# What is Complex numbers: Definition and 730 Discussions

In mathematics, a complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is a symbol called the imaginary unit, and satisfying the equation i2 = −1. Because no "real" number satisfies this equation, i was called an imaginary number by René Descartes. For the complex number a + bi, a is called the real part and b is called the imaginary part. The set of complex numbers is denoted by either of the symbols

C

{\displaystyle \mathbb {C} }
or C. Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers and are fundamental in many aspects of the scientific description of the natural world.Complex numbers allow solutions to all polynomial equations, even those that have no solutions in real numbers. More precisely, the fundamental theorem of algebra asserts that every polynomial equation with real or complex coefficients has a solution which is a complex number. For example, the equation

(
x
+
1

)

2

=

9

{\displaystyle (x+1)^{2}=-9}

has no real solution, since the square of a real number cannot be negative, but has the two nonreal complex solutions −1 + 3i and −1 − 3i.
Addition, subtraction and multiplication of complex numbers can be naturally defined by using the rule i2 = −1 combined with the associative, commutative and distributive laws. Every nonzero complex number has a multiplicative inverse. This makes the complex numbers a field that has the real numbers as a subfield. The complex numbers form also a real vector space of dimension two, with {1, i} as a standard basis.
This standard basis makes the complex numbers a Cartesian plane, called the complex plane. This allows a geometric interpretation of the complex numbers and their operations, and conversely expressing in terms of complex numbers some geometric properties and constructions. For example, the real numbers form the real line which is identified to the horizontal axis of the complex plane. The complex numbers of absolute value one form the unit circle. The addition of a complex number is a translation in the complex plane, and the multiplication by a complex number is a similarity centered at the origin. The complex conjugation is the reflection symmetry with respect to the real axis. The complex absolute value is a Euclidean norm.
In summary, the complex numbers form a rich structure that is simultaneously an algebraically closed field, a commutative algebra over the reals, and a Euclidean vector space of dimension two.

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1. ### A "spiral" in the Complex plane

I understand that the "spiral" converges to 1+i-1/2-i/3!+1/4!+i/5!-1/6!-i/7!... . It splits into two: one for Re, 1-1/2+1/4!-1/6!..., and the other for Im, 1-1/3!+1/5!-1/7!... . Any hints on how to compute them?
2. ### What's wrong in this manipulation of a square root of a negative number?

For context, I have a 2x2 system of linear first order differential equations $$\vec{x}'=A\vec{x}$$ where $$A=\begin{bmatrix} 0 & 1\\c&-2\end{bmatrix}$$ and the characteristic polynomial is $$\lambda^2+2\lambda-c=0$$ The two eigenvalues are ##-1\pm\sqrt{1+c}##. Suppose ##c<-1##. Then...
3. ### I Formal definition of multiplication for real and complex numbers

I know that the definition of multiplication for integers is just repeated addition. For example, 5 times 3 means 5 + 5 + 5, but what about if we want to extend this definition to real or complex numbers ? Like for example, what does pi times e mean ? How are we supposed to add pi to itself e...
4. ### Find the modulus and the argument of ##\dfrac{2}{(4-2i)^2}##

In my lines i have, ##(4-2i)^2 = (4-2i)(4-2i)## ##r^2 = 4^2 + (-2)^2 = 20## ##r \cos θ = 4## and ##r\sin θ = -2## ##\tan θ =-\dfrac{1}{2}## ##⇒θ = 5.82## radians. Therefore, ##|(4-2i)^2| = \sqrt{20} ×\sqrt{20} = 20## Argument = ##5.82 + 5.82 = 11.64##. also ##|2|## = ##2## and argument =...
5. ### Deduce orthogonality relations for sine and cosine w/ Euler's Formula

To solve part (a), we write ##e^{inx}e^{-imx}=e^{ix(n-m)}##. If ##m=n## then this expression is 1, and so the integral of 1 from 0 to ##2\pi## is ##2\pi##. If ##m\neq n## then we use Euler's formula and integrate. The result is zero. My question is how do we solve part (b) using part (a)? I...
6. ### How to find the cube roots of a real number not equal to 1?

There is that term ##5^{1/3}## but that is exactly what we're trying to find.
7. ### What is a point of using complex numbers here?

Firstly, the exercise itself is not difficult: On one hand, $$|(a + ib)(c + id)|^2 = |a + ib|^2|c + id|^2 = (a^2 + b^2) (c^2 + d^2) = MN.$$ On the other hand, ##(a + ib)(c + id) = p+ iq## for some integers p and q, and so $$|(a + ib)(c + id)|^2 = |p + iq|^2 = p^2 + q^2.$$ Thus, ##MN = p^2 +...
8. ### B Understanding the Relationship Between i*cos and sin in Circuit Analysis

In circuit analysis, everything seems to work out when you set i*cos = sin. But thats not a legitimate equation, so why does that work? Is there a proof that this is a real equation?
9. ### B Getting from complex domain to real domain

Hi! I am ok with understanding Euler's formula and how its proven. It is basic mathematic operations that are made possible by the characteristics of i, cos, sin, and exp. What still makes me uncomfortable is the jump we make at the very beginning or end of calculations, basically Acosx <==>...
10. ### B What are complex numbers good for?

I was thinking of investigating field theory because i like reading about quantum interpretations. What role does complex numbers have in physics? I have a hard time seeing why properties of a point in that field are not just multi dimensional properties on some parameter space? Why start...
11. ### I Formula for the propagation of complex errors

If I have 2 measurements ##x = (3.0 ± 0.1), y = (-2.0 ± 0.1)## and want to calculate how the error propagates when calculating a function from those values this formula should be used: ##f(x, y) = f(x, y) ± \sqrt {(\frac{\partial f}{\partial x}*\Delta x)^2+(\frac{\partial f}{\partial y}*\Delta...
12. ### I Fourier Transformation on a Ring

Hello, I have a question about the FFT and would like to share my thoughts with you. The background is a problem 30.2-6 from the legendary algorithms book by Cormen. It says that instead of doing an n element FFT over the field of complex numbers, we can use ##\mathbb{Z}_m##, where ##m = 2^{t...
13. ### Use of i and j in complex numbers

Is there any particular reason as to why certain texts use ##j## and others ##i## when looking at complex numbers? Maths is a relatively easy subject but at times made difficult with all this mix-up... i tend to use a lot of my time in trying to understand author's language and this is also...
14. ### Adding square roots of ##i## leading to different answers!

Statement of the problem : We have to find what is ##\sqrt{i} + \sqrt{-i}## First Attempt (Euler's Formula) : I use the Euler's formula (see Relevant Equations 1) above which yields ##i = e^{i\frac{\pi}{2}}##. Likewise ##-i = e^{i\left(-\frac{\pi}{2}\right)}##. Now I evaluate where, in the...
15. ### B Calculating non-dielectric reflectance without using complex numbers

I need a to calculate the fresnel reflection ratio of a non dielectric material given the incident angle, the refractive indexes of the incident and interfacing materials and the extinction coefficient of the interfacing material. I need to to this without directly using complex numbers, due to...
16. ### I How can you represent a point by "z = x + iy" as shown here?

Snapshot of Mary L. Boas' Mathematical Physics book So, the marked lines say If we think of P as the point z = x +iy in the complex plane, we could replace (2.3) by a single equation to describe the motion of P But, until now I have only learned of representing points in the form (x,y), now...
17. ### I Continuity of Quotient of Complex Values

Hey all, I have a very simple question regarding the quotient of complex values. Consider the function: $$f(a) = \sqrt{\frac{a-1i}{a+1i}}$$ where ##i## is the imaginary unit. When I evaluate f(0) in Mathematica, I get ##f(0) = 1i##, as expected. But if I evaluate at a very small value of ##a##...
18. ### I What Are Examples of Alternate Orderings in Mathematics?

In another thread This has me curious about "ordering other than our normal ordering." What does this mean? I take it that "normal ordering (of integers)" is ... 0, 1, 2, 3... Do mathematicians consider alternate orderings like ...0, 2, 1, 3... That doesnt seem to make sense to me, that's...
19. ### Solve the problem involving complex numbers

Hello guys, I am refreshing on complex numbers today; kindly see attached. ok for part (a) this is a circle with centre ##(\sqrt{3}, -1)## with radius =##1## thus, we shall have,
20. ### 3x3 matrix with complex numbers

The attempt at a solution: I tried the normal method to find the determinant equal to 2j. I ended up with: 2j = -4yj -2xj -2j -x +y then I tried to see if I had to factorize with j so I didn't turn the j^2 into -1 and ended up with 2 different options: 1) 0= y(-4j-j^2) -x(2j-1) -2j 2)...
21. ### I A question about Young's inequality and complex numbers

Let ##\Omega## here be ##\Omega=\sqrt{-u}##, in which it is not difficult to realize that ##\Omega ## is real if ##u<0##; imaginary, if ##u>0##. Now, suppose further that ##u=(a-b)^2## with ##a<0## and ##b>0## real numbers. Bearing this in mind, I want to demonstrate that ##\Omega## is real. To...
22. ### Find the GCD of the given complex numbers (Gaussian Integers)

Hello guys, I am able to follow the working...but i needed some clarification. By rounding to the nearest integer...did they mean? ##z=1.2-1.4i## is rounded down to ##z=1-i##? I can see from here they came up with simultaneous equation i.e ##(1-i) + (x+iy) = \dfrac{6}{5} - \dfrac{7i}{5}## to...
23. ### Using complex numbers to solve for a current in this circuit

First I solved 4+j3, which I squared 4 and 3 to equal 16 and 9 then I added them to get 25 and then I got the square root of 25 = 5. Then I plugged it back in to the equation. [50/(5)(50)+100] x 150 to get 50/350x 150= 1/7(150)= 21.42. I've attached the correct answer.
24. ### I Newton-Raphson Method With Complex Numbers

I'm trying to code Newton Raphson's method for finding zeros. I realize that even if the solution is real, it's possible for guesses to be complex. For example: $$y=\sqrt{x-6}-2$$ While 10 is a valid real root, for any guess less than 6, the result is complex. I tried to run the code allowing...
25. ### Complex numbers problem |z| - iz = 1-2i

Here is my attempt(photo below), but somehow the solution in the textbook is z= 2 - (3/2)i, and mine is z=(-3/2) +2i. Can someone please tell me where I am making a mistake? I suppose it's something with x being part of the real part of the 1st complex number and x being part of an imaginary...
26. ### How Can I Solve a System of Equations With Complex Numbers?

How can I solve a system of equations with complex numbers 2z+w=7i zi+w=-1 I have tried substituting z with a+bi and I have tried substituting w=7i-2z but didn't get anything useful. Edit: also, I've tried, multiplying lower eq. with -1 so that I can cancel w but I get stuck with 2z and zi and...
27. ### Determine if the given set is Bounded- Complex Numbers

My interest is only on part (a). Wah! been going round circles to try understand why the radius = ##2##. I know that the given sequence is both bounded and monotonic. I can state that its bounded above by ##1## and bounded below by ##0##. Now when it comes to the radius=##2##, i can also say...
28. ### Solve ##z^2(1-z^2)=16## using Complex numbers

The problem is as shown...all steps are pretty easy to follow. I need help on the highlighted part in red. How did they come to; ##z^4+8z^2+16-9z^2=0## or is it by manipulating ##-z^2= 8z^2-9z^2?## trial and error ...
29. ### B Phase Difference of Current & Voltage: Capacitors, Inductors & Complex Numbers

how does capacitors and inductors cause phase difference between current and voltage? how does complex number come into play to explain the relation between phase of current and voltage?
30. ### I Real ODE yields real solution through complex numbers

Hello, I'm posting here since what follows is not about homework, but constitutes a personal research which underlies some more general questions. As with the infamous "casus irreducibilis" (i.e. finding the real roots of a cubic function sometimes requires intermediate calculations with...
31. ### Prove by induction the sum of complex numbers is complex number

See the work below: I feel like it that I did it correctly. I feel like I skip a step in my induction. Please point any errors.
32. ### Can we use criss-cross approach with complex number equations?

I am not sure why criss-cross approach would work here, but it seems to get the answer. What would be the reason why we could use this approach? $$\frac {z-1} {z+1} = ni$$ $$\implies \frac {z-1} {z+1} = \frac {ni} {1}$$ $$\implies {(z-1)} \times 1= {ni} \times {(z+1)}$$
33. ### Need help with a question about powers of complex numbers

(z-3)3=-8, solve for z. I'm new to complex numbers, so I'm stuck on this basic problem: how do you find all real and non-real solutions in the equality, (z-3)^3=-8? Thanks a bunch.
34. ### Find ##z## in the form ##a+bi## under Complex Numbers

For part (a), ##z##=##\dfrac {3+i}{3-i}## ⋅##\dfrac {3+i}{3+i}## ##z##=##\dfrac {4}{5}##+##\dfrac {3}{5}i## part (b) no problem as long as one understands the argand plane... For part (c) Modulus of ##z=1## and Modulus of ##z-z^*##=##\frac{6}{5}i##
35. ### Prove that ##12≤OP≤13## in the problem involving complex numbers

Find the question below; note that no solution is provided for this question. My approach; Find part of my sketch here; * My diagram may not be accurate..i just noted that, ##OP## takes smallest value of ##12## when ##|z+5|=|z-5|## i.e at the end of its minor axis and greatest value ##13##...
36. ### Prove that ##c^2+d^2=1## in the problem involving complex numbers

Easy questions, just a lot of computation... $$\frac {z}{z^*}=\frac {a+bi}{a-bi} ×\frac {a+bi}{a+bi}$$ $$c+di=\frac {a^2-b^2}{a^2+b^2}+\frac {2abi}{a^2+b^2}$$ $$⇒c^2= \frac {a^4-2a^2b^2+b^4}{(a^2+b^2)^2}$$ $$⇒d^2= \frac {4a^2b^2}{(a^2+b^2)^2}$$ Therefore...
37. ### Solve this pair of simultaneous equations involving complex numbers

$$(1+i)z+(2-i)w=3+4i$$ $$iz+(3+i)w=-1+5i$$ ok, multiplying the first equation by##(1-i)## and the second equation by ##i##, we get, $$2z+(1-3i)w=7+i$$ $$-z+(-1+3i)w=-5-i$$ adding the two equations, we get ##z=2##, We know that, $$iz+(3+i)w=-1+5i$$ $$⇒2i+(3+i)w=-1+5i$$...
38. ### Proving geometric sum for complex numbers

I went ahead and tried to prove by induction but I got stuck at the base case for ## N =1 ## ( in my course we don't define ## 0 ## as natural so that's why I started from ## N = 1 ## ) which gives ## \sum_{k=0}^1 z_k = 1 + z = 1+ a + ib ## . I need to show that this is equal to ## \frac{1-...

40. ### MHB Complex numbers such that modulus less than or equal to 1.

https://www.physicsforums.com/attachments/11378
41. ### Engineering Solving Problems Involving Complex Vectors

Hi Here is my attempt at a solution for problems 1) and 2) that can be found within the summary. Problem 1) a = 3-2i b= -6-4i c= 4+ 6i d= -4+3i Now, to calculate each vector modulus, I applied the following formula: $$\left| Vector modulus \right| = \sqrt {(a^2 + b^2) }$$ where a = real part...
42. ### A Interpretation of Lagrangian solution (complex numbers)

Hi Guys Finally after a great struggle I have managed to develop and solve my Lagrangian system. I have checked it numerous times over and over and I believe that everything is correct. I have attached a PDF which has the derivation of the system. Please ignore all the forcing functions and...
43. ### Analysis 1 Homework Help with Complex Numbers

I need help actually creating the proof. I've done the scratch needed for the problem, it's just forming the proof that I need help in. Bar(a+bi/c+di)= (a-bi) / (c-di) Bar ((a+bi/c+di)*(c-di/c-di)) = ((a-bi/c-di)*(c+di/c+di)) Bar((ac+bd/c^2 +d^2)+(i(bc-ad)/c^2+d^2)) =...
44. ### B Real numbers and complex numbers

To find √(-2)√(-3). Method 1. √(-2)√(-3) = √( (-2)(-3) ) = √(6). Method 2. √(-2)√(-3) = √( (-1)(2) )√( (-1)(3) ) = √((-1)√(2)√(-1)√(3) = i√(2)i√(3) = (i)(i)√(2)√(3) = -1√( (2)(3) ) =-√6. Why don't the two methods give the same answer? Thanks for any help.
45. ### e to the pi i for dummies

This Mathologer video explains e raised to the i pi in a way that even Homer Simpson can understand.
46. ### Scholarship exam exercise about complex numbers - Can't solve

Hello, I have this (I am solving scholarship exams)math problem and I don't quite know what to do with it , Could You please help? The exercise is about complex numbers and it says: Calculate in the algebraic form(a+bi) I thought on applying substitution since -1=i^2 and z is the real part but...
47. ### I Questions about the arg of complex numbers

Hi PF community, I'm reading about complex numbers and i have some questions about the argument of a complex number that i can't solve with Google or reading again the same page. Well, my first doubt is about , i can't understand where come this and why there is some random integer, i...
48. ### Using complex numbers to model 3 phase AC

Assume a transformer as above, with 230V L-N, and I want to work out the L-L voltage. A phasor diagram will show me that the voltages are 120° out of phase. (230∠0°) + (230∠120°) = (230cos0 + j230sin0) + (230cos120 + j230sin120) = 230 + (-115 + j199.2) 115 + j199.2 = 230∠60 What I’m looking...
49. ### RC circuit using complex numbers

The impedance Z = R -j/wC + ##\frac{1}{\frac{1}{R} - \frac{\omega C}{j}}## But,1/wC=R So, solving this, I find: Z= 3R/2(1-j) |Z| =##\frac{3R}{\sqrt 2}## I =##\frac{V_i \sqrt 2} {3R}## Vi - IR-IXc =Vo Solving this, ##Vo = V_i -\frac {V_i \sqrt 2}{3} - \frac{V_i \sqrt 2}{3R} \frac{-j}{wC}##...
50. ### TI-83+ Graphing Calculator giving wrong answer? (Complex numbers)

I think the solution should be: METHOD #1: \begin{align} (\sqrt[4] {-1})^4 & = (-1)^{\frac 4 4} \nonumber \\ & = (-1)^1 \text{, can reduce 4/4 since base is a constant and not a variable in ℝ} \nonumber \\ & = -1 \nonumber \end{align} Alternatively, METHOD #2 for same answer is...