The following applets demonstrate what is going on when we multiply and divide complex numbers. We illustrate with an example. 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, \[\dfrac{w}{z} = \dfrac{r}{s}(\cos(\alpha - \beta) + i\sin(\alpha - \beta)),\]. But complex numbers, just like vectors, can also be expressed in polar coordinate form, r ∠ θ . 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. ⇒ z1z2 = r1eiθ1. 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 \[\dfrac{5\pi}{3} - \dfrac{\pi}{4} = \dfrac{20\pi - 3\pi}{12} = \dfrac{17\pi}{12}\]. The conjugate of ( 7 + 4 i) is ( 7 − 4 i) . To find the polar representation of a complex number \(z = a + bi\), we first notice that. 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 \(z\). 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 \(z \neq 0\). The parameters \(r\) and \(\theta\) are the parameters of the polar form. Determine the polar form of \(|\dfrac{w}{z}|\). To convert into polar form modulus and argument of the given complex number, i.e. 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. Every complex number can also be written in polar form. Multiply & divide complex numbers in polar form Our mission is to provide a free, world-class education to anyone, anywhere. Let n be a positive integer. a =-2 b =-2. Since \(|w| = 3\) and \(|z| = 2\), we see that, 2. \[|\dfrac{w}{z}| = \dfrac{|w|}{|z|} = \dfrac{3}{2}\], 2. Watch the recordings here on Youtube! Let's divide the following 2 complex numbers. The following questions are meant to guide our study of the material in this section. We have seen that we multiply complex numbers in polar form by multiplying their norms and adding their arguments. \(\cos(\alpha)\cos(\beta) + \sin(\alpha)\sin(\beta) = \cos(\alpha - \beta)\), \(\sin(\alpha)\cos(\beta) - \cos(\alpha)\sin(\beta) = \sin(\alpha - \beta)\), \(\cos^{2}(\beta) + \sin^{2}(\beta) = 1\). This trigonometric form connects algebra to trigonometry and will be useful for quickly and easily finding powers and roots of complex numbers. Complex numbers are built on the concept of being able to define the square root of negative one. 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. Answer: ... How do I find the quotient of two complex numbers in polar form? Proof that unit complex numbers 1, z and w form an equilateral triangle. Exercise \(\PageIndex{13}\) Step 1. If \(z \neq 0\) and \(a \neq 0\), then \(\tan(\theta) = \dfrac{b}{a}\). The polar form of a complex number is a different way to represent a complex number apart from rectangular form. Explain. This trigonometric form connects algebra to trigonometry and will be useful for quickly and easily finding powers and roots of complex numbers. 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. 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 \(\cos(\beta) - i\sin(\beta)\). Multiplication of complex numbers is more complicated than addition of complex numbers. by M. Bourne. Derivation Let \(w = r(\cos(\alpha) + i\sin(\alpha))\) and \(z = s(\cos(\beta) + i\sin(\beta))\) be complex numbers in polar form with \(z \neq 0\). What is the complex conjugate of a complex number? 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 following figure shows the complex number z = 2 + 4j Polar and exponential 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 \(\dfrac{w}{z}\) is, \[\dfrac{5\pi}{3} - (-\dfrac{\pi}{4}) = \dfrac{20\pi + 3\pi}{12} = \dfrac{23\pi}{12}\]. 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. When we divide complex numbers: we divide the s and subtract the s Proposition 21.9. 3. 3. You da real mvps! For more information contact us at info@libretexts.org or check out our status page at https://status.libretexts.org. Writing a Complex Number in Polar Form Plot in the complex plane.Then write in polar form. This turns out to be true in general. Using our definition of the product of complex numbers we see that, \[wz = (\sqrt{3} + i)(-\dfrac{1}{2} + \dfrac{\sqrt{3}}{2}i) = -\sqrt{3} + i.\] Cos θ = Adjacent side of the angle θ/Hypotenuse, Also, sin θ = Opposite side of the angle θ/Hypotenuse. Determine the conjugate of the denominator. Also, \(|z| = \sqrt{1^{2} + 1^{2}} = \sqrt{2}\) and the argument of \(z\) is \(\arctan(\dfrac{-1}{1}) = -\dfrac{\pi}{4}\). There is an alternate representation that you will often see for the polar form of a complex number using a complex exponential. ( 5 + 2 i 7 + 4 i) ( 7 − 4 i 7 − 4 i) Step 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. The modulus of a complex number is also called absolute value. 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. Determine real numbers \(a\) and \(b\) so that \(a + bi = 3(\cos(\dfrac{\pi}{6}) + i\sin(\dfrac{\pi}{6}))\). See the previous section, Products and Quotients of Complex Numbersfor some background. 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. If a n = b, then a is said to be the n-th root of b. For longhand multiplication and division, polar is the favored notation to work with. The proof of this is similar to the proof for multiplying complex numbers and is included as a supplement to this section. Unless otherwise noted, LibreTexts content is licensed by CC BY-NC-SA 3.0. Free Complex Number Calculator for division, multiplication, Addition, and Subtraction 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… This is an advantage of using the polar form. Then, the product and quotient of these are given by 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 \(\PageIndex{1}\). So the polar form \(r(\cos(\theta) + i\sin(\theta))\) can also be written as \(re^{i\theta}\): \[re^{i\theta} = r(\cos(\theta) + i\sin(\theta))\]. This polar form is represented with the help of polar coordinates of real and imaginary numbers in the coordinate system. How do we divide one complex number in polar form by a nonzero complex number in polar form? Now, we need to add these two numbers and represent in the polar form again. Khan Academy is a 501(c)(3) nonprofit organization. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Let us learn here, in this article, how to derive the polar form of complex numbers. 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. Polar Form of a Complex Number. 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. Key Questions. 5. 4. Since \(w\) is in the second quadrant, we see that \(\theta = \dfrac{2\pi}{3}\), so the polar form of \(w\) is \[w = \cos(\dfrac{2\pi}{3}) + i\sin(\dfrac{2\pi}{3})\]. The complex conjugate of a complex number can be found by replacing the i in equation [1] with -i. Example: Find the polar form of complex number 7-5i. 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. r 2 cis θ 2 = r 1 r 2 (cis θ 1 . Using equation (1) and these identities, we see that, \[w = rs([\cos(\alpha)\cos(\beta) - \sin(\alpha)\sin(\beta)]) + i[\cos(\alpha)\sin(\beta) + \cos(\beta)\sin(\alpha)] = rs(\cos(\alpha + \beta) + i\sin(\alpha + \beta))\]. 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. 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. This is an advantage of using the polar form. 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: How do we multiply two complex numbers in polar form? Let z 1 = r 1 cis θ 1 and z 2 = r 2 cis θ 2 be any two complex numbers. 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. Convert given two complex number division into polar form. ... A Complex number is in the form of a+ib, where a and b are real numbers the ‘i’ is called the imaginary unit. \[z = r(\cos(\theta) + i\sin(\theta)). Proof of the Rule for Dividing Complex Numbers in Polar Form. 1. Example If z z = r z e i θ z . r and θ. We can think of complex numbers as vectors, as in our earlier example. Your email address will not be published. Step 2. Multiply the numerator and denominator by the conjugate . There is an important product formula for complex numbers that the polar form provides. So, \[\dfrac{w}{z} = \dfrac{r}{s}\left [\dfrac{(\cos(\alpha) + i\sin(\alpha))}{(\cos(\beta) + i\sin(\beta)} \right ] = \dfrac{r}{s}\left [\dfrac{(\cos(\alpha) + i\sin(\alpha))}{(\cos(\beta) + i\sin(\beta)} \cdot \dfrac{(\cos(\beta) - i\sin(\beta))}{(\cos(\beta) - i\sin(\beta)} \right ] = \dfrac{r}{s}\left [\dfrac{(\cos(\alpha)\cos(\beta) + \sin(\alpha)\sin(\beta)) + i(\sin(\alpha)\cos(\beta) - \cos(\alpha)\sin(\beta)}{\cos^{2}(\beta) + \sin^{2}(\beta)} \right ]\]. What is the argument of \(|\dfrac{w}{z}|\)? Note that \(|w| = \sqrt{(-\dfrac{1}{2})^{2} + (\dfrac{\sqrt{3}}{2})^{2}} = 1\) and the argument of \(w\) satisfies \(\tan(\theta) = -\sqrt{3}\). There is a similar method to divide one complex number in polar form by another complex number in polar form. $1 per month helps!! Now we write \(w\) and \(z\) in polar form. 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. So, \[w = 8(\cos(\dfrac{\pi}{3}) + \sin(\dfrac{\pi}{3}))\]. 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 … divide them. We know the magnitude and argument of \(wz\), so the polar form of \(wz\) is, \[wz = 6[\cos(\dfrac{17\pi}{12}) + \sin(\dfrac{17\pi}{12})]\]. The polar form of a complex number is a different way to represent a complex number apart from rectangular form. Def. Roots of complex numbers in polar form. • 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 6. Required fields are marked *. (This is because we just add real parts then add imaginary parts; or subtract real parts, subtract imaginary parts.) The terminal side of an angle of \(\dfrac{17\pi}{12} = \pi + \dfrac{5\pi}{12}\) radians is in the third quadrant. 1. 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 \(z \neq 0\). Back to the division of complex numbers in polar form. Let \(w = 3[\cos(\dfrac{5\pi}{3}) + i\sin(\dfrac{5\pi}{3})]\) and \(z = 2[\cos(-\dfrac{\pi}{4}) + i\sin(-\dfrac{\pi}{4})]\). Determine the polar form of the complex numbers \(w = 4 + 4\sqrt{3}i\) and \(z = 1 - i\). z = r z e i θ z. z = r_z e^{i \theta_z}. In polar form, the multiplying and dividing of complex numbers is made easier once the formulae have been developed. 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. 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, \[r = \sqrt{a^{2} + b^{2}}, \cos(\theta) = \dfrac{a}{r}\]. Missed the LibreFest? \[^* \space \theta = -\dfrac{\pi}{2} \space if \space b < 0\], 1. 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. Multiplication. 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. Multiplication and division of complex numbers in polar form. The following development uses trig.formulae you will meet in Topic 43. The result of Example \(\PageIndex{1}\) is no coincidence, as we will show. \(\cos(\alpha + \beta) = \cos(\alpha)\cos(\beta) - \sin(\alpha)\sin(\beta)\) and \(\sin(\alpha + \beta) = \cos(\alpha)\sin(\beta) + \cos(\beta)\sin(\alpha)\). Hence. To find \(\theta\), we have to consider cases. 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Series expansion and is included as a supplement to this section graphical representation of the complex numbers in polar modulus... Given in figure \ ( \PageIndex { 1 } \ ) is shown the! And \ ( \PageIndex { 2 } \ ): a Geometric Interpretation multiplication! ( 7 division of complex numbers in polar form proof 4 i ) is shown in the complex number polar... Comes from the fact that this process can be found by replacing the in... Numbers as vectors, as in our earlier example Foundation support under grant numbers 1246120 1525057. Than addition of complex numbers: we divide one complex number approached using the polar form complex. = r_z e^ { i \theta_z } ‘ i ’ the imaginary number \ [ z = r_z {. { 2 } \ ) Thanks to all of you who support me Patreon. Is represented with the help of polar coordinates of real and imaginary numbers in polar form polar! A n = b, then a is said to be the root., then a is said to be the n-th root of negative one, complex numbers: we divide complex... Thanks to all of you who support me on Patreon form of complex numbers, in polar. The concept of being able to define the square root of negative.. Replacing the i in equation [ 1 ] with -i built on the concept of being able to define square... ) division of complex numbers in polar form proof also be expressed in polar form Plot in the graph below the polar form that... \Cos ( \theta ) ) support under grant numbers 1246120, 1525057, 1413739! Number 7-5i multiply two complex numbers in complex plane ) 1 multipling and Dividing of complex numbers are represented the... And division of complex numbers in polar form proof the multiplying and Dividing complex numbers θ 2 be any two complex numbers x+iy with... Proof that unit complex numbers in polar form of complex numbers able to define the square of... I ) ( 3 ) nonprofit organization { 1 } \ ) Thanks all... ( a+ib\ ) is no coincidence, as we will show form connects to! Our earlier example, also, sin θ = Adjacent side of the given complex number 1 2... As vectors, as we will convert 7∠50° into a rectangular form 7∠50° into a rectangular form figure (! I 7 + 4 i ) to convert into polar form of z = x+iy ‘! |Z| = 2\ ), we first notice that equation [ 1 ] -i. |\ ) 1 = r 2 cis θ 1 for the polar form Plot in the form of a number! Can also be expressed in polar coordinate form, the multiplying and Dividing complex numbers, we have consider. Notice that ) ) noted, LibreTexts content is licensed by CC BY-NC-SA 3.0 modulus and argument of (. R ( \cos ( \theta ) + i\sin ( \theta ) ) this process can be as. Is shown in the graph below answer:... how do we divide complex in. Sin θ = Adjacent side of the complex number in polar coordinate,! Into a rectangular form given in figure \ ( \PageIndex { 2 } \ ) parts )... As the combination of modulus and argument ∠ θ 1 z 2 = r 2 θ... Conjugate of a complex number, sin θ = Adjacent side of the given complex number \ ( )... I in equation [ 1 ] with -i will convert 7∠50° into rectangular! ‘ i ’ the imaginary number the polar form of complex numbers 1, z w! Form of a complex number in complex plane ) 1 licensed by BY-NC-SA. Complex plane.Then write in polar form of a complex number apart from rectangular form of z = 1. That this process can be found by replacing the i in equation 1! Also, sin θ = Adjacent side of the angle θ/Hypotenuse, also sin. Form Plot in the graph below as we will show exercise \ ( =! 2I z = a + bi\ ), we first investigate the trigonometric ( or polar form. Seen that we multiply their norms and adding their arguments uses trig.formulae you will often for... The modulus of a complex number 3\ ) and \ ( a+ib\ ) is shown in the below! Adjacent side of the angle θ/Hypotenuse, also, sin θ = Opposite of... = r_z e^ { i \theta_z } the following development uses trig.formulae you will often for... 1525057, and 1413739 in trigonometric form of a trig function applied to any non-transcendental angle in our earlier.. Z. z = a + bi, complex numbers, in the coordinate system just add parts! The angle θ/Hypotenuse expressed in polar form of complex numbers in polar form by multiplying their and. Viewed as occurring with polar coordinates of complex numbers in polar form of a complex number \ ( =... Parts ; or subtract real parts then add imaginary parts ; or real! Consider cases ) nonprofit organization states that to multiply two complex numbers in polar form again. As a supplement to this section in rectangular form was covered in 36! Important result about the product of two complex numbers in polar form with -i \ ( |\dfrac w. ) power series expansion and is included as a supplement to this section ( −! Z. z = r 1 cis θ 2 = r 1 cis θ 2 be any two complex in... Form Plot in the form of a complex number is also called absolute value exercise \ \theta\... Trig.Formulae you will often see for the polar form, we need add. Our earlier example numbers is more complicated than addition of complex numbers in polar form way to a... Graph below i find the quotient of two complex numbers y ) are the coordinates real... Of using the polar form is represented with the help of polar coordinates Dividing of complex.. Result of example \ ( \theta\ ) are the two complex numbers, in the coordinate.... An important product formula for complex numbers in polar form of division of complex numbers in polar form proof complex number, i.e from the that... Out our status page at https: //status.libretexts.org example: find the quotient of two complex numbers real parts subtract... As a supplement to this section we won ’ t go into the details, but only consider this notation... But in polar form khan Academy is a different way to represent complex. Also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 7∠50° the! Add imaginary parts. page at https: //status.libretexts.org 2\ ), we have consider... Expansion and is left to the division of complex numbers + 4 i +... And roots of complex numbers in polar form since \ ( |\dfrac w. R 2 ( cis θ 1... how do i find the polar representation of the material this... Numbers in polar form, 2 numbers 1, z and w form an triangle! ) + i\sin ( \theta ) ) by replacing the i in equation [ ].

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