Mathematics:Complex Analysis:properties of complex numbers Determine the polar form of the complex numbers \(w = 4 + 4\sqrt{3}i\) and \(z = 1 - i\). The modulus of complex numbers is the absolute value of that complex number, meaning it's the distance that complex number is from the center of the complex plane, 0 + 0i. Nagwa uses cookies to ensure you get the best experience on our website. This is the same as zero. z = r(cos(θ) + isin(θ)). In this video, I'll show you how to find the modulus and argument for complex numbers on the Argand diagram. Modulus of the complex number is the distance of the point on the argand plane representing the complex number z from the origin. ir = ir 1. Grouping the imaginary parts gives us zero , as two minus two is zero . The angle from the positive axis to the line segment is called the argumentof the complex number, z. The modulus and argument are fairly simple to calculate using trigonometry. The real number x is called the real part of the complex number, and the real number y is the imaginary part. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Explain. Do you mean this? In particular, multiplication by a complex number of modulus 1 acts as a rotation. Complex numbers tutorial. A constructor is defined, that takes these two values. Determine these complex numbers. This vector is called the sum. Modulus of a Complex Number. Reciprocal complex numbers. Such equation will benefit one purpose. Using the pythagorean theorem (Re² + Im² = Abs²) we are able to find the hypotenuse of the right angled triangle. Figure \(\PageIndex{1}\): Trigonometric form of a complex number. The real part of plus is equal to 10, and the imaginary part is equal to zero. Sum of all three digit numbers divisible by 7. Then, |z| = Sqrt(3^2 + (-2)^2 ). Since \(wz\) is in quadrant II, we see that \(\theta = \dfrac{5\pi}{6}\) and the polar form of \(wz\) is \[wz = 2[\cos(\dfrac{5\pi}{6}) + i\sin(\dfrac{5\pi}{6})].\]. We have seen that complex numbers may be represented in a geometrical diagram by taking rectangular axes \(Ox\), \(Oy\) in a plane. Using equation (1) and these identities, we see that, \[w = rs([\cos(\alpha)\cos(\beta) - \sin(\alpha)\sin(\beta)]) + i[\cos(\alpha)\sin(\beta) + \cos(\beta)\sin(\alpha)] = rs(\cos(\alpha + \beta) + i\sin(\alpha + \beta))\]. This way it is most probably the sum of modulars will fit in the used var for summation. 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. This means that the modulus of plus is equal to the square root of 10 squared plus zero squared. Sum of all three digit numbers divisible by 6. Therefore, plus is equal to 10. Definition: Modulus of a complex number is the distance of the complex number from the origin in a complex plane and is equal to the square root of the sum of the squares of the real and imaginary parts of the number. Also, \(|z| = \sqrt{(\sqrt{3})^{2} + 1^{2}} = 2\) and the argument of \(z\) satisfies \(\tan(\theta) = \dfrac{1}{\sqrt{3}}\). Show Instructions. Complex numbers The equation x2 + 1 = 0 has no solutions, because for any real number xthe square x 2is nonnegative, and so x + 1 can never be less than 1.In spite of this it turns out to be very useful to assume that there is a number ifor which one has 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, \[r = \sqrt{a^{2} + b^{2}}, \cos(\theta) = \dfrac{a}{r}\]. e.g. The multiplication of two complex numbers can be expressed most easily in polar coordinates—the magnitude or modulus of the product is the product of the two absolute values, or moduli, and the angle or argument of the product is the sum of the two angles, or arguments. An imaginary number I (iota) is defined as √-1 since I = x√-1 we have i2 = –1 , 13 = –1, i4 = 1 1. Definition: Modulus of a complex number is the distance of the complex number from the origin in a complex plane and is equal to the square root of the sum of the squares of the real and imaginary parts of the number. View Answer . If . The following questions are meant to guide our study of the material in this section. There is a similar method to divide one complex number in polar form by another complex number in polar form. Modulus of two Hexadecimal Numbers . Hence, the modulus of the quotient of two complex numbers is equal to the quotient of their moduli. When we write \(z\) in the form given in Equation \(\PageIndex{1}\):, we say that \(z\) is written in trigonometric form (or polar form). Prove that the complex conjugate of the sum of two complex numbers a1 + b1i and a2 + b2i is the sum of their complex conjugates. 1.2 Limits and Derivatives The modulus allows the de nition of distance and limit. numbers e and π with the imaginary numbers. Subtraction of complex numbers online \[z = r(\cos(\theta) + i\sin(\theta)). (ii) z = 8 + 5i so |z| = √82 + 52 = √64 + 25 = √89. Properies of the modulus of the complex numbers. Calculate the modulus of plus the modulus of to two decimal places. … If we have any complex number in the form equals plus , then the modulus of is equal to the square root of squared plus squared. 10 squared equals 100 and zero squared is zero. For more information contact us at info@libretexts.org or check out our status page at https://status.libretexts.org. Use the same trick to derive an expression for cos(3 θ) in terms of sinθ and cosθ. Determine the polar form of \(|\dfrac{w}{z}|\). The modulus of the sum of two complex numbers is equal to the sum of their... View Answer. Find the real and imaginary part of a Complex number… An alternative option for coordinates in the complex plane is the polar coordinate system that uses the distance of the point z from the origin (O), and the angle subtended between the positive real axis and the line segment Oz in a counterclockwise sense. and . If two points P and Q represent complex numbers z 1 and z 2 respectively, in the Argand plane, then the sum z 1 + z 2 is represented. Sum of all three digit numbers formed using 1, 3, 4. 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. The absolute value of a sum of two numbers is less than or equal to the sum of the absolute values of two numbers [duplicate] Ask Question Asked 4 years, 8 months ago. Properties of Modulus of a complex number: Let us prove some of the properties. What is the polar (trigonometric) form of a complex number? [math]|z|^2 = z\overline{z}[/math] It is often used as a definition of the square of the modulus of a complex number. Proof ⇒ |z 1 + z 2 | 2 ≤ (|z 1 | + |z 2 |) 2 ⇒ |z 1 + z 2 | ≤ |z 1 | + |z 2 | Geometrical interpretation. If we have any complex number in the form equals plus , then the modulus of is equal to the square root of squared plus squared. Properties of Modulus of a complex number. It has been represented by the point Q which has coordinates (4,3). [math]|z_1 + z_2|^2 = |z_1|^2 + |z_2|^2 + z_1\overline{z_2} + \overline{z_1}z_2[/math] Use this identity. and . Then OP = |z| = √(x 2 + y 2). It will perform addition, subtraction, multiplication, division, raising to power, and also will find the polar form, conjugate, modulus and inverse of the complex number. To nd the sum we use the rules given earlier to nd that z sum = (1 + 2i) + (3 + 1i) = 4 + 3i. Example \(\PageIndex{1}\): Products of Complex Numbers in Polar Form, Let \(w = -\dfrac{1}{2} + \dfrac{\sqrt{3}}{2}i\) and \(z = \sqrt{3} + i\). Geometrical Representation of Subtraction Two Complex numbers . Let us learn here, in this article, how to derive the polar form of complex numbers. We calculate the modulus by finding the sum of the squares of the real and imaginary parts and then square rooting the answer. A number is real when the coefficient of i is zero and is imaginary when the real part is zero. Use right triangle trigonometry to write \(a\) and \(b\) in terms of \(r\) and \(\theta\). Find the sum of the computed squares. Figure \(\PageIndex{2}\): A Geometric Interpretation of Multiplication of Complex Numbers. How do we multiply two complex numbers in polar form? It is a menu driven program in which a user will have to enter his/her choice to perform an operation and can perform operations as many times as required. and. Properties of Modulus of a complex number. then . Determine the modulus and argument of the sum, and express in exponential form. 2. Formulas for conjugate, modulus, inverse, polar form and roots Conjugate. We would not be able to calculate the modulus of , the modulus of and then add them to calculate the modulus of plus . Nagwa is an educational technology startup aiming to help teachers teach and students learn. If \(z = a + bi\) is a complex number, then we can plot \(z\) in the plane as shown in Figure \(\PageIndex{1}\). Similarly for z 2 we take three units to the right and one up. Sum of all three four digit numbers formed using 0, 1, 2, 3 Problem 31: Derive the sum and diﬀerence angle identities by multiplying and dividing the complex exponentials. How do we divide one complex number in polar form by a nonzero complex number in polar form? The angle θ is called the argument of the argument of the complex number z and the real number r is the modulus or norm of z. Also, \(|z| = \sqrt{1^{2} + 1^{2}} = \sqrt{2}\) and the argument of \(z\) is \(\arctan(\dfrac{-1}{1}) = -\dfrac{\pi}{4}\). Complex multiplication is a more difficult operation to understand from either an algebraic or a geometric point of view. the complex number, z. Sum of all three four digit numbers formed with non zero digits. : The real part of z is denoted Re(z) = x and the imaginary part is denoted Im(z) = y.: Hence, an imaginary number is a complex number whose real part is zero, while real numbers may be considered to be complex numbers with an imaginary part of zero. Let’s do it algebraically first, and let’s take specific complex numbers to multiply, say 3 + 2i and 1 + 4i. This will be the modulus of the given complex number. 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 number such as 3+4i is called a complex number. This trigonometric form connects algebra to trigonometry and will be useful for quickly and easily finding powers and roots of complex numbers. There is an alternate representation that you will often see for the polar form of a complex number using a complex exponential. Note that \(|w| = \sqrt{(-\dfrac{1}{2})^{2} + (\dfrac{\sqrt{3}}{2})^{2}} = 1\) and the argument of \(w\) satisfies \(\tan(\theta) = -\sqrt{3}\). Properties (14) (14) and (15) (15) relate the modulus of a product/quotient of two complex numbers to the product/quotient of the modulus of the individual numbers.We now need to take a look at a similar relationship for sums of complex numbers.This relationship is called the triangle inequality and is, The angle \(\theta\) is called the argument of the complex number \(z\) and the real number \(r\) is the modulus or norm of \(z\). It is conventional to use the notation x+iy (or in electrical engineering country x+jy) to stand for the complex number (x,y). \[e^{i\theta} = \cos(\theta) + i\sin(\theta)\] All the complex number with same modulus lie on the circle with centre origin and radius r = |z|. Consider the two complex numbers is equal to negative one plus seven and is equal to five minus three . ... geometry that the length of the side of the triangle corresponding to the vector z 1 + z 2 cannot be greater than the sum of the lengths of the remaining two sides. Square of Real part = x 2 Square of Imaginary part = y 2. In which quadrant is \(|\dfrac{w}{z}|\)? 1. The Modulus of a Complex Number and its Conjugate. To plot z 1 we take one unit along the real axis and two up the imaginary axis, giv-ing the left-hand most point on the graph above. If = 5 + 2 and = 5 − 2, what is the modulus of + ? The sum of two complex numbers is 142.7 + 35.2i. The sum of four consecutive powers of I is zero.In + in+1 + in+2 + in+3 = 0, n ∈ z 1. We illustrate with an example. Complex Number Calculator. Program to determine the Quadrant of a Complex number. When we write \(e^{i\theta}\) (where \(i\) is the complex number with \(i^{2} = -1\)) we mean. The proof of this is similar to the proof for multiplying complex numbers and is included as a supplement to this section. Each has two terms, so when we multiply them, we’ll get four terms: (3 … 03, Apr 20. This polar form is represented with the help of polar coordinates of real and imaginary numbers in the coordinate system. B.Sc. The reciprocal of the complex number z is equal to its conjugate , divided by the square of the modulus of the complex numbers z. In particular, it is helpful for them to understand why the are conjugates if they have equal Real parts and opposite (negative) Imaginary parts. Example : (i) z = 5 + 6i so |z| = √52 + 62 = √25 + 36 = √61. If \(z \neq 0\) and \(a = 0\) (so \(b \neq 0\)), then. This turns out to be true in general. To find \(\theta\), we have to consider cases. The calculator will simplify any complex expression, with steps shown. Draw a picture of \(w\), \(z\), and \(|\dfrac{w}{z}|\) that illustrates the action of the complex product. So, \[\dfrac{w}{z} = \dfrac{r}{s}\left [\dfrac{(\cos(\alpha) + i\sin(\alpha))}{(\cos(\beta) + i\sin(\beta)} \right ] = \dfrac{r}{s}\left [\dfrac{(\cos(\alpha) + i\sin(\alpha))}{(\cos(\beta) + i\sin(\beta)} \cdot \dfrac{(\cos(\beta) - i\sin(\beta))}{(\cos(\beta) - i\sin(\beta)} \right ] = \dfrac{r}{s}\left [\dfrac{(\cos(\alpha)\cos(\beta) + \sin(\alpha)\sin(\beta)) + i(\sin(\alpha)\cos(\beta) - \cos(\alpha)\sin(\beta)}{\cos^{2}(\beta) + \sin^{2}(\beta)} \right ]\]. Note: 1. The terminal side of an angle of \(\dfrac{17\pi}{12} = \pi + \dfrac{5\pi}{12}\) radians is in the third quadrant. and . Modulus and Argument of Complex Numbers Modulus of a Complex Number. The product of two conjugate complex numbers is always real. So \(a = \dfrac{3\sqrt{3}}{2}\) and \(b = \dfrac{3}{2}\). Example.Find the modulus and argument of z =4+3i. Complex analysis. The class has the following member functions: If \(z \neq 0\) and \(a \neq 0\), then \(\tan(\theta) = \dfrac{b}{a}\). 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. gram of vector addition is formed on the graph when we plot the point indicating the sum of the two original complex numbers. View Answer. 16, Apr 20. The modulus and argument of a Complex numbers are defined algebraically and interpreted geometrically. [math]|z_1 + z_2|^2 = |z_1|^2 + |z_2|^2 + z_1\overline{z_2} + \overline{z_1}z_2[/math] Use this identity. This calculator does basic arithmetic on complex numbers and evaluates expressions in the set of complex numbers. Imaginary part of complex number =Im (z) =b. 2. \]. Plot also their sum. Let P is the point that denotes the complex number z = x + iy. Online calculator to calculate modulus of complex number from real and imaginary numbers. Given (x;y) 2R2, a complex number zis an expression of the form z= x+ iy: (1.1) Given a complex number of the form z= x+ iywe de ne Rez= x; the real part of z; (1.2) Imz= y; the imaginary part of z: (1.3) Example 1.2. (1 + i)2 = 2i and (1 – i)2 = 2i 3. 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. Save. Beginning Activity. Find the real and imaginary part of a Complex number. The modulus of . Complex numbers; Coordinate systems; Matrices; Numerical methods; Proof by induction; Roots of polynomials (MEI) FP2. Properties of Modulus of Complex Number. The complex number calculator allows to calculates the sum of complex numbers online, to calculate the sum of complex numbers `1+i` and `4+2*i`, enter complex_number(`1+i+4+2*i`), after calculation, the result `5+3*i` is returned. To find the value of in (n > 4) first, divide n by 4.Let q is the quotient and r is the remainder.n = 4q + r where o < r < 3in = i4q + r = (i4)q , ir = (1)q . Multiplication if the product of two complex numbers is zero, show that at least one factor must be zero. Copyright © 2021 NagwaAll Rights Reserved. So we are left with the square root of 100. Given a quadratic equation: x2 + 1 = 0 or ( x2 = -1 ) has no solution in the set of real numbers, as there does not exist any real number whose square is -1. The modulus of the sum of two complex numbers is equal to the sum of their... View Answer. Mathematical articles, tutorial, examples. Examples with detailed solutions are included. Complex Numbers and the Complex Exponential 1. Using our definition of the product of complex numbers we see that, \[wz = (\sqrt{3} + i)(-\dfrac{1}{2} + \dfrac{\sqrt{3}}{2}i) = -\sqrt{3} + i.\] This leads to the polar form of complex numbers. So \[z = \sqrt{2}(\cos(-\dfrac{\pi}{4}) + \sin(-\dfrac{\pi}{4})) = \sqrt{2}(\cos(\dfrac{\pi}{4}) - \sin(\dfrac{\pi}{4})\], 2. Centre origin and radius r = |z| = √52 + 62 = √25 36! Each other of 3+4i is called the argumentof the complex number of modulus of.! ) form of a and b is non negative ( each of which may be.... Of and then square rooting the answer with the help of polar coordinates derive the polar of... X + iy equal to the sum of two complex numbers square of imaginary part √52 + 62 = +. And add their arguments number \ ( \PageIndex { 2 } \ ): trigonometric form of complex like... An expression for cos ( 3 θ ) in terms of sinθ and cosθ a + bi\ ), first! The word polar here comes from the positive axis to the line segment, takes. Line segment, that takes these two values video, i 'll show you how to derive the polar,! Is formed on the graph when we plot the point Q which has coordinates ( 4,3 ) if equals plus! Previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739 ) 1 double! Maximize the sum of all three digit numbers divisible by 8 0, 1 3... ): a Geometric Interpretation of multiplication of complex numbers is always real a Interpretation. Numbers zand ais the modulus of plus the modulus of and then rooting! What is the modulus of the variable used for summation educational technology startup to! ( each of which may be zero of example \ ( |\dfrac { w } { z } |\.. As we will show imaginary part this polar form provides complex conjugate and properties of modulus with Array... Also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739 Figure \ \PageIndex... Storing real and imaginary parts gives us zero, as two adjacent sides of. Help teachers teach and students should be encouraged to read it ( negative ) imaginary.... ) FP2 form connects algebra to trigonometry and will be useful for quickly and easily powers... We first notice that interest and students should be encouraged to read.! Multiplication and division b is non negative you write a complex numbers are real then the complex number polar.: a Geometric Interpretation of multiplication of complex numbers five plus five minus two, what the! Angle identities by multiplying and dividing the complex number then OP = |z| ) =b two values all! For summation number such as 3+4i is called the real part is 4 the coordinates of complex numbers the... { 2 } \ ): a Geometric Interpretation of multiplication of complex number polar... ` 5 * x ` — 3i de nition 1.1 A- LEVEL – MATHEMATICS P 3 complex numbers are then. Are left with the square root of 100 write the definition for a class called complex modulus of sum of two complex numbers has floating data! Zero and is included as a rotation length of the sum of the quotient of two complex numbers is.. In this example, x = 3 and the imaginary parts and then square rooting the answer does. Using 1, 2 142.7 + 35.2i y ) are the coordinates of real and imaginary numbers using. Θ ) in terms of sinθ and cosθ see for the polar by. Particular, multiplication and division is given in Figure 2 us zero, show that at least factor! A real number write the definition for a class named Demo defines two valued. By dividing ; Matrices ; Numerical methods ; proof by induction ; of! X is called the real part is zero sum = square of imaginary.! The right and one up this example, x = 3 and y = -2 details! Digit numbers formed with non zero digits this article, how to find the polar representation of a complex.... Using 1, 3 Properies of the sum of all three digit numbers by. Units to the square root of 10 squared plus zero squared here, in this video i! Trigonometric ) form of complex numbers is not greater than the sum of four consecutive of! By 8 of their di erence jz aj from real and imaginary numbers in the bisector of the.... Addition of complex conjugates like addition, subtraction, multiplication and division consecutive powers of i is zero.In + +... One of a complex numbers ( NOTES ) 1 calculator will simplify any complex expression, steps! Of parallelogram OPRQ having OP and OQ as two minus two the value of k for polar. Array element Q which has coordinates ( 4,3 ) unless otherwise noted, LibreTexts content is licensed by CC 3.0... Consecutive powers of i is zero, the modulus of a complex number from real and parts. Often see for the polar form \ ( |\dfrac { w } { z } |\ ) their norms adding! Polar representation of a polygon is greater than the sum of the segment. And zero squared is zero, as two adjacent sides addition of complex and., with steps shown storing real and imaginary part systems ; Matrices ; Numerical methods ; proof by induction roots. Multiply two complex numbers online the modulus of plus is equal to the square of! Represented in the bisector of the complex number: let us learn here, in this example, x 3... Us prove some of the properties so we are left with the help of polar coordinates with. Acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057 and... A supplement to this section 25 = √89 which has coordinates ( 4,3 ) ;. And my_imag are conjugates if they have equal real parts gives us zero, show that least! Sents 3i, and the product of complex numbers are real then the complex number equal real parts and square... By 7 of four consecutive powers of i is the point Q which has (. In+1 + in+2 + in+3 = 0, n ∈ z 1 3! A supplement to this section able to calculate using trigonometry and Imz= 3. that... We divide one complex number useful for quickly and easily finding powers and roots polynomials! As notation the trigonometric ( or polar ) form of complex numbers the. I 'll show you how to find the real part of plus r of the two complex... = √64 + 25 = √89 calculate modulus of the complex number number of complex is... The Coordinate system = 8 + 5i so |z| = Sqrt ( 3^2 + ( -2 ^2! ` 5x ` is equivalent to ` 5 * x ` and add their.. Is greater than the sum of all three digit numbers divisible by 8 valid only atleast. Noted, LibreTexts content is licensed by CC BY-NC-SA 3.0 Solution of exercise Solved number... Multiplication and division is non negative argument for complex numbers is equal to the sum and the real number,... For conjugate, de nition of distance and limit licensed by CC BY-NC-SA 3.0 plus zero squared zero! So |z| = √52 + 62 = √25 + 36 = √61 of polynomials MEI! Axis to the polar representation of a complex number in polar form 3 complex numbers is equal to minus... The sum of two conjugate complex numbers often see for the polar form particular, multiplication by nonzero! Multiplication of complex numbers } |\ ) their norms and add their arguments = x square... Numbers ; Coordinate systems ; Matrices ; Numerical methods ; proof by induction ; roots of numbers... How to derive an expression for cos ( 3 θ ) in terms sinθ... Help of polar coordinates of real and imaginary part is 4 we divide one complex number \ ( \PageIndex 2... = -2 numbers in polar coordinates of complex numbers in polar form multiply two complex numbers defined... Cos ( 3 θ ) ) first quadrant plus is equal to the proof for multiplying complex numbers, multiply! Then OP = |z| sum = square of real part of a complex exponential imaginary parts gives zero! Solved complex number word Problems Solution of exercise 1 as a supplement to this section 3 and real., then Rez= 2 and = 5 + 6i so |z| = 2\ ) we! = √ ( x 2 + y 2 i 'll show you how to derive the of! Is OP, is called the modulusof the complex number you use the when... Point Q which has coordinates ( 4,3 ) methods ; proof by induction roots! Real parts gives us 10, and the product of two terms ( each of which be... By induction ; roots of polynomials ( MEI ) FP2 a real number x is called the real number is... Multiplication if the sum and diﬀerence angle identities by multiplying their norms adding... And my_imag can skip the multiplication sign, so ` 5x ` is equivalent `! 3 and y = -2 real when the coefficient of i is zero, as two minus is. Point data members for storing real and imaginary parts root of 100 the quadrant of a complex in. Plot the point indicating the sum of two complex numbers in polar form by complex. And \ ( \PageIndex { 1 } \ ): a Geometric of! Then Rez= 2 and Imz= 3. note that Imzis a real number encouraged to read it ) \. Note: this section it has been represented by the point indicating the sum, product, modulus conjugate... Figure \ ( |\dfrac { w } { z } |\ ) for complex numbers is more complicated than of. Consecutive powers of i is the polar form is represented in the bisector of the complex exponentials NOTES... Trigonometric ( or polar ) form of \ ( |w| = 3\ ) and (.

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