complex numbers Definition and Topics - 57 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. M

    Engineering Manipulating 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...
  2. L

    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)) =...
  3. e to the pi i for dummies

    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.
  4. K

    I Show that ##\mathbb{C}## can be obtained as 2 × 2 matrices

    I have this problem in my book: Show that ##\mathbb{C}## can be obtained as 2 × 2 matrices with coefficients in ##\mathbb{R}## using an arbitrary 2 × 2 matrix ##J## with a characteristic polynomial that does not contain real zeros. In the picture below is the given solution for this: I...
  5. A

    Exponential Wavefunction for Infinite Potential Well Problem

    Using the boundary conditions where psi is 0, I found that k = n*pi/a, since sin(x) is zero when k*a = 0. I set up my normalization integral as follows: A^2 * integral from 0 to a of (((exp(ikx) - exp(-ikx))*(exp(-ikx) - exp(ikx)) dx) = 1 After simplifying, and accounting for the fact that...
  6. M

    Find the set of points that satisfy:|z|^2 + |z - 2*i|^2 =< 10

    Hello everyone, I've been struggling quite a bit with this problem, since I'm not sure how to approach it correctly. The inequality form reminds me of the equation of a circle (x^2 + y^2 = r^2), but I have no idea how to be sure about it. Would it help just to simplify the inequality in terms...
  7. redtree

    I The domain of the Fourier transform

    Given the domain of the integral for the Fourier transform is over the real numbers, how does the Fourier transform transform functions whose independent variable is complex? For example, given \begin{equation} \begin{split} \hat{f}(k_{\mathbb{C}}) &= \int_{\mathbb{R}} f(z_{\mathbb{C}})...
  8. S

    I Principal difference between complex numbers and 2D vectors revisited

    I know this topic was raised many times at numerous forums and I read some of these discussions. However, I did not manage to find an answer for the following principal question. I gather one deals with the same set in both cases equipped it with two different structures (it is obvious if one...
  9. Mutatis

    Write ##5-3i## in the polar form ##re^\left(i\theta\right)##

    Homework Statement Write ##5-3i## in the polar form ##re^\left(i\theta\right)##. Homework Equations $$ |z|=\sqrt {a^2+b^2} $$ The Attempt at a Solution First I've found the absolute value of ##z##: $$ |z|=\sqrt {5^2+3^2}=\sqrt {34} $$. Next, I've found $$ \sin(\theta) = \frac {-3} {\sqrt...
  10. Cantor080

    I Existence of experience related to i

    If we have solution of an equation as x=1, it may be expressing, depending on context, 1 apple, 1 excess certain thing, etc. And, if we have solution of an equation as x=-1, it may be expressing, depending on context, 1 deficient apple, 1 deficient certain thing, etc. Is there any experience...
  11. T

    Solving ##z^{1+i}=##

    Homework Statement Find ##z## in ##z^{1+i}=4##. Is my solution correct Homework Equations ##\log(z_1 z_2)=\log(z_1)+\log(z_2)## such that ##z_1, z_2\in \{z\in\Bbb{C} : (z=x+iy) \land (x\in\Bbb{R}) \land -\infty \lt y \lt +\infty\}## ##re^{i\theta}=r(\cos\theta + i\sin\theta)## The Attempt at a...
  12. ubergewehr273

    Problem involving complex numbers

    Homework Statement Refer given image. Homework Equations Expansion of determinant. w^2+w+1=0 where w is cube root of 1. The Attempt at a Solution Expanding the determinant I got cw^2+bw+a-c=0. Well after that I have no idea how to proceed.
  13. pairofstrings

    I Please help me understand Complex Numbers

    Hi. If you have seen the above image which shows a parabola then you can also see that there is a colored portion of the parabola that have solution in "another dimension" - the "another dimension" can give me new numbers to form a solution of a function like f(x) = x2 + 1. 1. Is this "another...
  14. T

    Complex number - calculation

    Hi I was hoping some of you would give me a clue on how to solve this complex number task. z = (1 +(√3 /2) + i/2)^24 → x=(1 +(√3 /2), y= 1/2 I think there must be some nice looking way to solve it. My way was to calculate |z| which was equal to [√(3+2√3)]/2 → cosθ = x/|z|, sinθ= y/|z| After...
  15. M

    B Complex numbers imaginary part

    Hello everyone. Iam reading about complex numbers at the moment ad Iam quite confused. I know how to use them but Iam not getting a real understanding of what they actually are :-( What exactly is the imaginary part of a complex number? I read that it could in example be phase.... Thanks in...
  16. Mr Real

    I Constant raised to complex numbers

    It's not a homework question. I just thought up a method of finding answers to problems where a number is raised to a complex number and I need to know if I am right. If we have to find e^(i), can we do it by; first squaring it to get, e^(-1) which is 1/e and then taking its square root to get...
  17. CheesyPeeps

    Express z in the form a+ib

    Homework Statement I've used z* to mean z conjugate. Given the equation z + 2iz* = 8 + 7i, express z in the form a + ib. From SQA Advanced Higher Mathematics 2005 Exam Paper Homework Equations n/a The Attempt at a Solution I substituted a+ib and its conjugate in for z and z*, which, after...
  18. Poetria

    Complex exponentials - homework

    Could you give me a hint how to attack this problem? Find a complex number z = a+i*b such that f(t)=Re e^(z*t) where f(t)=cos(2*pi*t) I have begun as follows: e^((a+i*b)*t)=e^(a*t)*(cos(b)+i*sin(b)) Re e^(z*t)= e^(a*t)*cos(b) What to do now?
  19. K

    Complex Numbers problem

    Homework Statement If Z1+Z2+Z3=0 and Z1*Z2 + Z2*Z3 + Z3*Z1=0 and Z1, Z2, Z3 are all complex, what is the value of (|z1|+|z2|+|z3|)/(|z1*z2|+|z2*z3|+|z3*z1|) Homework Equations The Attempt at a Solution I tried to multiply the equations by the product of all conjugates and reach some...
  20. TheChemist_

    Determining graphical set of solutions for complex numbers

    Homework Statement So we have been doing complex numbers for about 2 weeks and there is this one equation I just can't solve. It's about showing the set of solutions in graphical form (on "coordinate" system with the imaginary and the real axis). So here is the equation: Homework Equations...
  21. J

    I Problem with this estimation lemma example

    I have been trying to show that $$\lim_{U\rightarrow\infty}\int_C \frac{ze^{ikz}}{z^2+a^2}dz = 0 $$ Where $$R>2a$$ and $$k>0$$ And C is the curve, defined by $$C = {x+iU | -R\le x\le R}$$ I have tried by using the fact that $$|\int_C \frac{ze^{ikz}}{z^2+a^2}dz| \le\int_C...
  22. whatphysics

    How do you work out simultaneous eqns w/ complex numbers & phasor

    I'm having trouble figuring out to get the answers from the 2 equations. The phasors and complex numbers confuse me. Do I need to change the phasor form? How do I go about doing this thanks! (Not homework question im trying to figure this for my exam!)
  23. R

    Stuck on expressing a complex number in the form (a+bi)

    Homework Statement Express the complex number (−3 +4i)3 in the form a + bi Homework Equations z = r(cos(θ) + isin(θ)) The Attempt at a Solution z = -3 + 4i z3 = r3(cos(3θ) + isin(3θ)) r = sqrt ((-3)2 + 42) = 5 θ = arcsin(4/5) = 0.9273 ∴ z3 = 53(cos(3⋅0.9273) + isin(3⋅0.9273)) a = -117 b...
  24. C

    Nonzero complex numbers

    Homework Statement Consider 3 nonzero complex numbers $$z_1,z_2,z_3$$ each satisfying $$z^2=i \bar{z}$$. We are supposed to find $$z_1+z_2+z_3, z_1z_2z_3, z_1z_2+z_2z_3+z_3z_1$$. The answers- 0, purely imaginary , purely real respectively. Homework Equations The Attempt at a Solution I...
  25. terhje

    Complex numbers. write equation on form "a+bi"

    Homework Statement Write this complex number in the form "a+bi" a) cos(-pi/3) + i*sin(-pi/3) b) 2√2(cos(-5pi/6)+i*sin(-5pi/6)) Homework Equations my only problem is that im getting + instead of - on the cosinus side.(real number) The Attempt at a Solution a) pi/3 in the unit circle is 1/2 for...
  26. Rectifier

    Complex numbers - factors

    The problem The following equation ##z^4-2z^3+12z^2-14z+35=0## has a root with the real component = 1. What are the other solutions? The attempt This means that solutions are ##z = 1 \pm bi##and the factors are ##(z-(1-bi))(z-(1+bi)) ## and thus ## (z-(1-bi))(z-(1+bi)) =...
  27. Ma Xie Er

    A Basic Spectral Analysis proof help

    I'm reading "Time Series Analysis and Its Applications with R examples", 3rd edition, by Shumway and Stoffer, and I don't really understand a proof. This is not for homework, just my own edification. It goes like this: Σt=1n cos2(2πtj/n) = ¼ ∑t=1n (e2πitj/n - e2πitj/n)2 = ¼∑t=1ne4πtj/n + 1 + 1...
  28. 5

    Complex polynomial help

    Homework Statement [/B] Suppose q(z) = z^3 − z^2 + rz + s, is a complex polynomial with 1 + i and i as zeros. Find r and s and the third complex zero. The Attempt at a Solution [/B] (z-(1+i)(z-i) = Z^2-z-1-2iz+i (Z^2-z-1-2iz+i)(z+d) = Z^3+z^2(d-1-zi)-z(d+1+2di-i)-d(1-i) Z^2 term...
  29. 5

    Complex numbers in the form a+bi

    Homework Statement How would I go about solving 1/z=(-4+4i) The answer that I keep on getting is wrong The Attempt at a Solution [/B] What I did z=1/(-4+4i)x(-4-4i)/(-4-4i) z=(-4-4i)/(16+16i-16i-16i^2) z=(-4-4i)/32 z=-1/8-i/8 This is the answer that I got however it says that it is...
  30. hamad12a

    How to find the third root of z^3=1?

    Homework Statement in a given activity: solve for z in C the equation: z^3=1 Homework Equations prove that the roots are 1, i, and i^2 The Attempt at a Solution using z^3-1=0 <=> Z^3-1^3 == a^3-b^3=(a-b)(a^2+2ab+b^2) it's clear the solution are 1 and i^2=-1 but i didn't find "i" as a solution...
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