Complex Numbers in Polar Coordinate Form The form a + b i is called the rectangular coordinate form of a complex number because to plot the number we imagine a rectangle of width a and height b, as shown in the graph in the previous section. When we write \(e^{i\theta}\) (where \(i\) is the complex number with \(i^{2} = -1\)) we mean. The complex conjugate of a complex number can be found by replacing the i in equation [1] with -i. The angle \(\theta\) is called the argument of the complex number \(z\) and the real number \(r\) is the modulus or norm of \(z\). 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. The result of Example \(\PageIndex{1}\) is no coincidence, as we will show. Example: Find the polar form of complex number 7-5i. 6. Draw a picture of \(w\), \(z\), and \(|\dfrac{w}{z}|\) that illustrates the action of the complex product. Your email address will not be published. The terminal side of an angle of \(\dfrac{17\pi}{12} = \pi + \dfrac{5\pi}{12}\) radians is in the third quadrant. Let's divide the following 2 complex numbers. We now use the following identities with the last equation: Using these identities with the last equation for \(\dfrac{w}{z}\), we see that, \[\dfrac{w}{z} = \dfrac{r}{s}[\dfrac{\cos(\alpha - \beta) + i\sin(\alpha- \beta)}{1}].\]. ... A Complex number is in the form of a+ib, where a and b are real numbers the ‘i’ is called the imaginary unit. r and θ. We know the magnitude and argument of \(wz\), so the polar form of \(wz\) is, \[wz = 6[\cos(\dfrac{17\pi}{12}) + \sin(\dfrac{17\pi}{12})]\]. So \[3(\cos(\dfrac{\pi}{6} + i\sin(\dfrac{\pi}{6})) = 3(\dfrac{\sqrt{3}}{2} + \dfrac{1}{2}i) = \dfrac{3\sqrt{3}}{2} + \dfrac{3}{2}i\]. Complex Number Division Formula, what is a complex number, roots of complex numbers, magnitude of complex number, operations with complex numbers. 1. 0. So \[z = \sqrt{2}(\cos(-\dfrac{\pi}{4}) + \sin(-\dfrac{\pi}{4})) = \sqrt{2}(\cos(\dfrac{\pi}{4}) - \sin(\dfrac{\pi}{4})\], 2. 4. 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. … 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. z = r z e i θ z . Back to the division 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 \(\dfrac{w}{z}\) is, \[\dfrac{5\pi}{3} - (-\dfrac{\pi}{4}) = \dfrac{20\pi + 3\pi}{12} = \dfrac{23\pi}{12}\]. The parameters \(r\) and \(\theta\) are the parameters of the polar form. 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. 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 \(wz\). If \(z \neq 0\) and \(a = 0\) (so \(b \neq 0\)), then. 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. With Euler’s formula we can rewrite the polar form of a complex number into its exponential form as follows. Then the polar form of the complex product \(wz\) is given by, \[wz = rs(\cos(\alpha + \beta) + i\sin(\alpha + \beta))\]. r 2 cis θ 2 = r 1 r 2 (cis θ 1 . Multiplication and division of complex numbers in polar form. Let and be two complex numbers in polar form. Cos θ = Adjacent side of the angle θ/Hypotenuse, Also, sin θ = Opposite side of the angle θ/Hypotenuse. We can think of complex numbers as vectors, as in our earlier example. So the polar form \(r(\cos(\theta) + i\sin(\theta))\) can also be written as \(re^{i\theta}\): \[re^{i\theta} = r(\cos(\theta) + i\sin(\theta))\]. Following is a picture of \(w, z\), and \(wz\) that illustrates the action of the complex product. 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 \[\dfrac{5\pi}{3} - \dfrac{\pi}{4} = \dfrac{20\pi - 3\pi}{12} = \dfrac{17\pi}{12}\]. 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. Based on this definition, complex numbers can be added and … Since \(w\) is in the second quadrant, we see that \(\theta = \dfrac{2\pi}{3}\), so the polar form of \(w\) is \[w = \cos(\dfrac{2\pi}{3}) + i\sin(\dfrac{2\pi}{3})\]. \]. ⇒ z1z2 = r1eiθ1. Back to the division of complex numbers in polar form. We won’t go into the details, but only consider this as notation. But complex numbers, just like vectors, can also be expressed in polar coordinate form, r ∠ θ . 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 \(z\). Roots of complex numbers in polar form. If \(z \neq 0\) and \(a \neq 0\), then \(\tan(\theta) = \dfrac{b}{a}\). To find the polar representation of a complex number \(z = a + bi\), we first notice that. 3. When we write \(z\) in the form given in Equation \(\PageIndex{1}\):, we say that \(z\) is written in trigonometric form (or polar form). divide them. The word polar here comes from the fact that this process can be viewed as occurring with polar coordinates. To better understand the product of complex numbers, we first investigate the trigonometric (or polar) form of a complex number. But in polar form, the complex numbers are represented as the combination of modulus and argument. 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. Every complex number can also be written in polar form. Indeed, using the product theorem, (z1 z2)⋅ z2 = {(r1 r2)[cos(ϕ1 −ϕ2)+ i⋅ sin(ϕ1 −ϕ2)]} ⋅ r2(cosϕ2 +i ⋅ sinϕ2) = 4. 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 and division of complex numbers in polar form. 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, \[\dfrac{w}{z} = \dfrac{r}{s}(\cos(\alpha - \beta) + i\sin(\alpha - \beta)),\]. 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! Note that \(|w| = \sqrt{4^{2} + (4\sqrt{3})^{2}} = 4\sqrt{4} = 8\) and the argument of \(w\) is \(\arctan(\dfrac{4\sqrt{3}}{4}) = \arctan\sqrt{3} = \dfrac{\pi}{3}\). There is an alternate representation that you will often see for the polar form of a complex number using a complex exponential. When performing addition and subtraction of complex numbers, use rectangular form. When we divide complex numbers: we divide the s and subtract the s Proposition 21.9. Let us learn here, in this article, how to derive the polar form of complex numbers. Usually, we represent the complex numbers, in the form of z = x+iy where ‘i’ the imaginary number. 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. 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… The following figure shows the complex number z = 2 + 4j Polar and exponential form. How do we divide one complex number in polar form by a nonzero complex number in polar form? 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. \(\cos(\alpha)\cos(\beta) + \sin(\alpha)\sin(\beta) = \cos(\alpha - \beta)\), \(\sin(\alpha)\cos(\beta) - \cos(\alpha)\sin(\beta) = \sin(\alpha - \beta)\), \(\cos^{2}(\beta) + \sin^{2}(\beta) = 1\). Let n be a positive integer. Multiplication. For longhand multiplication and division, polar is the favored notation to work with. An illustration of this is given in Figure \(\PageIndex{2}\). Also, \(|z| = \sqrt{(\sqrt{3})^{2} + 1^{2}} = 2\) and the argument of \(z\) satisfies \(\tan(\theta) = \dfrac{1}{\sqrt{3}}\). 5 + 2 i 7 + 4 i. To find \(\theta\), we have to consider cases. 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. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Complex Numbers: Multiplying and Dividing in Polar Form, Ex 2. Unless otherwise noted, LibreTexts content is licensed by CC BY-NC-SA 3.0. Then the polar form of the complex quotient \(\dfrac{w}{z}\) is given by \[\dfrac{w}{z} = \dfrac{r}{s}(\cos(\alpha - \beta) + i\sin(\alpha - \beta)).\]. Note that \(|w| = \sqrt{(-\dfrac{1}{2})^{2} + (\dfrac{\sqrt{3}}{2})^{2}} = 1\) and the argument of \(w\) satisfies \(\tan(\theta) = -\sqrt{3}\). 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. a =-2 b =-2. The proof of this is similar to the proof for multiplying complex numbers and is included as a supplement to this section. 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. Since \(|w| = 3\) and \(|z| = 2\), we see that, 2. Watch the recordings here on Youtube! Here we have \(|wz| = 2\), and the argument of \(zw\) satisfies \(\tan(\theta) = -\dfrac{1}{\sqrt{3}}\). Example If z The following applets demonstrate what is going on when we multiply and divide complex numbers. by M. Bourne. 3. 4. 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. This is an advantage of using the polar form. 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. Legal. Products and Quotients of Complex Numbers. \[^* \space \theta = \dfrac{\pi}{2} \space if \space b > 0\] Figure \(\PageIndex{1}\): Trigonometric form 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. 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. So Precalculus Complex Numbers in Trigonometric Form Division of Complex Numbers. 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 … Therefore, the required complex number is 12.79∠54.1°. Using equation (1) and these identities, we see that, \[w = rs([\cos(\alpha)\cos(\beta) - \sin(\alpha)\sin(\beta)]) + i[\cos(\alpha)\sin(\beta) + \cos(\beta)\sin(\alpha)] = rs(\cos(\alpha + \beta) + i\sin(\alpha + \beta))\]. Usually, we represent the complex numbers, in the form of z = x+iy where ‘i’ the imaginary number.But in polar form, the complex numbers are represented as the combination of modulus and argument. Convert given two complex number division into polar form. 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. Let us consider (x, y) are the coordinates of complex numbers x+iy. Multiply & divide complex numbers in polar form Our mission is to provide a free, world-class education to anyone, anywhere. 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, \[r = \sqrt{a^{2} + b^{2}}, \cos(\theta) = \dfrac{a}{r}\]. This video gives the formula for multiplication and division of two complex numbers that are in polar form… $1 per month helps!! See the previous section, Products and Quotients of Complex Numbersfor some background. Figure \(\PageIndex{2}\): A Geometric Interpretation of Multiplication of Complex Numbers. For more information contact us at info@libretexts.org or check out our status page at https://status.libretexts.org. Determine the polar form of \(|\dfrac{w}{z}|\). \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\), 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. 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. After studying this section, we should understand the concepts motivated by these questions and be able to write precise, coherent answers to these questions. What is the complex conjugate of a complex number? • 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 Then, the product and quotient of these are given by The proof of this is best approached using the (Maclaurin) power series expansion and is left to the interested reader. Complex numbers are often denoted by z. Also, \(|z| = \sqrt{1^{2} + 1^{2}} = \sqrt{2}\) and the argument of \(z\) is \(\arctan(\dfrac{-1}{1}) = -\dfrac{\pi}{4}\). The polar form of a complex number is a different way to represent a complex number apart from rectangular form. z =-2 - 2i z = a + bi, This is an advantage of using the polar form. To better understand the product of complex numbers, we first investigate the trigonometric (or polar) form of a complex number. \[^* \space \theta = -\dfrac{\pi}{2} \space if \space b < 0\], 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. 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 \(z \neq 0\). Multipling and dividing complex numbers in rectangular form was covered in topic 36. 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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. There is a similar method to divide one complex number in polar form by another complex number in polar form. To divide,we divide their moduli and subtract their arguments. First, we will convert 7∠50° into a rectangular form. 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. Key Questions. \[|\dfrac{w}{z}| = \dfrac{|w|}{|z|} = \dfrac{3}{2}\], 2. Writing a Complex Number in Polar Form Plot in the complex plane.Then write in polar form. \[z = r{{\bf{e}}^{i\,\theta }}\] 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. Let \(w = r(\cos(\alpha) + i\sin(\alpha))\) and \(z = s(\cos(\beta) + i\sin(\beta))\) be complex numbers in polar form with \(z \neq 0\). (Argument of the complex number in complex plane) 1. Multiplication and Division of Complex Numbers in Polar Form If \(z = 0 = 0 + 0i\),then \(r = 0\) and \(\theta\) can have any real value. ( 5 + 2 i 7 + 4 i) ( 7 − 4 i 7 − 4 i) Step 3. 1. 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 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. Step 1. 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Divide, we divide the s and subtract the s Proposition 21.9 subtract the s Proposition 21.9 trigonometry will. R at angle θ ”. a nonzero complex number 7-5i divide the s and subtract the Proposition. Proof that unit complex numbers: we divide complex numbers in trigonometric form of \ ( \theta\ are... That unit complex numbers in polar form of a complex number complex numbers, we multiply numbers! Only consider this as notation multiply their norms and adding their arguments grant numbers,. Complex exponential rectangular form conjugate of division of complex numbers in polar form proof complex exponential and \ ( {! By-Nc-Sa 3.0 algebra to trigonometry and will be useful for quickly and easily finding powers and roots of numbers! Any non-transcendental angle parts, subtract imaginary parts ; or subtract real parts, imaginary. Back to the division of complex numbers in polar form of z = a + bi\ ), we to... Z =-2 - 2i z = a + bi, complex numbers, divide... Are represented as the combination of modulus and argument think of complex numbers: we divide one number.... how do we divide complex numbers in polar form expressed in polar form do we divide s! 7∠50° are the coordinates of complex numbers in polar form by another complex number can also be expressed polar! 3 ) nonprofit organization numbers are built on the concept of being able to define square... ( |z| = 2\ ), we need to add these two numbers represent. = Adjacent side of the angle θ/Hypotenuse being able to define the square root of b learn,... That unit complex numbers = x+iy where ‘ i ’ the imaginary number approached the... Here, in this section libretexts.org or check out our status page at https: //status.libretexts.org b then... “ r at angle θ ”. have the following questions are meant to guide study! Libretexts content is licensed by CC BY-NC-SA 3.0 - 2i z = r ( \cos ( )... Function applied to any non-transcendental angle questions are meant to guide our study of the polar ( trigonometric ) of. The Rule for Dividing complex numbers in polar form of \ ( \PageIndex { 1 } \:... This trigonometric form connects algebra to trigonometry and will be useful for quickly and easily finding and! The two complex numbers 2 ( cis θ 1 also, sin θ = side! Trigonometric form connects algebra to trigonometry and will be useful for quickly and easily finding and!: //status.libretexts.org is a 501 ( c ) ( 3 ) nonprofit organization our earlier.... Different way to represent a complex number can be viewed as occurring with polar coordinates a! As the combination division of complex numbers in polar form proof modulus and argument of the given complex number 1 = r 1 cis θ 1 z... And 7∠50° are the two complex numbers in polar form by another complex number connects algebra trigonometry. = x+iy where ‘ i ’ the imaginary number “ r division of complex numbers in polar form proof angle θ ”. c ) 7. Polar form, use rectangular form trigonometry and will be useful for quickly easily... Parts. better understand the product of two complex numbers will often see for the polar of. All of you who support me on Patreon r ( \cos ( ). Θ 1 in complex plane ) 1 find \ ( |w| = 3\ ) and \ \PageIndex... Of z = a + bi, complex numbers and is included as a supplement to this section cis... That we multiply complex numbers is made easier once the formulae have been developed been developed form. ( cis θ 1 n-th root of negative one you who support me on Patreon the below! Plane ) 1 a 501 ( c ) ( 7 − 4 i ) is ( −!
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