Example #1: Solve for "x" and "y": 3 4 21 16− x+=−iy i real parts imaginary parts −=321x 416iy i=− x = -7 y = -4 Thus x = -7 and y = -4 A complex number is any number that can be written as abi+ , where a and b Thus in reality there are 4 equations and 4 variables to be found (each equation has a real and imaginary part as does each variable). Here, a and b are real numbers and i is the imaginary number. 5 1 Solving Linear Systems Of Equations Using Substitution Good Overview And Examples Of The Substi College Algebra Help Systems Of Equations College Algebra If all the coefficients are real, the root will be real. COMPLEX NUMBERS In the next section we will solve quadratic equations, which have a term raised to the second power (for example, x^2- 4x + 3 = 0).Solutions of quadratic equations may not be real numbers. Bombelli used this to solve the equation x 3 = 15x + 4 to get the solution Now, the square root of -121 is not a real number; it's neither positive, negative, nor zero. Practice this with polynomials, and isolate the variables to solve the equations. Imaginary numbers are based on the mathematical number $$ i $$. In this case, it's $$ z^3 - 3z^2 + 6z - 4 = (z - 1)(z - 1 + \sqrt{3}i)(z - 1 - \sqrt{3}i). I know it's possible via rref () but cSolve () is nice and clean. So we want to find all of the real and/or complex roots of this equation right over here. 5 1 Solving Linear Systems Of Equations Using Substitution Good Overview And Examples Of The Substi College Algebra Help Systems Of Equations College Algebra rsin rcos x r rei y z= x+iy= rcos +ir sin = r(cos i ) = rei (3:6) This is the polar form of a complex number and x+ iyis the rectangular form of the same number. I am getting an equation with this form . . (−3 −9i)(1+10i) ( − 3 . I know this software, Algebrator which has helped a lot of beginners clear their concepts. 2.4849+0.1552i. Sometimes an equation will have multiples of an unknown and other numbers, eg \ (3x + 2 = 8\). Using solve() to find positive real solutions to a complex equation. $$ So you can see the solution of the equation easily from this representation. You can solve for Ix in terms of Iy and then do substitution. Browse more Topics under Complex Numbers And Quadratic Equations. The method involves the polar form of a complex number.. Casio purposely dumbs-down their scientific calculators so that you will buy their more expensive graphing calcs. A quick example of what I am trying to do ill use "<" to represent an angle for a number in polar format i.e. A complex equation is an equation that involves complex numbers when solving it. It's All about complex conjugates and multiplication. Watch this tutorial to see the quadratic formula be used to find the complex solutions to a quadratic equation. /Promoting Scientific Calculators for the Benefit of M. Then define the coefficients x= [real (A) imag (A) real (B) imag (B)] in your linear equations and solve for them as. Definition (Imaginary unit, complex number, real and imaginary part, complex conjugate). The complex number equation calculator returns the complex values for which the quadratic equation is zero. It's just algebra, you have two equations in two unknowns. thank you. In equations of this type, your aim is to get all the \ ( {x}\)s (or . Let's examine a few of these cases: 1. $$ i \text { is defined to be } \sqrt{-1} $$ From this 1 fact, we can derive a general formula for powers of $$ i $$ by looking at some examples. These manipulations can be addition, subtraction . Sometimes logarithmic equations are more complex. So, by the identity of complex numbers, we'll get : $$\begin{cases} 4(m+n)=16 \\ -5m = 15\end{cases}$$ can you solve that system of linear equations now and yield your solution ? Let's get organized: A number of the form , where a and b are real numbers, is called a complex number.Here are some examples: The number a is called the real part of a+bi, the number b is called the imaginary part of a+bi.. Luckily, algebra with complex numbers works very predictably, here are some examples: complex-numbers cubics . where Z is a complex number. Use cSolve () as you would normally use the solve (). yis the . In this section we discuss the solution to homogeneous, linear, second order differential equations, ay'' + by' + c = 0, in which the roots of the characteristic polynomial, ar^2 + br + c = 0, are complex roots. solve for x and y 2x-4y+3xi-12i=0 . When in the standard form \(a\) is called the real part of the complex number and \(b\) is called the imaginary part of the complex number. Transcribed image text: Finding nth roots of complex numbers (or equivalently solving the equation zn = a, where a is a given complex number, n is an integer, and z needs to be found) is easy with polar forms: How many roots are there in the first place? Basically you have four equations in four unknowns once you separate real/imag components. The addition of complex numbers makes a significant difference in mathematics. strange behaviour when solving equations symbolically. a phasor 5.5<25volts, and j to be the square root of -1 for a number in . Complex Numbers and the Complex Exponential 1. Solve complex equations step-by-step. The only two roots of this quadratic equation right here are going to turn out to be complex, because when we evaluate this, we're going to get an imaginary number. It can't do these. Get step-by-step solutions from expert tutors as fast as 15-30 minutes. To make this work we de ne ias the square root of 1: i2 = 1 so x2 = i2; x= i: A general complex number is written as z= x+ iy: xis the real part of the complex number, sometimes written Re(z). Therefore, the real and imaginary parts will be zero and we have and. Hope this helps. - Solve the given equation in the complex number system. I cannot figure out how to solve an equation involving complex number in R. I hope anybody can help me Perform the indicated operation and write your answer in standard form. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Real Solution of x^3+8 == 0? The general problem, of course, that's a different thing. Complex Numbers and Quadratic Equations. In this section we will solve systems of two linear differential equations in which the eigenvalues are complex numbers. Combine this with the complex exponential and you have another way to represent complex numbers. Solution of Complex Quadratic Equations. Here's another one: Solve. Here is an example: Trying to find X and Y (40 + 50i)(_X_) - 70(_Y_) = 130 -40(_X_) + (170 - 50i)_Y_ = 0 I can do two equations two unknowns on my TI-84 without complex numbers using matricies. You want to direct the calculator to compute complex answers, in either rectangular notation for impedance phasors (e.g. Examine the following example: x 2 = − 11 x = − 11 11 ⋅ − 1 = 11 ⋅ i i 11. We bow to this nice of How To Solve Complex Numbers graphic could possibly be the most trending topic behind we allowance it in google improvement or facebook. It's less of a programming problem and more of an algebra with complex variables problem. The best you can do is rewrite to express y as a function of x. The . Solving Equations in Complex Numbers 1. The standard form of a complex number is \[a + bi\] where \(a\) and \(b\) are real numbers and they can be anything, positive, negative, zero, integers, fractions, decimals, it doesn't matter. I have two equations and two unknowns. To solve this equation just enter the expression x^2+1=0 and press calculate button. Variable are allowed input of complex numbers. Using actual numbers instead of variables, consider the example of (3+3i) + (5-2i). Solving More Complex Logarithmic Equations. The equation has two solutions which may be identical or different. exp(az) = 1 + cz which I wanna solve for z, where c is a complex number, so expectedly z is complex also. So we're essentially going to get two complex numbers when we take the positive and negative version of this root. Its submitted by direction in the best field. Without the ability to take the square root of a negative . Complex numbers help in the solution of quadratic equations with the help of the Quadratic formula. I have used this software a couple of times when I was in high school and I recommend it to every novice . Explanation: . To solve for , we must first solve the equation with the complex number for and .We therefore need to match up the real portion of the compex number with the real portions of the expression, and the imaginary portion of the complex number with the imaginary portion of the expression. Of course this is not very helpful if you have 10 complex equations (20 equations in 20 unknowns). So let's say we want to solve the equation x to the third power is equal to 1. There will be many solutions that will satisfy it. Solving Complex Algebraic Equations (Solve for x) In this video, we will be learning how to solve for x (or another variable) in complex algebraic equations using inverse operations. Complex numbers enable us to solve equations that we wouldn't be able to otherwise solve. Here are a number of highest rated How To Solve Complex Numbers pictures upon internet. Example 1. Wow, thanks. $\begingroup$ Solving two-by-two systems "by hand" is easily done either by elimination or by Cramer's Rule. How To Divide Complex Numbers Precalculus Tips Complex Numbers Precalculus Maths Exam . My thoughts are to do this: p=[4,0,-11,0,7-3i] roots(p) However, the answer I get is different from the actual answer, which is . each other. A complex number can be represented in the form of a+bi, which is the combination of both the real numbers and the imaginary numbers. Your first 5 questions are on us! We can solve quadratic equations with complex coefficients using the quadratic formula. I want to solve the following equations: 0.38 + 0.28j=x+((?0.31 + 0.54j)^2*y)/(1-y^2) 0.38 + 0.27j=y+((?0.31 + 0.54j)^2*x)/(1-x^2) where both x and y are complex numbers. COMPLEX NUMBERS AND DIFFERENTIAL EQUATIONS 3 3. A real number only carries one piece of information: its location on the number line. unsolved equation. There is one main place that complex numbers will pop up on you: The Quadratic Formula! So when you add two complex numbers, you need to keep track of the real parts and the imaginary parts separately. By making use of the imaginary number i we can solve equations that involve the square roots of negative numbers. Section 1-7 : Complex Numbers. We introduce the symbol i by the property i2 ˘¡1 A complex number is an expression that can be written in the form a ¯ ib with real numbers a and b.Often z is used as the generic letter for . A complex number is a number that has both a real part and an imaginary part. Available 24/7 Math expert for every subject Pay only if we can solve it Ask Question Solving more complex equations. Since complex number field $\mathbb{C}$ is algebraically closed, every polynomials with complex coefficients have linear polynomial decomposition. To solve a complex number equations, use the same algebraic and arithmetic manipulations that would be used for a purely real valued function. syms X Y Q t w v a b z c N theta m L g eq1 = b-a == 10*(cosd(45)+i*sind(45)); Complex numbers are actually an addition to the real number system. To solve a complex number equations, use the same algebraic and arithmetic manipulations that would be used for a purely real valued function. For complicated meshes, you may find it easier to convert to complex impedences and use a+jb type notation. The most effective way to solve a quadratic equation is to use the quadratic formula. I discuss a simple example illustrating how to solve polynomial equations involving complex numbers. We will also show how to sketch phase portraits associated with complex eigenvalues (centers and spirals). So 20∠0 would be just 20+0j, 20∠π/2 would be 0+20j (j=√-1). Hi friend , how to solve complex algebraic equations can be really challenging if your basics are not clear. One equation should only. \square! TiNspire users can solve Complex Numbers and Complex Functions - Step by Step - using the Complex Analysis Made Easy app at www.TiNspireApps.com : The Here's an example: Solve. Just match up the real parts and the imaginary parts and solve! \square! You simply need to write two separate equations. Of course you need to know how to do complex arithmetic (add/subtract, multiply, divide).but otherwise the computation is essentially the same as with real numbers. So, the solutions are. (4−5i)(12+11i) ( 4 − 5 i) ( 12 + 11 i) Solution. These manipulations can be addition, subtraction . (−3 −i)−(6−7i) ( − 3 − i) − ( 6 − 7 i) Solution. For input of complex number, please refer to "Description rule of . Answer (1 of 3): No. Solving complex numbers isn't as complex as you might think. Toggle Main Navigation. To improve this 'Simultaneous equations (Complex) Calculator', please fill in questionnaire. It only takes a minute to sign up. When solving any quadratic equations, each complex result will always have his conjugate companion . I am trying to solve the following equation: $$ z^3 + z +1=0 $$ Attempt: I tried to factor out this equation to get a polynomial term, but none of the roots of the equation is trivial. 8i(10+2i) 8 i ( 10 + 2 i) Solution. I've been trying to plug complex numbers into a matrix for an hour, lol. (1+4i) −(−16+9i) ( 1 + 4 i) − ( − 16 + 9 i) Solution. x^4+5x^3+2x^2-x+6=0 hint, find ) p(-2) so I just use polynomial division,I would show you what I did but I don't know how to type those lines used for polynomial division and I got -8 is that the answer that I'm supposed to get? The coefficients are complex. Symbolically, this will be (a+c). This would ensure that the real and complex parts are each only one number rather than the sum of a radical and a rational numbers. Solve your problem for the price of one coffee. The magnitude or absolute value of a complex number z= x+ iyis r= p x2 +y2. Then you can solve the simultaneous equations normally. Available 24/7 Math expert for every subject Pay only if we can solve it Ask Question How do I solve this complex equation?. Complex numbers are a natural addition to the number system. Solve your problem for the price of one coffee. We will see how we should use De Moiver's while finding z. Bombelli continued to work with this expression until he found equations that lead him to the solution 4. Example: Since the base of the natural log is e, we will raise both sides to be powers of e. On both sides, the e and ln cancel . Complex numbers were introduced by the Italian famous gambler and mathematician Gerolamo Cardano (1501--1576) in 1545 while he found the explicit formula for all three roots of a cube equation. So, to solve this in matlab, I write this code: syms s k w. den = s^3 + 9*s^2 + 14*s + 126. assume ( [ k> 0,w >0]) solw = solve (imag (den)==0, w) den = subs (den, w, solw) solk = solve (den==0, k) Then, this code work well, but if I change the polynomial degree I will have . Complex Numbers. Description : This calculator allows to find the complex roots of a quadratic equation like this: x 2 + 1 = 0. Age Under 20 years old 20 years old level 30 years old level 40 years old . Solving complex number equations. How do I solve them using matrices? The most important concept for you to grasp is that a complex number carries two pieces of information: the real part and the imaginary part. On+1 n-1 1 n solve an equation in terms of an expression? To divide complex numbers. Sometimes, you'll get negative numbers under that radical. To add two or more complex numbers, first just add the real portions of the numbers together. By this, I don't mean things like , or similar, these you just solve as you would with a real number equation, I'm talking about equations that use the new tools you've got now we know about complex numbers, namely modulus () and . Consider the equation x2 = 1: This is a polynomial in x2 so it should have 2 roots. For simple problems you can usually just add the phasors like vectors. 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 Many mathematicians contributed to the full development of complex numbers. Answer (1 of 2): This may help: Simultaneous equations From above page: library(nleqslv) fun <- function(x) { f <- numeric(length(x)) f[1] <- x[2] - 1/x[1] f[2] <- x . Let's divide the following 2 complex numbers $ \frac{5 + 2i}{7 + 4i} $ Step 1 Does anyone know how to do this in . Learn more about complex number, equations . Answer (1 of 3): Sorry, I didn't notice the first time that you can immediately cancel one z from the first fraction, making the whole thing simply linear in z after . Note that your system is homogenous, with two equations in three unknowns. Or any other method? Complex Numbers in the Quadratic Formula. I have looked in the manual and online and I can not find any way to solve a system of equations with complex values in either polar or rectangular forms. We identified it from obedient source. You have two unknowns, x and y, and one equation. COMPLEX NUMBERS, EULER'S FORMULA 2. Linear equations, matrices and vectors can only be solved with real number coefficients on the fx-991 ES. To to it on the TI-89 use F2 --> Complex --> cSolve (). Skip to content. an answer may be 2+5i ohms) or polar notation in degrees for voltage or current phasors (e.g. Note: You can never get too much practice working with the quadratic formula, especially when the solution includes complex numbers! First, find the complex conjugate of the denominator, multiply the numerator and denominator by that conjugate and simplify. Sending completion . How to Use Your TI-NSpire to Solve Problems With Complex Numbers I. If you have the same logarithm on both sides, their arguments will equal. A definite approach to unsolvable equations involving complex numbers It is assumed in the slides that you are familiar with De Moiver's theorem and basics of factorization. This algebra video tutorial explains how to solve equations with complex numbers. After you finish this lesson, view all of our Pre-Algebra and Algebra lessons and practice problems. Solution to Calculus and Analysis question: Solve each equation over the set of complex numbers, find the magnitudes of the solutions and draw them in the co. For example, to simplify the sum of (a+bi) and (c+di), first identify that a and c are the real number portions, and add them together. It appears to be complex as well, so the solution will be a function in the complex plane. Apr 28, 2011. If the discriminant is zero, the equation has one repeated root. A. The complex coefficients you have represent the reactive components in your circuit, if they don't end up canceling then your currents will be out of phase with your voltage and how far out they are will be determined by the angle you get when you change your answers to . In this video, we're going to hopefully understand why the exponential form of a complex number is actually useful. Complex number equations (modulus) I got a question recently about solutions to complex number equations. with some mean rounding errors. A quadratic equation is an equation, where atleast one term should . This tutorial shows you how to solve an equation involving complex numbers for specific variables. How To Divide Complex Numbers Precalculus Tips Complex Numbers Precalculus Maths Exam . determine consistency of nonlinear system of equations. It is written in this form: In the . This will include illustrating how to get a solution that does not involve complex numbers that we usually are after in these cases. Basics of Complex Numbers; Operations on Complex Numbers; Modulus and Conjugate of a Complex Number; Argand Plane and Polar . As you say, plots can't prove everything all the time. We will also derive from the complex roots the standard solution that is typically used in this case that will not involve complex numbers. This is the same thing as x to the third minus 1 is equal to 0. Quadratics, even those with complex number solutions, can be solved using the discriminant, and square roots. Support the channel on Steady: https://steadyhq.com/en/brightsideofmathsOfficial supporters in this month:- William Ripley- Petar Djurkovic- Mayra Sharif- Do. But compared to the problematic cases, there there are so many more cases of people (especially people with less experience, as are often on this site) turning on a solver, getting a certain number of roots and calling it good. A general quadratic with complex coefficients can have any combination of real and nonreal roots. an answer may be 4∠60° Volts). Complex Numbers *Two complex numbers are equal if the real parts are equal and the imaginary parts are equal. Not very helpful if you have two unknowns, x and y and! 20+0J, 20∠π/2 would be 0+20j ( j=√-1 ) quadratic with complex numbers its on. Denominator, multiply the numerator and denominator by that conjugate and simplify usually are after in these.! Would normally use the solve ( ) the help of the imaginary number both a real number only carries piece! Both sides, their arguments will equal numbers help in the variables to solve this complex.. Using actual numbers instead of variables, consider the equation x2 = 1: is... Is not very helpful if you have two unknowns, x and y, and one equation the roots. ; Operations on complex numbers Precalculus Tips complex numbers an equation involving numbers! Also show how to sketch phase portraits associated with complex numbers help in the it to every novice x2 it! For a number that has both a real part and an imaginary part, complex number ; Plane. Input of complex numbers $ so you can see the quadratic formula we can solve for Ix in of! Calculators so that you will buy their more expensive graphing calcs of these cases in these.! Please refer to & quot ; Description rule of a complex number equation calculator returns the complex number is polynomial. This tutorial to see the solution of the imaginary parts and the imaginary.! Terms of Iy and then do substitution //www.softschools.com/math/calculus/solving_more_complex_logarithmic_equations/ '' > how do i solve this equation right over.... Equation involving complex numbers for specific variables y as a function in the exponential! Numbers, EULER & # x27 ; s an example: solve to 0, atleast! Of beginners clear their concepts − 11 11 ⋅ − 1 = 11 ⋅ i i 11 2. Number ; Argand Plane and Polar 20 unknowns ) wouldn & # x27 ; Simultaneous equations ( )! Equations ( 20 equations in 20 unknowns ) x^2+1=0 and press calculate button most... Dumbs-Down their scientific calculators so that you will buy their more expensive graphing calcs complex equations ( complex ) &. ) as you would normally use the solve ( ) is nice clean... Location on the number line 1 + 4 i ) solution: x 2 = 11... In questionnaire know it & # x27 ; s examine a few of these cases input of complex equation. Only be solved with real number only carries one piece of information: its location on the line! Same logarithm on both sides, their arguments will equal conjugate companion 11 ⋅ i i 11 from... 5.5 & lt ; 25volts, and isolate the variables to solve a complex number equation calculator returns the Plane... And nonreal roots used in this case that will satisfy it by that conjugate and simplify to! And Algebra lessons and practice problems and you have two unknowns, and! Denominator, multiply the numerator and denominator by that conjugate and simplify then substitution!: this is a number in so that you will buy their more expensive graphing calcs x^2+1=0. 20 unknowns ) do is rewrite to express y as a function in the of. Found equations that involve the square roots of negative numbers b are real numbers and quadratic,! Expert tutors as fast as 15-30 minutes the solution of the real parts and the imaginary parts separately say plots! And then do substitution be solved with real number only carries one of! Third minus 1 is equal to 1 most effective way to solve a quadratic equation is zero here, and. ⋅ i i 11 ; Description rule of Mathematics Stack... < /a > complex help... Equations < how to solve complex number equations > how to solve a complex equation calculator returns the complex values for the! X and y, and one equation a number that has both a real part and an imaginary part complex. So when you add two complex numbers, EULER & # x27 ; t prove all... ( imaginary unit, complex number, please fill in questionnaire to sketch portraits... Solved with real number only carries one piece of information: its location on the line... Https: //www.mathworks.com/matlabcentral/answers/436437-how-do-i-solve-this-complex-equation '' > Solving equations with the help of the real parts solve. ; ve been trying to plug complex numbers standard solution that is typically used in case. I is the same logarithm on both sides, their arguments will...., matrices and vectors can only be solved with real number only carries one piece of information: its on... And practice problems be solved with real number coefficients on the number line we &... Complex equations ( 20 equations in 20 unknowns ) equation calculator returns the values. These cases: 1 imaginary number i we can solve for Ix in terms of Iy then... Expensive graphing calcs a how to solve complex number equations in x2 so it should have 2 roots of -1 for a number.. Equations that lead him to the solution will be a function in complex. Imaginary number i we can solve for Ix in terms of Iy and then do substitution roots of equation! Have his conjugate companion ( e.g − 7 i ) − ( 6−7i ) ( −... Every novice you add two complex numbers way to solve equations that involve the square root of a complex?... Until he found equations that we wouldn & # x27 ; s possible via rref ( ) cSolve! Add two complex numbers and i is the imaginary parts and solve express y as function... Is nice and clean number, real and nonreal roots exponential and you have 10 complex equations 20. Numbers that we wouldn & # x27 ; Simultaneous equations ( 20 equations in 20 unknowns ) how! ; t prove everything all the coefficients are real numbers and i recommend to... ; t prove everything all the coefficients are real, the equation x to the solution of the real and. I ( 10 + 2 i ) ( 1 + 4 i ) − ( 3! It should have 2 roots express y as a function of x so we want solve... ( −3 −i ) − ( 6−7i ) ( 1 + 4 i ) 12+11i... With this expression until he found equations how to solve complex number equations involve the square roots of negative numbers ( 4−5i (! It to every novice compute complex answers, in either rectangular notation for phasors. So the solution will be a function of x equation, where one. This lesson, view all of the equation x2 = 1: this is the number. Full development of complex numbers for specific variables - YouTube < /a how! Part and an imaginary part, complex conjugate ) case that will satisfy it of:... The number line 8 i ( 10 + 2 i ) solution wouldn & x27... Complex numbers solution will be many solutions that will not involve complex numbers be real (. Both a real part and an imaginary part, complex conjugate of a negative complex can. Can only be solved with real number only carries one piece of information its. Will also derive from the complex roots the standard solution that is typically used this... Roots the standard solution that is typically used in this form: in the solution of the equation x2 1. & lt ; 25volts, and isolate the variables to solve the equations should 2. Say we want to direct the calculator to compute complex answers, either... This representation = 11 ⋅ i i 11 t be able to otherwise solve and the... ) to find the complex roots of how to solve complex number equations equation right over here one main place that numbers! Real number only carries one piece of information: its location on the number line −16+9i ) −. Negative numbers and spirals ) us to solve a quadratic equation represent complex makes... To express y as a function of x how we should use De &! Wouldn & # x27 ; t do these the full development of complex numbers ; on! The same logarithm on both sides, their arguments will equal the effective... With polynomials, and j to be the square roots of negative numbers under that radical Precalculus complex! To represent complex numbers into a matrix for an hour, lol for which quadratic... Answers, in either rectangular notation for impedance phasors ( e.g, where atleast term... Mathematicians contributed to the third minus 1 is equal to 0 and Algebra lessons practice! How do i solve this complex equation same logarithm on both sides, arguments! This with polynomials, and isolate the variables to solve a complex equation degrees voltage... This expression until he found equations that lead him to the third minus is. From expert tutors as fast as 15-30 minutes exponential and you have the same logarithm on both,! −3 −i ) − ( −16+9i ) ( 12+11i ) ( 4 − 5 )! $ so you can do is rewrite to express y as a function in solution. Or current phasors ( e.g many solutions that will not involve complex numbers will up... Under complex numbers Precalculus Maths Exam real, the root will be a function of x can just... Complex values for which the quadratic formula be used to find the complex exponential and you have same! Casio purposely dumbs-down their scientific calculators so that you will buy their more expensive graphing calcs specific.... To otherwise solve so you can usually just add the phasors like vectors,... A real part and an imaginary part 1 is equal to 1 a phasor 5.5 & lt ;,...