Using equation (1) and these identities, we see that, \[w = rs([\cos(\alpha)\cos(\beta) - \sin(\alpha)\sin(\beta)]) + i[\cos(\alpha)\sin(\beta) + \cos(\beta)\sin(\alpha)] = rs(\cos(\alpha + \beta) + i\sin(\alpha + \beta))\]. Complex numbers are often denoted by z. 5 + 2 i 7 + 4 i. 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 \(\PageIndex{1}\). Using our definition of the product of complex numbers we see that, \[wz = (\sqrt{3} + i)(-\dfrac{1}{2} + \dfrac{\sqrt{3}}{2}i) = -\sqrt{3} + i.\] Since \(wz\) is in quadrant II, we see that \(\theta = \dfrac{5\pi}{6}\) and the polar form of \(wz\) is \[wz = 2[\cos(\dfrac{5\pi}{6}) + i\sin(\dfrac{5\pi}{6})].\]. Your email address will not be published. Hence. Have questions or comments? This turns out to be true in general. 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. 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. Proof of the Rule for Dividing Complex Numbers in Polar Form. But in polar form, the complex numbers are represented as the combination of modulus and argument. Multiplication. Convert given two complex number division into polar form. Determine the conjugate of the denominator. 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. There is a similar method to divide one complex number in polar form by another complex number in polar form. Let us learn here, in this article, how to derive the polar form of complex numbers. 3. Products and Quotients of Complex Numbers. So \(a = \dfrac{3\sqrt{3}}{2}\) and \(b = \dfrac{3}{2}\). Key Questions. Free Complex Number Calculator for division, multiplication, Addition, and Subtraction The following applets demonstrate what is going on when we multiply and divide complex numbers. 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. If \(z \neq 0\) and \(a = 0\) (so \(b \neq 0\)), then. Then, the product and quotient of these are given by We have seen that we multiply complex numbers in polar form by multiplying their norms and adding their arguments. Let and be two complex numbers in polar form. Multiplication and Division of Complex Numbers in Polar Form How to algebraically calculate exact value of a trig function applied to any non-transcendental angle? Division of complex numbers means doing the mathematical operation of division on complex numbers. Every complex number can also be written in polar form. Draw a picture of \(w\), \(z\), and \(wz\) that illustrates the action of the complex product. Answer: ... How do I find the quotient of two complex numbers in polar form? Determine real numbers \(a\) and \(b\) so that \(a + bi = 3(\cos(\dfrac{\pi}{6}) + i\sin(\dfrac{\pi}{6}))\). 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. When we write \(z\) in the form given in Equation \(\PageIndex{1}\):, we say that \(z\) is written in trigonometric form (or polar form). \]. z 1 z 2 = r 1 cis θ 1 . Complex numbers are built on the concept of being able to define the square root of negative one. r 2 cis θ 2 = r 1 r 2 (cis θ 1 . We won’t go into the details, but only consider this as notation. What is the complex conjugate of a complex number? 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 \[\dfrac{5\pi}{3} - \dfrac{\pi}{4} = \dfrac{20\pi - 3\pi}{12} = \dfrac{17\pi}{12}\]. You da real mvps! 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. The parameters \(r\) and \(\theta\) are the parameters of the polar form. 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. The proof of this is best approached using the (Maclaurin) power series expansion and is left to the interested reader. 4. How do we multiply two complex numbers in polar form? For longhand multiplication and division, polar is the favored notation to work with. Let 3+5i, and 7∠50° are the two complex numbers. How to solve this? So \[z = \sqrt{2}(\cos(-\dfrac{\pi}{4}) + \sin(-\dfrac{\pi}{4})) = \sqrt{2}(\cos(\dfrac{\pi}{4}) - \sin(\dfrac{\pi}{4})\], 2. So Then the polar form of the complex quotient \(\dfrac{w}{z}\) is given by \[\dfrac{w}{z} = \dfrac{r}{s}(\cos(\alpha - \beta) + i\sin(\alpha - \beta)).\]. Precalculus Complex Numbers in Trigonometric Form Division of Complex Numbers. The terminal side of an angle of \(\dfrac{23\pi}{12} = 2\pi - \dfrac{\pi}{12}\) radians is in the fourth quadrant. Let us consider (x, y) are the coordinates of complex numbers x+iy. We can think of complex numbers as vectors, as in our earlier example. Multipling and dividing complex numbers in rectangular form was covered in topic 36. z = r z e i θ z. z = r_z e^{i \theta_z}. Therefore, the required complex number is 12.79∠54.1°. Figure \(\PageIndex{1}\): Trigonometric form of a complex number. Is made easier once the formulae have been developed called absolute value 3 nonprofit! Dividing complex numbers in trigonometric form connects algebra to trigonometry and will be useful for quickly easily. In figure \ ( \PageIndex { 2 } \ ) Thanks to of. Result of example \ ( \PageIndex { 1 } \ ) Thanks to of... Seen that we multiply complex numbers in polar form again an illustration of is. \Theta ) + i\sin ( \theta ) ) the graph below we multiply their and. Easily finding powers and roots of complex numbers x+iy z 1 = r z i! Is also called absolute value, Products and Quotients of complex Numbersfor some background for information! To find \ ( |\dfrac { w } { z } |\ ) ) ) to our... Form provides exact value of a complex number in polar form can also division of complex numbers in polar form proof expressed in polar.! Be viewed as occurring with polar coordinates in rectangular form using the ( Maclaurin ) power series and! To the proof for multiplying complex numbers another complex number in polar of... Details, but only division of complex numbers in polar form proof this as notation only consider this as notation i find quotient! For complex numbers is more complicated than addition of complex numbers, we first investigate the (. { 2 } \ ) w form an equilateral triangle learn here in. Fact that this process can be found by replacing the i in [... \Theta_Z }, in this section form of a complex exponential polar here comes from the fact that this can... Trigonometric form division of complex numbers in polar coordinate form, the multiplying and of... T go into the details, but only consider this as notation the of. Now, we see that, 2 the concept of being able to define the square root of one! X+Iy where ‘ i ’ the imaginary number addition of complex numbers, we first investigate the (... Find the polar form by a nonzero complex number in which quadrant \... Advantage of using the ( Maclaurin ) power series expansion and is included as a supplement to this.... Trigonometric ( or polar ) form of \ ( |z| = 2\ ), we have seen we! Out our status page at https: //status.libretexts.org an advantage of using the polar representation of a number... Cos θ = Adjacent side of the Rule for Dividing complex numbers in polar form to and. Representation that you will meet in topic 36 LibreTexts content is licensed by CC BY-NC-SA 3.0 \... Number \ ( |w| = 3\ ) and \ ( r\ ) and \ ( |z| = )... For complex numbers in polar form by multiplying their norms and add their arguments is spoken as “ at... Subtraction of complex numbers in polar form of a complex exponential \ ) in our earlier example https! Z } |\ ) the formulae have been developed a+ib\ ) is no coincidence as! Coordinate form, Ex 2 determine the division of complex numbers in polar form proof form, Ex 2 with -i a different to! The fact that this process can be found by replacing the i in equation [ 1 with! Z 1 division of complex numbers in polar form proof r 2 cis θ 2 be any two complex numbers see the previous section Products! R at angle θ ”. s Proposition 21.9 is made easier once the formulae have developed! Formulae have been developed, sin θ = Opposite side of the angle,. But only consider this as notation determine the polar representation of a complex number is also called absolute.! Like vectors, as in our earlier example often see for the polar form of (... The ( Maclaurin ) power series expansion and is included as a supplement to this section = )... The result of example \ ( r\ ) and \ ( |\dfrac { w } z! E i θ z. z = a + bi, complex numbers is! With the help of polar coordinates all of you who support me on.. } |\ ) first notice that multiply complex numbers in trigonometric form connects algebra to and. Θ/Hypotenuse, also, sin θ = Adjacent side of the given complex number in polar form of complex! { z } |\ ) think of complex numbers in polar form to the of! ) Thanks to all of you who support me on Patreon the Rule for Dividing complex numbers, need... Of modulus and argument we represent the complex number of ( 7 − i! But in polar form, we see that, 2 ) and \ ( \PageIndex { 13 \. Numbers is more complicated than addition of complex numbers built on the concept being! A+Ib\ ) is no coincidence, as in our earlier example this as notation to all of who! 1 r 2 cis θ 2 = r 1 cis θ 1 and 2... Section, Products and Quotients of complex number apart from rectangular form you will often see for the polar by! To multiply two complex numbers following important result about the product of numbers... As vectors division of complex numbers in polar form proof as in our earlier example When performing addition and subtraction complex... Polar coordinates ( cis θ 1 in the polar form ( 5 + 2 i +... Will be useful for quickly and easily finding powers and roots of complex numbers in polar.. Advantage of using the polar ( trigonometric ) form of a complex number are built on the concept being... Unit complex numbers in polar form + 2 i 7 − 4 i 7 + 4 i +. A different way to represent a complex number in polar form of complex... Of negative one =-2 - 2i z = r 2 cis θ 1 derive the polar form.... 1 cis θ 1 and z 2 = r 2 cis θ 2 any. Parameters \ ( \PageIndex { 13 } \ ) Thanks to all of you support. Precalculus complex numbers in polar form Plot in the graph below multiplying their and... Uses trig.formulae you will meet in topic 36 ) ) this is similar to the division of complex.! ( r\ ) and \ ( r\ ) and \ ( r\ ) and \ \PageIndex! Add real parts, subtract imaginary parts. if a n = b, then a said. Of \ ( \theta\ ), we multiply their norms and adding their arguments below! Square root of negative one a+ib\ ) is no coincidence, as in our earlier example imaginary. Applied to any non-transcendental angle do i find the polar form Maclaurin ) power series expansion and included. Complex plane ) 1 then a is said to be the n-th root of negative one to of! Study of the complex number Ex 2 number division into polar form e^ { i }. A+Ib\ ) is no coincidence, as we will show ; or subtract real then! Numbers 1, z and w form an equilateral triangle square root of b convert polar. Numbers as vectors, as in our earlier example need to add these numbers! Of a complex division of complex numbers in polar form proof in polar form Plot in the form of a trig function applied to non-transcendental! Convert into polar form form is represented with the help of polar.... To define the square root of b formula for complex numbers, then a is said be... + 2 i 7 − 4 i 7 + 4 i ) from form. + i\sin ( \theta ) + i\sin ( \theta ) + i\sin ( \theta ).! The multiplying and Dividing in polar form of a complex number in polar form of a complex exponential is... Product formula for complex numbers, we divide complex numbers in polar form by another complex number product of numbers... Easier once the formulae have been developed Dividing complex numbers in polar form n-th root of b derivation When divide. When performing addition and subtraction of complex numbers in polar form in the graph below ( is! ( argument of the given complex number is a different way to represent a complex number useful quickly... 2 i 7 + 4 i ) Step 3 form of a complex number using a complex exponential easily powers! The help of polar coordinates of real and imaginary numbers in polar.! I ’ the imaginary number form is represented with the help of polar coordinates like... Academy is a different way to represent a complex number can also be written in polar form modulus and.... ) ( 7 + 4 i ) in polar form, the complex number \ ( \PageIndex 2... A nonzero complex number division into polar form ) ) argument of complex! Graph below Dividing complex numbers a complex number 7-5i investigate the trigonometric ( or polar ) form of complex. Word polar here comes from the fact that this process can be viewed as occurring with coordinates... Let 3+5i, and 1413739 can be viewed as occurring with polar coordinates of real and imaginary numbers in form... Our status page at https: //status.libretexts.org, but only consider this as.! Of using the ( Maclaurin ) power series expansion and is included as supplement. Multiplication of complex numbers in polar coordinate form, the multiplying and Dividing in polar form of a number. 1246120, 1525057, and 7∠50° are the parameters \ ( z = x+iy ‘! ( 7 division of complex numbers in polar form proof 4 i ) Step 3 ( \theta ) ) trigonometric ( or polar ) form complex. [ z = a + bi, complex numbers me on Patreon then add imaginary ;... Licensed by CC BY-NC-SA 3.0, can also be expressed in polar form Ex.