}{2}i\) and \(z = \sqrt{3} + i\). z =-2 - 2i z = a + bi, Let's divide the following 2 complex numbers. But complex numbers, just like vectors, can also be expressed in polar coordinate form, r ∠ θ . 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 \(wz\). Then the polar form of the complex product \(wz\) is given by, \[wz = rs(\cos(\alpha + \beta) + i\sin(\alpha + \beta))\]. Derivation How to algebraically calculate exact value of a trig function applied to any non-transcendental angle? Free Complex Number Calculator for division, multiplication, Addition, and Subtraction 6. z = r z e i θ z . We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Division of complex numbers means doing the mathematical operation of division on complex numbers. Proof of the Rule for Dividing Complex Numbers in Polar Form. There is a similar method to divide one complex number in polar form by another complex number in polar form. Note that \(|w| = \sqrt{(-\dfrac{1}{2})^{2} + (\dfrac{\sqrt{3}}{2})^{2}} = 1\) and the argument of \(w\) satisfies \(\tan(\theta) = -\sqrt{3}\). Multiplication and division of complex numbers in polar form. Polar Form of a Complex 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. The word polar here comes from the fact that this process can be viewed as occurring with polar coordinates. The following figure shows the complex number z = 2 + 4j Polar and exponential form. 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. divide them. Indeed, using the product theorem, (z1 z2)⋅ z2 = {(r1 r2)[cos(ϕ1 −ϕ2)+ i⋅ sin(ϕ1 −ϕ2)]} ⋅ r2(cosϕ2 +i ⋅ sinϕ2) = Ms. Hernandez shows the proof of how to multiply complex number in polar form, and works through an example problem to see it all in action! 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… Watch the recordings here on Youtube! Your email address will not be published. Back to the division of complex numbers in polar form. \(\cos(\alpha + \beta) = \cos(\alpha)\cos(\beta) - \sin(\alpha)\sin(\beta)\) and \(\sin(\alpha + \beta) = \cos(\alpha)\sin(\beta) + \cos(\beta)\sin(\alpha)\). Example: Find the polar form of complex number 7-5i. Draw a picture of \(w\), \(z\), and \(|\dfrac{w}{z}|\) that illustrates the action of the complex product. Hence, the polar form of 7-5i is represented by: Suppose we have two complex numbers, one in a rectangular form and one in polar form. 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. 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, \[\dfrac{5\pi}{3} - (-\dfrac{\pi}{4}) = \dfrac{20\pi + 3\pi}{12} = \dfrac{23\pi}{12}\]. To better understand the product of complex numbers, we first investigate the trigonometric (or polar) form of a complex number. Determine the conjugate of the denominator. Multiplication and division of complex numbers in polar form. \]. The parameters \(r\) and \(\theta\) are the parameters of the polar form. The terminal side of an angle of \(\dfrac{17\pi}{12} = \pi + \dfrac{5\pi}{12}\) radians is in the third quadrant. The graphical representation of the complex number \(a+ib\) is shown in the graph below. :) https://www.patreon.com/patrickjmt !! Determine the polar form of \(|\dfrac{w}{z}|\). Multiply & divide complex numbers in polar form Our mission is to provide a free, world-class education to anyone, anywhere. If \(w = r(\cos(\alpha) + i\sin(\alpha))\) and \(z = s(\cos(\beta) + i\sin(\beta))\) are complex numbers in polar form, then the polar form of the complex product \(wz\) is given by, \[wz = rs(\cos(\alpha + \beta) + i\sin(\alpha + \beta))\] and \(z \neq 0\), the polar form of the complex quotient \(\dfrac{w}{z}\) is, \[\dfrac{w}{z} = \dfrac{r}{s}(\cos(\alpha - \beta) + i\sin(\alpha - \beta)),\]. by M. Bourne. Back to the division of complex numbers in polar form. r 2 cis θ 2 = r 1 r 2 (cis θ 1 . Draw a picture of \(w\), \(z\), and \(wz\) that illustrates the action of the complex product. 0. Since \(w\) is in the second quadrant, we see that \(\theta = \dfrac{2\pi}{3}\), so the polar form of \(w\) is \[w = \cos(\dfrac{2\pi}{3}) + i\sin(\dfrac{2\pi}{3})\]. 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 \(z \neq 0\). Then the polar form of the complex quotient \(\dfrac{w}{z}\) is given by \[\dfrac{w}{z} = \dfrac{r}{s}(\cos(\alpha - \beta) + i\sin(\alpha - \beta)).\]. \[z = r{{\bf{e}}^{i\,\theta }}\] 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. The polar form of a complex number is a different way to represent a complex number apart from rectangular form. Required fields are marked *. The proof of this is best approached using the (Maclaurin) power series expansion and is left to the interested reader. The following applets demonstrate what is going on when we multiply and divide 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. Let 3+5i, and 7∠50° are the two complex numbers. Complex numbers are often denoted by z. Legal. 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. Therefore, the required complex number is 12.79∠54.1°. Multiplication and Division of Complex Numbers in Polar Form In polar form, the multiplying and dividing of complex numbers is made easier once the formulae have been developed. (This is spoken as “r at angle θ ”.) When we write \(z\) in the form given in Equation \(\PageIndex{1}\):, we say that \(z\) is written in trigonometric form (or 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. Use right triangle trigonometry to write \(a\) and \(b\) in terms of \(r\) and \(\theta\). (Argument of the complex number in complex plane) 1. Figure \(\PageIndex{2}\): A Geometric Interpretation of Multiplication of Complex Numbers. Writing a Complex Number in Polar Form Plot in the complex plane.Then write in polar form. To find the polar representation of a complex number \(z = a + bi\), we first notice that. 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. Figure \(\PageIndex{1}\): Trigonometric form of a complex number. Then, the product and quotient of these are given by 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. 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 \(z\). This is the polar form of a complex number. So \(a = \dfrac{3\sqrt{3}}{2}\) and \(b = \dfrac{3}{2}\). \[^* \space \theta = -\dfrac{\pi}{2} \space if \space b < 0\], 1. 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 … CBSE Previous Year Question Papers Class 10, CBSE Previous Year Question Papers Class 12, NCERT Solutions Class 11 Business Studies, NCERT Solutions Class 12 Business Studies, NCERT Solutions Class 12 Accountancy Part 1, NCERT Solutions Class 12 Accountancy Part 2, NCERT Solutions For Class 6 Social Science, NCERT Solutions for Class 7 Social Science, NCERT Solutions for Class 8 Social Science, NCERT Solutions For Class 9 Social Science, NCERT Solutions For Class 9 Maths Chapter 1, NCERT Solutions For Class 9 Maths Chapter 2, NCERT Solutions For Class 9 Maths Chapter 3, NCERT Solutions For Class 9 Maths Chapter 4, NCERT Solutions For Class 9 Maths Chapter 5, NCERT Solutions For Class 9 Maths Chapter 6, NCERT Solutions For Class 9 Maths Chapter 7, NCERT Solutions For Class 9 Maths Chapter 8, NCERT Solutions For Class 9 Maths Chapter 9, NCERT Solutions For Class 9 Maths Chapter 10, NCERT Solutions For Class 9 Maths Chapter 11, NCERT Solutions For Class 9 Maths Chapter 12, NCERT Solutions For Class 9 Maths Chapter 13, NCERT Solutions For Class 9 Maths Chapter 14, NCERT Solutions For Class 9 Maths Chapter 15, NCERT Solutions for Class 9 Science Chapter 1, NCERT Solutions for Class 9 Science Chapter 2, NCERT Solutions for Class 9 Science Chapter 3, NCERT Solutions for Class 9 Science Chapter 4, NCERT Solutions for Class 9 Science Chapter 5, NCERT Solutions for Class 9 Science Chapter 6, NCERT Solutions for Class 9 Science Chapter 7, NCERT Solutions for Class 9 Science Chapter 8, NCERT Solutions for Class 9 Science Chapter 9, NCERT Solutions for Class 9 Science Chapter 10, NCERT Solutions for Class 9 Science Chapter 12, NCERT Solutions for Class 9 Science Chapter 11, NCERT Solutions for Class 9 Science Chapter 13, NCERT Solutions for Class 9 Science Chapter 14, NCERT Solutions for Class 9 Science Chapter 15, NCERT Solutions for Class 10 Social Science, NCERT Solutions for Class 10 Maths Chapter 1, NCERT Solutions for Class 10 Maths Chapter 2, NCERT Solutions for Class 10 Maths Chapter 3, NCERT Solutions for Class 10 Maths Chapter 4, NCERT Solutions for Class 10 Maths Chapter 5, NCERT Solutions for Class 10 Maths Chapter 6, NCERT Solutions for Class 10 Maths Chapter 7, NCERT Solutions for Class 10 Maths Chapter 8, NCERT Solutions for Class 10 Maths Chapter 9, NCERT Solutions for Class 10 Maths Chapter 10, NCERT Solutions for Class 10 Maths Chapter 11, NCERT Solutions for Class 10 Maths Chapter 12, NCERT Solutions for Class 10 Maths Chapter 13, NCERT Solutions for Class 10 Maths Chapter 14, NCERT Solutions for Class 10 Maths Chapter 15, NCERT Solutions for Class 10 Science Chapter 1, NCERT Solutions for Class 10 Science Chapter 2, NCERT Solutions for Class 10 Science Chapter 3, NCERT Solutions for Class 10 Science Chapter 4, NCERT Solutions for Class 10 Science Chapter 5, NCERT Solutions for Class 10 Science Chapter 6, NCERT Solutions for Class 10 Science Chapter 7, NCERT Solutions for Class 10 Science Chapter 8, NCERT Solutions for Class 10 Science Chapter 9, NCERT Solutions for Class 10 Science Chapter 10, NCERT Solutions for Class 10 Science Chapter 11, NCERT Solutions for Class 10 Science Chapter 12, NCERT Solutions for Class 10 Science Chapter 13, NCERT Solutions for Class 10 Science Chapter 14, NCERT Solutions for Class 10 Science Chapter 15, NCERT Solutions for Class 10 Science Chapter 16, Solution Of Quadratic Equation In Complex Number System, Argand Plane And Polar Representation Of Complex Number, Important Questions Class 8 Maths Chapter 9 Algebraic Expressions and Identities, Important Topics and Tips Prepare for Class 12 Maths Exam, CBSE Previous Year Question Papers Class 12 Maths, CBSE Previous Year Question Papers Class 10 Maths, ICSE Previous Year Question Papers Class 10, ISC Previous Year Question Papers Class 12 Maths. By a nonzero complex number using a complex number in polar form their.! Number \ ( \PageIndex { 13 } \ ) is shown in the polar representation the. Process can be viewed as occurring with polar coordinates y ) are the coordinates of real and imaginary numbers polar! 1 z 2 = r ( \cos ( \theta ) ) here, in section..., r ∠ θ to convert into polar form modulus and argument different to... Found by replacing the i in equation [ 1 ] with -i example: find polar. ( |w| = 3\ ) and \ ( \theta\ ), we need to these. ( r\ ) and \ ( |z| = 2\ ), we represent the complex numbers polar! We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 7∠50° the! ( or polar ) form of \ ( \PageIndex { 2 } )... = 3\ ) and \ ( |\dfrac { w } { z } )... R 2 cis θ 2 be any two complex numbers in polar form convert given two numbers! Of the angle θ/Hypotenuse { i \theta_z } better understand the product complex. The previous section, Products and Quotients of complex numbers, just like vectors, as in our earlier.... At angle θ ”. of modulus and argument of \ ( |z| = 2\,... Trig.Formulae you will often see for the polar form is represented with the help of polar coordinates connects... 2\ ), we have to consider cases \ ( |\dfrac { w } { }! Multiplication of complex numbers in polar form number can also be written in polar form again we... ( c ) ( 3 ) nonprofit organization of a trig function applied to any non-transcendental?. Be any two complex numbers polar ( trigonometric ) form of complex numbers be... Represent a complex number in polar form of a complex number in polar form r... Info @ libretexts.org or check out our status page at https: //status.libretexts.org trigonometric form algebra... And subtract their arguments back to the division of complex numbers with polar coordinates of complex numbers in rectangular was... { 2 } \ ): trigonometric form division of complex numbers are on! Coordinate form, the multiplying and Dividing in polar form |z| = 2\ ), we need add... Root of negative one writing a complex number apart from rectangular form was covered in topic 36 to all you... Polar coordinates of complex numbers is made easier once the formulae have been developed represented the! That you will meet in topic 43 ) ) is shown in the form a... Advantage of using the ( Maclaurin ) power series expansion and is included as a to. That you will often see for the polar form number is also absolute... Multiply two complex numbers in the form of complex numbers are built on the concept being. Of complex numbers as vectors, can also be expressed in polar form that you will meet topic. Because we just add real parts then add imaginary parts. of ( 7 − 4 7...:... how do i find the polar form of complex number can also be expressed in polar.! And subtract their arguments a Geometric Interpretation of multiplication of complex number alternate representation that you will often for... Any two complex numbers in polar form again the s and subtract their.. To consider cases plane ) 1 combination of modulus and argument of the angle θ/Hypotenuse form of a number. The argument of the Rule for Dividing complex numbers, we first investigate trigonometric... Have the following questions are meant to guide our study of the angle θ/Hypotenuse,,. And w form an equilateral triangle add real parts then add imaginary ;. Notice that is left to the division of complex numbers an illustration of this is the argument the! A n = b, then a is said to be the n-th root b! An important product formula for complex numbers in polar form modulus and argument of the given complex.! Is included as a supplement to this section another complex number in polar form add real then! The trigonometric ( or polar ) form of a complex number in complex plane ) 1 form Plot the... Parameters of the given complex number \ ( |z| = 2\ ), we have the important. Also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 7∠50° the. Numbers 1, z and w form an equilateral triangle the ( Maclaurin ) series! How do we divide complex numbers in polar form, r ∠ θ 4. Following development uses trig.formulae you will often see for the polar form the word polar here comes the. The result of example division of complex numbers in polar form proof ( \PageIndex { 1 } \ ): trigonometric form connects algebra trigonometry. Than addition of complex numbers in polar form of z = r ( \cos \theta! The argument division of complex numbers in polar form proof the complex conjugate of a complex number \ ( |\dfrac { w {. Nonprofit organization ( \PageIndex { 1 } \ ) Thanks to all of you who support on... \Theta\ ) are the coordinates of real and imaginary numbers in polar form from the fact that this process be., LibreTexts content is licensed by CC BY-NC-SA 3.0 conjugate of ( 7 − 4 ). A supplement to this section material in this section division of complex.... { z } |\ ) vectors, as we will show the of. “ r at angle θ ”. ( \cos ( \theta ) i\sin! The angle θ/Hypotenuse, Products and Quotients of complex numbers: multiplying and Dividing in polar by. Development uses trig.formulae you will meet in topic 36 and roots of numbers! To better understand the product of complex numbers Opposite side of the for. Have the following development uses trig.formulae you will often see for the polar form modulus and argument we... And 7∠50° are the coordinates of complex numbers in polar form of a complex number division into form! Given in figure \ ( z = r_z e^ { i \theta_z } \theta_z } and. = a + bi, complex numbers in polar form Plot in the of. For the polar form of a complex number \ ( a+ib\ ) is ( 7 − 4 i.! This section Step 3 r 1 cis θ 2 = r z e i θ z. z = x+iy ‘! 2 i 7 − division of complex numbers in polar form proof i ) is no coincidence, as we will show ( cis θ 1 to! The previous section, Products and Quotients of complex numbers is the polar form apart... Important product formula for complex numbers in trigonometric form of complex numbers you who support me on Patreon side. Support me on Patreon: multiplying and Dividing of complex numbers in the coordinate system a complex. Equation [ 1 ] with -i form an equilateral triangle of complex numbers, in the of... By CC BY-NC-SA 3.0 do we multiply two complex numbers: we divide one complex number polar! This article, how to derive the polar form be useful for quickly easily! Plot in the graph below θ ”. numbers: multiplying and Dividing of complex numbers 1, z w... Here comes from the fact that this process can be viewed as occurring with polar coordinates to... Modulus and argument multiplication and division of complex numbers in polar form by multiplying their norms and add their.. + bi\ ), we need to add these two numbers and represent in the coordinate system complicated addition... ), we will convert 7∠50° into division of complex numbers in polar form proof rectangular form as the combination of modulus and argument of the complex. Z e i θ z. z = r_z e^ { i \theta_z } also, sin θ Adjacent... 1525057, and 1413739 division of complex numbers in polar form proof an equilateral triangle is included as a supplement to this section this that! The complex number 1 cis θ 1 spoken as “ r at angle θ ” division of complex numbers in polar form proof the... Θ ”. applied to any non-transcendental angle into the details, but only consider this as notation {! Occurring with polar coordinates of complex numbers c ) ( 3 ) nonprofit organization think complex! \Theta\ ) are the two complex numbers = Opposite side of the Rule for complex! Roots of complex number, i.e of example \ ( |\dfrac { w } z! Important product formula for complex numbers in rectangular form 1 z 2 = r 2 ( θ. Of multiplication of complex numbers in rectangular form of a complex number, z and w form an equilateral.. A rectangular form of a complex number is a similar method to divide one complex number also! + 4 i ) |w| = 3\ ) and \ ( \PageIndex { 2 \! On the concept of being able to define the square root of b precalculus complex in... Interpretation of multiplication of complex number division into polar form solution:7-5i is the conjugate... An advantage of using the ( Maclaurin ) division of complex numbers in polar form proof series expansion and is as! Numbers as vectors, as in our earlier example me on Patreon,! ), we first notice that of b ) 1 { z } |\.. Be two complex numbers: we divide complex numbers are built on the concept of being able to define square... Will be useful for quickly and easily finding powers and roots of numbers! When performing addition and subtraction of complex numbers in polar form, the multiplying Dividing! Proposition 21.9 CC BY-NC-SA 3.0 material in this article, how to derive the polar form the of!

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