What is Complex numbers: Definition and 715 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.
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...
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##...
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...
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,
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)...
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...
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...
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.
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...
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...
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...
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...
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 ...
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?
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...
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?
$$\frac {z-1} {z+1} = ni$$
$$\implies \frac {z-1} {z+1} = \frac {ni} {1}$$
$$\implies {(z-1)} \times 1= {ni} \times {(z+1)}$$
(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.
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##
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##...
$$(1+i)z+(2-i)w=3+4i$$
$$iz+(3+i)w=-1+5i$$
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, $$iz+(3+i)w=-1+5i$$
$$⇒2i+(3+i)w=-1+5i$$...
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-...
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...
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...
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)) =...
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...
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...
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...
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...
Anyone know what the simplest possible self-contained numeral system for complex numbers would be, analogous to signed ternary for integers? My guess would be quarter-imaginary base (https://en.wikipedia.org/wiki/Quater-imaginary_base.)
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...
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...
Solve ##Z^2\bar{Z}=8i##
i am confused on how to proceed
i have tried to substitute ##z=a+ib## solve the conjugate and the square, then separate the real from the imaginary and put all in a system, but becomes too complicated
Summary:: Hello, my question asks if the complex exponential equation 4e^(-j) is equal to 4 ∠-180°. I tried to use polar/rectangular conversions: a+bj=c∠θ with c=(√a^2 +b^2) and θ=tan^(-1)[b/a]
4e^(-j)=4 ∠-180°
c=4, 4=(√a^2 +b^2)
solving for a : a=(√16-b^2)
θ=tan^(-1)[b/a]= -1
b/(√16-b^2)=...
Let $z_1=18+83i,\,z_2=18+39i$ and $z_3=78+99i$, where $i=\sqrt{-1}$. Let $z$ be the unique complex number with the properties that
$\dfrac{z_3-z_1}{z_2-z_1}\cdot \dfrac{z-z_2}{z-z_3}$ is a real number and the imaginary part of $z$ is the greatest possible. Find the real part of $z$.
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...
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}})...
Hello! :smile:
I am locked in an exercise.
I must find (and graph) the complex numbers that verify the equation:
##z^2=\bar z^2 ##
If ##z=x+iy## then:
##(x+iy)^2=(x-iy)^2 ##
and operating and simplifying,
##4.x.yi=0 ##
and here I don't know how to continue...
can you help me with ideas?
thanks!
reducing it to various forms: for example, the one in the title, or 2*pi*k(ln m) = a(ln(n/m)), and so forth. My gut feeling is that it is true (that no such foursome exists), but manipulations have not got me anywhere. Anyone push me in the right direction? I am probably overlooking something...
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...