real and imaginary part calculator

Andrew Levido (andrew.levido@gmail.com) earned a bachelors degree in Electrical Engineering in Sydney, Australia, in 1986. The imaginary part calculator allows you to calculate online the imaginary part of a complex number. If the first number is A = x + yi and the second number is B = m + ni, then the sum of two complex numbers is: $$ A + B = x + yi + m + ni = (x + m) + (y + n) * I $$. Constants. Ditch The Switch: A Soft Latching Circuit Roundup. Hackaday (blog), June 24, 2019. https://hackaday.com/2019/06/24/ditch-the-switch-a-soft-latching-circuit-roundup/. However, the naive tan 1 definition does obscure the fact that a complex number with negative real part and negative imaginary part lies in the 3rd quadrant: the range of t a n 1 is [ 2, + 2] which is only radians or 180 degrees wide - it covers only half of the phase space. The logarithm of a complex numbers (also known as the complex logarithm) can be worked out the following way: One critical remark: since the phases 1\varphi_11 and 1+2k\varphi_1+2k\pi1+2k are equivalent, the complex logarithm has an infinite number of solutions, and we can write the general result as: ln(z1)+i(1+2k)\ln(|z_1|) + i(\varphi_1+2k\pi)ln(z1)+i(1+2k). In real life, where are complex numbers used? Numerical sequences | The value of real part: 1 The value of imaginary part after pressing [SHIFT][=] (Re<->Im): 1.732050808 (i . This means the current peaks 90 before the voltage does. Wireless Product Regulations (EE Tip #123). In mathematics we call it the coefficient of i or j. i = (-1) 1/2 Square root of negative number is not possible, so for calculation purpose, -1is named as imaginary and call it iota (i or j). Complex numbers | To solve a math equation, you need to find the value of the variable that makes the equation true. Domestic customers usually pay only for Real Power, but many commercial customers are charged for Apparent Power, driving them to spend a lot of money trying to correct the power factor. Select the following three, so the imaginary part vanishes: . Search Share. real_part online. The real DFT transforms an N point time domain signal into two N/2 + 1 point frequency domain signals. Do you remember how to add or subtract two 2D vectors? Complex Number Calculator. Real part: x = Re z = -1. Example: re (23i) = 2. imaginary part of complex number. Andrew has maintained a hands-on interest in electronics, particularly embedded systems, power electronics, and control theory in his free time. Here, a is the real part which is written as Re(z). You can use this complex number calculator as an imaginary number calculator - just input the real component equal to 0. Addition: (3+2i)+(5-3i). Example: im (23i) = 3i. Another way to write two parts of a complex number is Re\mathrm{Re}Re and Im\mathrm{Im}Im so that Re(z)=a\mathrm{Re}(z)=aRe(z)=a, and Im(z)=b\mathrm{Im}(z)=bIm(z)=b. If omitted, suffix is assumed to be "i". We can equally describe the same vector by its projection on the horizontal and vertical axes as shown. Real part of complex number Complex Number Calculator. The two signals in the frequency domain are called the real part and the imaginary part, holding the amplitudes of the cosine waves and sine waves, respectively. First, enter an expression with real and imaginary numbers. Site map But they both have shortcuts in the MATH menu under the CPX section. To add or subtract two complex numbers, just add or subtract the corresponding real and imaginary parts. Output: Real part: 3, Imaginary part: 4. Get in touch with us and let's discuss your ideas. This category only includes cookies that ensures basic functionalities and security features of the website. The left-hand side of Figure 4 shows that we can describe a vector as an arrow with a length corresponding to its magnitude and an angle from horizontal corresponding to its phase. Cartesian coordinates: Cartesian form of imaginary number: z = -1. Instructions :: All Functions re, real part of complex number. With this function, the calculator allows the online calculation of the imaginary part of a complex number. This is similar to combining "like terms" when you add polynomials together: (3 x + 2) + (5 x + 7) = 8 x + 9. A complex number is a number in the form of a sum of a real part and an imaginary part a + bi.The symbol i or j in electrical engineering (electrical engineers think differently from the rest of the world!) We will show you how to Find the real part and imaginary part calculator in this blog post. Of course, the division is possible only if G0G \neq 0G=0. Choose Find All Complex 587 Math Tutors . Example: conj (23i) = 2 + 3i. A complex number such as 5+2i is made up of two parts, a real part 5, and an imaginary part 2. Recall that the square root of a number is the number that when multiplied by itself yields the original number. One of the possible value If a = 2 and b = 3. If the voltage and current are out of phase by 90 as for a purely reactive load, the power will only have an imaginary component, and the average power will be zero. COMPLEX number (inumber) in excel derived for mathematical number having real and imaginary coefficients. Everybody needs a calculator at some point, get the ease of calculating anything from the source of calculator-online.net. An imaginary number is the square root of a negative number. Complex numbers can also be entered in the polar form r∠ , where r is the signed module (or length) and is the argument (or angle) of the complex number. In fact, there are also numbers with more imaginary parts: check our quaternion calculator. The calculator also converts a complex number into angle notation (phasor notation), exponential, or polar . However, there's no real number that gives the square root of a negative number. In mathematics, a complex number is defined as a combination of real and imaginary numbers. (In computer programming, the "atan2" function accounts for these multi-quadrant issues and . Math is all about solving equations and finding the right answer. imaginary_part button already appears, the result 7 is returned. a x+iy = a x .a iy = a x. e iyloga. imaginary_part online. Instructions :: All Functions re, real part of complex number. Complex numbers can be entered in the rectangular (standard) form a + bi, where a and b are the real part and imaginary part, respectively. Step 2: Click the blue arrow to submit. The real and imaginary parts of the Fourier coefficients c k are written in this unusual way for convenience in defining the classic Fourier series. Complex Number Calculator The Complex Number Calculator solves complex equations and gives real and imaginary solutions. Math Games, Copyright (c) 2013-2023 https://www.solumaths.com/en, solumaths : mathematics solutions online | Write down any arithmetic combination of complex numbers and the calculator will simplify them, with the steps shown. Input: z = 6 - 8i. Therefore, the two components of the vector are it's real part and it's imaginary part. But there's one last question: how "big" is a complex number? to arbitrary complicated forms such as Point B. Out of these cookies, the cookies that are categorized as necessary are stored on your browser as they are essential for the working of basic functionalities of the website. z : When b=0, z is real, when a=0, we say that z is pure imaginary. Lets suppose we have a complex number in the form z=x+jy (x being your real and y your imaginary part). Other calculators. The real part of the complex number is represented by x, and the imaginary part of the complex number is represented by y. This point is -1 - 4i . Returns the smallest (closest to negative infinity) value that is not less than the argument and is an integer. By taking a step-by-step approach, you can more easily see what's going on and how to solve the problem. Send feedback | Visit Wolfram|Alpha. Its pretty clear that the average power is now zeroweird, since the magnitudes of voltage and current are identical. Writing z = a + ib where a and b are real is called algebraic form of a Use this free circumference calculator to find the area, circumference and diameter of a circle. Note that there is a minus sign in the real part since, at some point, we faced a multiplication of two imaginary numbers iii\!\cdot\!iii, which equals 1-11 by definition. The conjugate of a complex number is defined as: As we can see, the conjugate of a complex number doesn't affect the real part, while the imaginary part has the opposite sign as the original. Step 2: Click the blue arrow to submit. We are here to help you with all of your homework needs! Now what if we have a capacitive load such as in Figure 2. If you have a question, our experts will have an answer for you in no time. Fractions | No matter what else is going on in your life, always remember to focus on your job. This is what is going on with our power. real_part button already appears, the result 1 is returned. Knowing that we can write any complex number using its polar coordinates on that plane: Here sin and cos are basic trigonometric functions. The Complex Number Calculator solves complex equations and gives real and imaginary, Scientific complex number calculator for evaluating mathematical expressions of real, imaginary and complex numbers in rectangular and polar forms showing, It will perform addition, subtraction, multiplication, division, raising to power, and also will find the polar form, conjugate, modulus, and inverse of the. By. On my TI-83, MATH has a whole key on its own. Yes. Perform operations on complex numbers step by step. The modulus (or magnitude) is the length (absolute value) in the complex plane, qualifying the complex number z= a+ib z = a + i b (with a a the real part and b b the imaginary part), it is denoted |z| | z | and is equal to |z|= a2+b2 | z | = a 2 + b 2. For calculating real part of the complex number following z=1+7i, complex number You can also try integral long division calculator and integral convergence calculator offered by this site to use freely. To multiply complex numbers the imaginary number calculator use formula as: F * G = | z_1| * exp (i_1) * | z^2| * exp (i^2) = | z_1 * z^2| * exp (i(_1 + ^2)), we see: A * B = | z_1 * z^2| and arg (A * B) = _1 +^2. To add and subtract complex numbers, you just combine the real parts and the imaginary parts, like this: (5 + 3 i) + (2 + 8 i) = (5 + 2) + (3 + 8) i = 7 + 11 i. Embed this widget . Fast and easy to navigate, gets straight to the point and in a way helps you figure out your own question and then answers it. Complex numbers | Fractions | This corresponds to the average power. Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site As you would expect, the sinusoidal voltage across the resistor (green trace) results in a sinusoidal current with the same phase (blue trace). Figure out math questions Math is a subject that can be difficult to understand, but with practice and patience, anyone can learn to figure out math problems. We Answer! sin(1+2i) / ln(3+4i) + atan(1+3i) 4(1.8), which contains non-algebraic functions and complex numbers in rectangular and polar forms. Similarly, b is the imaginary part written as Im(z). Find more Mathematics widgets in. The first step to solving any problem is to scan it and break it down into smaller pieces. Then, it's pretty straightforward to achieve both addition and subtraction of complex numbers. But opting out of some of these cookies may have an effect on your browsing experience. Real numbers insert as 7.5, no 7,5 Constants pi - number Pi e - the base of natural logarithm i - complex number oo Read on to find the answer to the question: "what is a complex number" learn about the algebraic and polar forms of complex numbers, and master the skills of multiplying and dividing complex numbers. The real part is 2 and the imaginary part is 3, so the complex coordinate is (2, 3) where 2 is on the real (or horizontal) axis and 3 is on the imaginary (or vertical) axis. A complex number is a number of the form , where and are real numbers and is the imaginary unit , the square root of . Mathematical equations can often be confusing, but there are ways to make them clearer. | z_1| Exp * exp(-_1 * d) * exp (i(_1 * m + n * ln |z_1|)). We can also use polar coordinate notation to consider the above operations, such as A = |Z_1| * exp (i_1), B = | z^2| * exp (i^2). Although we are mixing two different notations, it's fine. I've been using math app since high school and even now it still answers every kind of college math problem I give it. COMPLEX (real_num, i_num, [suffix]) The COMPLEX function syntax has the following arguments: Real_num Required. One way to think about math equations is to think of them as a puzzle. (1+i)/(1-i), if the button real_part already appears, the result 0 is returned. In DC circuits the average value of voltage and current is equal to the instantaneous value so the average power is the same as the instantaneous power yielding the familiar relationship P = V I . It is represented as z = a + bi. Configurable Digital Gate Driver for SiC MOSFETs, By using our site, you acknowledge that you have read and understand our, Sign up for the Advertiser Notification Broadcast, https://hackaday.com/2019/06/24/ditch-the-switch-a-soft-latching-circuit-roundup/, https://circuits-diy.com/soft-latch-switch-circuit/. The suffix for the imaginary component of the complex number. By using simple language and breaking the equation down into smaller pieces, you can help your reader understand what the equation is trying to say. The formulas which transform complex numbers from Cartesian form to polar form are exactly the same as classical coordinate transformations: where z|z|z is the modulus/absolute value of the complex number, \varphi is known as an argument or phase (sometimes we use notation arg(z)=\mathrm{arg}(z) = \varphiarg(z)=), and tan\tantan is the tangent of the given argument. To calculate the imaginary part of the following complex expression z=`(1+i)/(1-i)`, Maths Exercise Generator | enter imaginary_part(`1+7i`) or directly 1+7i, if the You can get an expert answer to your question in real-time on JustAsk. This widget give some information about your complex number z. Find the real and imaginary parts of the complex number calculator - The real part calculator allows you to calculate online the real part of a complex number. At the end of this text, you can also find information about the properties of complex numbers (most of which are based on the conjugate or the absolute value of complex numbers) and even some practical applications of them. Complex numbers are sometimes really helpful with algebraic expressions, especially if they are related to trigonometric functions. HDL Code Generation Generate Verilog and VHDL code for FPGA and ASIC designs using HDL Coder. The proof goes as such: Euler's formula says that e^{i \theta} = \cos \theta + i \sin \theta When . You Ask? Is it? The only thing that has changed is the phase relationship. Addition: (3+2i)+(5-3i). I just explained how you guys should decide the real and imaginary p. So, product of a and c should be > 4. To get the imaginary solutions, 4ac should be > 16. Power companies would like all loads to be resistive (PF = 1) since they must size their transmission networks for the Apparent Power, but only Real Power does useful work at the point of consumption. The horizontal axis is the real (R)axis and the vertical axis is the imaginary (I) axis. As an imaginary unit, use, (1+i) (3+5i) = 1*3+1*5i+i*3+i*5i = 3+5i+3i-5 = -2+8, pow(1+2i,1/3)*sqrt(4) = 2.439233+0.9434225, pow(-5i,1/8)*pow(8,1/3) = 2.3986959-0.4771303, (6-5i)^(-3+32i) = 2929449.0399425-9022199.5826224, equation with complex numbers: (z+i/2 )/(1-i) = 4z+5i, system of equations with imaginary numbers: x-y = 4+6i; 3ix+7y=x+iy, multiplication of three complex numbers: (1+3i)(3+4i)(5+3i), Find the product of 3-4i and its conjugate. Math practice | The sum of a real number and an imaginary number is known as a complex number. Every complex number \(z\) can be represented as a vector in the Gaussian number plane. If you don't feel comfortable with it, just use our complex number calculator to ensure the outcome is correct. It will perform addition, Complex numbers in real life. The angle \varphi is measured from the X-axis counterclockwise, and it can vary from 000 to 22\pi2 or from -\pi to \pi (depending on convention since both are equivalent). Enter the equation for which you want to find all complex solutions. Input the complex binomial you would like to graph on the complex plane. And that's it. Multiplying complex numbers isn't that scary. They're written a + bi, where. The imaginary coefficient of the complex number. Definitions and Formulas. Real functions | You can check how we can do it in practice with the voltage divider calculator. This vector is uniquely defined by the real part and the imaginary part of the complex number \(z\). Added Aug 1, 2010 by Roman in Mathematics. The basic imaginary number is denoted with the letter iii (sometimes jjj e.g. Enter complex numbers expression: Real part: re(3+2i). When b=0b = 0b=0, the number is purely real, and if a=0a = 0a=0, we have a purely imaginary number. In the given expression a is the real part and b is the imaginary part of the complex number. Lets take a look at the calculation of theorem: A / B = (x + yi)/(m + ni) =, expand the numerator and denominator by combining the complex numbers of the numerator and denominator. Solving math problems can be fun and challenging! For example, you could rewrite i as a real part-- 0 is a real number-- 0 plus i. Imaginary Part of a Complex Number Description Determine the imaginary part of a complex number . I don't recommend it if you don't wanna LEARN how to resolve equations. You can use this complex number calculator as an imaginary number calculator - just input the real component equal to 0. Scientific complex number calculator for evaluating mathematical expressions of real, imaginary and complex numbers in rectangular and polar forms showing. We provide top-quality homework help to students of all levels. Math Calculators | He worked for several years in R&D for power electronics and telecommunication companies before moving into management roles. Great since I hate asking for help, this is a great automated math solving app, if I don't get it it simply tells me how to do it and shows me the answer, this app is cool especially when it comes to math it really helps me a lot to do my homework thank you so much math app i can spend my whole day answering my modules. Homework Support Solutions. enter imaginary_part(`(1+i)/(1-i)`) or directly Choose Find All Complex. Click "Submit." Plot will be shown with Real and Imaginary Axes. Remarks. Free Complex Numbers Calculator - Simplify complex expressions using algebraic rules step-by-step. (1+2i)-(3-4i), About this page: Cubic equations calculator; The calculator finds real and complex roots of cubic equations with real coefficients a, b, c and d: ax + bx + cx + d = 0 (1) using the . Arguably, the best app I have on my phone, dISAPPOINTING, great tool for checking your work, love this app, when it comes to maths sessions always used this. Expert teachers will give you an answer in real-time. Complex numbers calculation examples. Thus proving useful and efficient, it is indeed a helpful app. $$ A * B = (x + yi) * (m + ni) = x * m + x * n * i + y * m * i + y * n * i * i = (x * my * n) + (x * n + y * m) * i $$. Algebraic calculation | . Imaginary part: y = Im z = 0 Decide mathematic question. Necessary cookies are absolutely essential for the website to function properly. sin(1+2i) / ln(3+4i) + atan(1+3i) 4(1.8). Furthermore, this complex number calculator shows work step-by-step and converts the complex number results to standard, polar and other modular forms. RapidTables. We can't measure the real part or imaginary parts in isolation, because that would miss the big picture. Contact | Get the free Complex Number Calculator widget for your website, blog, Wordpress, Blogger, or iGoogle. As a powerful scientific complex number calculator it can also be used to calculate exponential, logarithmic, trigonometric, hyperbolic, ( Gamma ), ( psi) and ( zeta) functions with imaginary or complex number as arguments. This is a great app if your falling behind in high school math or calculus. Example: re(2-3i) = 2. im, imaginary part of complex number. In the second half-cycle, both are negative, so the resulting power is also positive. This website uses cookies to improve your experience while you navigate through the website. Discount calculator uses a product's original price and discount percentage to find the final price and the amount you save. Graphing Calculator | Copyright document.write(new Date().getFullYear()) KCK Media Corp.All Rights Reserved, https://circuitcellar.com/author/andrew-levido/. The imaginary number calculator makes the given expression simple with these steps: Every real number is a complex number, but its not compulsory each complex number is a real number. If a number is purely real or imaginary, set the other component equal to 0. Technically, any real number or imaginary number can be regarded as a complex number. You could re. Now, let's move to something more challenging - we want to find how multiplying complex numbers work. It is expressed as x + yi. Math practice | To answer the question of what is a complex number, we have first to ask: "What is an imaginary number?". The imaginary part; im(1+I) Modulus of a complex number; absolute(1+I) The argument; arg(1+I) . Math is the study of numbers, shapes, and patterns. If you're looking for the best homework solution, look no further than our website. Use this online complex number calculator to perform basic operations like multiplication and division with complex numbers. Unfortunately, many modern loads, particularly electric motors and switched-mode power supplies have very low power factor. Any real number is a complex number. $-6i$: This is an imaginary number that only contains an imaginary part, $-6i$. A complex number is the sum of an imaginary number and a real number, expressed as a + bi. Simple calculator; Percentage calculator; Examples: Input: z = 3 + 4i. Most loads in AC systems will have some resistive and some reactive component, so one number is not enough to describe power in these cases. The Typeset version of the Im command is I Obtain the Imaginary Part of a Complex Number Enter a complex number: Extract the imaginary part: Commands Used. GPU Code Generation Generate CUDA code for NVIDIA GPUs using GPU Coder. Site map If the phase angle was 90, the projection on the real axis would be zero, and the imaginary part would be equal to the magnitude. Thus, every real number is a complex number. Feel free to check Omni's trigonometric functions calculator if you need to quickly revise trigonometric functions. Analogically, we can find the difference between these numbers: and now Re(FG)=ac\mathrm{Re}(F - G) = a - cRe(FG)=ac, and Im(FG)=bd\mathrm{Im}(F - G) = b - dIm(FG)=bd. Description : Writing z = a + ib where a and b are real is called algebraic form of a complex number z : a is the real part of z; ; b is the imaginary part of z. The quantity cos is the Power Factor (PF) and varies from one for a perfectly resistive load, to zero for a perfectly reactive load. Numbers | Division: (3+2i)/(5-3i). This time, the real part can be written as Re(A * B) = x * m - y * n, and the imaginary part as Im(A * B) = x * n + y * m. Remember that complex number calculators use a negative sign in the real part because, at some point, we are faced with the product of two numbers i * i, which by definition is -1. Love the app I'm 10 and don't really know math but this helps me so much! A complex number calculator is a scientific calculator that is also capable of performing It looks like the second attempt is much simpler; thus sometimes it's worth considering changing the form of our expressions before starting a calculation. The S paramater data are complex numbers that have magnitude and phase, or stated another way, real and imaginary components. Similarly, the length of a vector in the two-dimensional Euclidean plane is the distance between its end and the origin of the coordinate system. The Complex Number Calculator solves complex equations and gives real and imaginary solutions. First, the imaginary numbers calculator finds a general formula for the complex power of two numbers, given as A * B. AB = (x + yi) (m + ni) = Since it is not clear how to extend this expression, the complex calculator use F as the polar form of a complex number. With Instant Expert Tutoring, you can get help from a tutor anytime, anywhere. Real and imaginary parts calculator can be found online or in math books. The set of all complex numbers is represented by Z C. The set of all imaginary numbers is expressed as Z C R. Complex numbers are also used to calculate the voltage, current, or resistance in an AC circuit (AC means alternating current). Real. Complex numbers calculation examples. Arshad, Areeba. The real part calculator allows you to calculate online the real part of a complex number. Imaginary part: im(3+2i). When performing simple operations with complex numbers, it's helpful to think about them as vectors. $-3 + 4i$ : $-3$ represents the real number part while $4i$ represents the imaginary part. For another, the sum of 3 + i and -1 + 2 i is 2 + 3 i. Math Calculators | Instructions :: All Functions re, real part of complex number. The real part R[z] of a complex number z=x+iy is the real number not multiplying i, so R[x+iy]=x. I will spare you the math, but the resulting power waveform is also a sinusoid (red trace), but with twice the frequency and offset above zero. If the angle is between the two, as in the figure, there will be a real and imaginary part to the quantity. in electronics), and is defined by: The definition of a complex number zzz is the combination of real aaa and imaginary bib\!\cdot\!ibi parts, so that: Here, both aaa and bbb are classically understood as real numbers. example. Imaginary part: y = Im z = 0 A complex number is the fancy name for numbers with both real and imaginary parts. real part of complex number. There are many ways to stay healthy and fit, but some methods are more effective than others. According to the notation in the previous section: However, use an online Composite Function Calculator that solves the composition of the functions from entered values of functions f(x) and g(x) at specific points. Over the years he has written a number of articles for various electronics publications and occasionally provides consulting services as time allows. For further details I can recommend for example: S. O. Kasap and P. Capper, Springer . Real and imaginary number calculator - We discuss how Real and imaginary number calculator can help students learn Algebra in this blog post. The i is called the iota, which has a value of $ \sqrt{-1} $. To calculate the imaginary part of the following complex expression z= 1 + i 1 - i, enter imaginary_part ( 1 + i 1 - i) or directly (1+i)/ (1-i), if the button imaginary_part already appears, the result 1 is returned. $5$: Since $5$ is a counting number and consequently, a real number, $5$ is still a complex number with its imaginary number part equal to $0$. amazing app, i know you might think it is just to cheat with but it actually shows you the answers, and even sometimes graphs with steps to how it got there. 16-4ac < 0. Also, we can significantly simplify multiple issues from electronics with complex numbers. We also use third-party cookies that help us analyze and understand how you use this website. ADVERTISMENT However, an online Scientific Notation Calculator allows you to add, subtract, multiply, and divide numbers in scientific notation. Example: re(2-3i) = 2. im, imaginary part of complex number. The lines. To find two parts of ( + i) x+iy . Equations | Algebraic calculation | Where Epsilon1 = n^2 - k^2, is a real part of dielectric function and Epsilon2 (imaginary part) = 2nk. b=5 V=a+b It's important to understand the difference between Python variables and SymPy symbols when using SymPy. You can probably tell by inspection that the average power in the resistor will be P = (Vpk Ipk). This pointer is uniquely defined by its length and the angle \(\) to the real axis (x). | Languages available : fr|en|es|pt|de, See intermediate and additional calculations, Calculate online with real_part (real part of complex number), Solving quadratic equation with complex number, Free online math quizzes on complex numbers. Math is a way of solving problems by using numbers and equations. Let's have a look at calculations with step-by-step hints: We obtain the following results: Re(F/G)=ac+bdc+d\mathrm{Re}(F / G) = \frac{a \cdot c + b \cdot d}{c + d}Re(F/G)=c+dac+bd, Im(FG)=bcadc+d\mathrm{Im}(F\!\cdot\!G) = \frac{b \cdot c - a \cdot d}{c + d}Im(FG)=c+dbcad. Is it possible to accomplish calculations of complex numbers specially in polar form with scientific calculators? Following with notation from the previous section, we can write: This time the real part can be written as Re(FG)=acbd\mathrm{Re}(F\!\cdot\!G) = a\!\cdot\!c - b\!\cdot\!dRe(FG)=acbd, and imaginary part as Im(FG)=ad+bc\mathrm{Im}(F\!\cdot\!G) = a\!\cdot\!d + b\!\cdot\!cIm(FG)=ad+bc.

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