Questions involving complex numbers: numbers of the form $a+bi$ where $i^2=-1$.
11
votes
4answers
738 views
Equation of the complex locus: $|z-1|=2|z +1|$
This question requires finding the Cartesian equation for the locus:
$|z-1| = 2|z+1|$
that is, where the modulus of $z -1$ is twice the modulus of $z+1$
I've solved this problem algebraically ...
11
votes
2answers
278 views
A property of roots of the truncated series for $\sin(x)$
Let $p_n(x) = \sum\limits_{k=0}^n \frac{(-1)^kx^{2k+1}}{(2k+1)!}$
In other words, $p_n$ is the polynomial made of the first $n$ terms of the Taylor expansion of $\sin(x)$ around $x = 0$.
...
10
votes
4answers
430 views
Which step in this process allows me to erroneously conclude that $i = 1$
I was playing around with imaginary numbers and exponents and came up with this:
$$ i = \sqrt{-1} $$
$$ \sqrt{-1} = (-1)^{1/2} $$
$$ (-1)^{1/2} = (-1)^{2/4} $$
$$ (-1)^{2/4} = ((-1)^{2})^{1/4} ...
10
votes
6answers
465 views
Complex roots of $z^3 + \bar{z} = 0$
I'm trying to find the complex roots of $z^3 + \bar{z} = 0$ using De Moivre.
Some suggested multiplying both sides by z first, but that seems wrong to me as it would add a root ( and I wouldn't know ...
10
votes
4answers
770 views
How do you integrate imaginary numbers?
How would you find, for instance, $\int_{0}^{4} i\> x dx$? Can you just treat $i$ as a constant, or do you have to do something more sophisticated?
Thanks!
10
votes
5answers
389 views
imaginary numbers - how can I understand them - especially as they occur in 'roots' of polynomials?
In another question here, about roots of equations being imaginary, the accepted answer
said something interesting about "imaginary" (as a technical word in math) not meaning "not real". I ...
10
votes
4answers
732 views
Why is it that Complex Numbers are algebraically closed?
I find it curious that Complex Numbers give enough flexibility to be algebraically closed, where the reals, rational numbers do not. For the reals it is easy to see that they cannot be used to solve ...
10
votes
3answers
198 views
What's $(-1)^{2/3}\; $?
I know that
$\left ( -1 \right )^{2/3}=\left ( \left ( -1 \right )^{2} \right )^{1/3}=1$
But Matlab computes this as $- 0.5 + 0.8660254038i$ a complex number.Why?
10
votes
2answers
170 views
Is it possible to construct an ordered field which is also algebraically closed?
It is well known that while the real numbers are totally ordered, they are not algebraically closed, and while the complex numbers are algebraically closed, they are not totally ordered. Is it ...
9
votes
7answers
1k views
Is there a formula for $(1+i)^n+(1-i)^n$?
I'm wondering if there is a formula for the value of $(1+i)^n+(1-i)^n$?
I calculated the first terms starting with $n=1$ to be, in order, $2$, $0$, $-4$, $-8$, $-8$, $0$, $16$, $\dots$
So it seems ...
9
votes
6answers
558 views
inequality involving complex exponential
Is it true that
$$|e^{ix}-e^{iy}|\leq |x-y|$$ for $x,y\in\mathbb{R}$? I can't figure it out. I tried looking at the series for exponential but it did not help.
Could someone offer a hint?
9
votes
5answers
820 views
How to compute $\sqrt{i + 1}$ [duplicate]
Possible Duplicate:
How do I get the square root of a complex number?
I'm currently playing with complex numbers and I realized that I don't understand how to compute $\sqrt{i + 1}$. My ...
9
votes
8answers
539 views
For complex $z$, $|z| = 1 \implies \text{Re}\left(\frac{1-z}{1+z}\right) = 0$
If $|z|=1$, show that: $$\mathrm{Re}\left(\frac{1 - z}{1 + z}\right) = 0$$
I reasoned that for $z = x + iy$, $\sqrt{x^2 + y^2} = 1\implies x^2 + y ^2 = 1$ and figured the real part would be:
...
9
votes
2answers
947 views
Do “imaginary” and “complex” angles exist?
During some experimentation with sines and cosines, its inverses, and complex numbers, I came across these results that I found quite interesting:
$ \sin ^ {-1} ( 2 ) \approx 1.57 - 1.32 i $
$ \sin ...
9
votes
3answers
203 views
Putting ${n \choose 0} + {n \choose 5} + {n \choose 10} + \cdots + {n \choose 5k} + \cdots$ in a closed form
As the title says, I'm trying to transform $\displaystyle{n \choose 0} + {n \choose 5} + {n \choose 10} + \cdots + {n \choose 5k} + \cdots$ into a closed form. My work:
$\displaystyle\left(1 + ...
9
votes
3answers
212 views
square root of $1/2 + \sqrt3/2?$
Playing with Maple, I noticed that it gives the square root of $c = 1+\frac{\sqrt3}{2}$ as equal to $a = \frac{1}{2}+\frac{\sqrt3}{2}$.
Indeed it checks out. But I got curious: how can I find that ...
9
votes
3answers
446 views
Why this proof $0=1$ is wrong?(breakfast joke)
We have $$e^{2\pi i n}=1$$
So we have $$e^{2\pi in+1}=e$$
which implies $$(e^{2\pi in+1})^{2\pi in+1}=e^{2\pi in+1}=e$$
Thus we have $$e^{-4\pi^{2}n^{2}+4\pi in+1}=e$$
This implies ...
9
votes
2answers
463 views
Suppose $a \in \mathbb{R}$, and $\exists n \in \mathbb{N}$, that $a^n \in \mathbb{Q}$, and $(a + 1)^n \in \mathbb{Q}$
1)
Suppose $a \in \mathbb{R}$, and $\exists n \in \mathbb{N}$, that $a^n \in \mathbb{Q}$, and $(a + 1)^n \in \mathbb{Q}$.
Prove: Is it true that $a \in \mathbb{Q}$?
2) Suppose $a \in \mathbb{C}$, ...
9
votes
1answer
346 views
Is $\mathbb{C}^*$ modulo the roots of unity isomorphic to $\mathbb{R}^+$?
A student came to me showing a question from his exam in basic group theory, in which they are asked to prove that $\mathbb{C}^*$ modulo the subgroup of roots of unity is isomorphic to $\mathbb{R}^+$ ...
9
votes
2answers
280 views
If the sequence $(a_n b_n)$ converges and $a_n \to 0$, when does $(b_n)$ converge too?
Given that the sequence $(a_n b_n)$ converges and $a_n \to 0$, are there conditions which can be placed on $(b_n)$ and/or $(a_n b_n)$ so that $(b_n)$ converges as well?
9
votes
3answers
306 views
Help me prove $\sqrt{1+i\sqrt 3}+\sqrt{1-i\sqrt 3}=\sqrt 6$
Please help me prove this Leibniz equation: $\sqrt{1+i\sqrt 3}+\sqrt{1-i\sqrt 3}=\sqrt 6$. Thanks!
8
votes
4answers
386 views
Is it correct to say that $\mathbb{R}$ has fewer elements than $\mathbb{C}$ if both are infinite?
My math teacher said that. I disagreed, but he said that I was wrong. But I'm not convinced - is it really right? Please notice that I'm not talking about $\mathbb{R}$ $⊂$ $\mathbb{C}$, but ...
8
votes
4answers
375 views
Is there a problem in defining a complex number by $ z = x+iy$?
The field $\mathbb{C} $ of complex numbers is well-defined by the Hamilton axioms of addition and product between complex numbers, i.e., a complex number $z$ is a ordered pair of real numbers $(x,y)$, ...
8
votes
3answers
300 views
Proof for law of complex exponents using only differential equation
I just read that an elegant proof exists that the law of exponents also holds for complex numbers ($a,b,z$ all complex): $$e^{a+b}=e^ae^b,$$ which only uses the definition that $$y=e^{zt}$$ is a ...
8
votes
3answers
243 views
How to prove $|z_1-z_2| \geq |z_1|-|z_2|$ in other way than this?
How to prove $|z_1-z_2| \geq |z_1|-|z_2|$ in other way than this? I mean I tried to find on the internet but could not find. I ask for more straighforward way than the proof that is presented for item ...
8
votes
5answers
638 views
Alternative to imaginary numbers?
In this video, starting at 3:45 the professor says
There are some superb papers written that discount the idea that we should ever use j (imaginary unit) on the grounds that it conceals some ...
8
votes
5answers
453 views
Is there any way to represent an imaginary number?
Is there any way to represent an imaginary number? Like the square root of -1? Is there any possible way to do this?
Sorry if you think this is a dumb question. I am a 7th grade student in ...
8
votes
2answers
213 views
Summation of complex numbers
This is a series problem where the terms are complex numbers. I am looking for a better approach to solving this problem.
If $\displaystyle z = \frac{1+i}{\sqrt2}$, Evaluate $1 + z + z^2 + ... + ...
8
votes
3answers
133 views
What is the right treatment for $0^i$?
I need to calculate a limit of a complex expression (had it in a physics research) that contains a term $(r-b)^p$ for $r\rightarrow b+$ where $r,b$ are reals, and $p$ is complex, let's suppose for ...
8
votes
2answers
249 views
If $0$, $z_1$, $z_2$ and $z_3$ are concyclic, then $\frac{1}{z_1}$,$\frac{1}{z_2}$,$\frac{1}{z_3}$ are collinear
If the complex numbers $0$, $z_1$, $z_2$ and $z_3$ are concyclic, prove that $\frac{1}{z_1}$,$\frac{1}{z_2}$,$\frac{1}{z_3}$ are collinear.
I really can't seem to get anywhere on this problem, ...
7
votes
7answers
1k views
What's the thing with $\sqrt{-1} = i$
What's the thing with $\sqrt{-1} = i$? Do they really teach this in the US? It makes very little sense, because $-i$ is also a square root of $-1$, and the choice of which root to label as $i$ is ...
7
votes
2answers
286 views
Limit of complex function
Im trying to find the limit of:
$$ \frac{\operatorname{Re}(z) \operatorname{Im}(z)}{z^2}$$
as z tends to zero.
7
votes
2answers
159 views
How does one find $z\in \mathbb{C}$ such that $\sin z=100?$
I am self-studying Complex Analysis and I am suppose to find $z\in \mathbb{C}$ such that $\sin z=100.$ I know that $$\sin z=\sin x \cosh y+i\cos x\sinh y$$
So I must have $\sin x \cosh y=100.$ I ...
7
votes
2answers
154 views
Invariant under transformation $i\mapsto -i$ implies real?
When one has an expression in terms of $i$, one can send $i$ to $-i$ and, if the expression remains unchanged, one can conclude that the expression is, in fact, real. Analogous statements hold for ...
7
votes
4answers
273 views
Has anyone talked themselves into understanding Euler's identity a bit?
What does the ratio of the circumference of a circle to its diameter have to do with the base of the natural logarithm and $\sqrt{-1}$?
7
votes
5answers
241 views
When are we (not) allowed to replace $x$ by $ix$?
It seems to be quite a common manipulation to replace $x$ by $ix$.
Every time I see it's being done in a textbook, I blindly trust the author without really understanding when are we allowed to do so ...
7
votes
3answers
129 views
How do I completely solve the equation $z^4 - 2z^3 + 9z^2 - 14z + 14 = 0$ where there is a root with the real part of $1$.
I would please like some help with solving the following equation:
$$z^4 - 2z^3 + 9z^2 - 14z + 14 = 0$$
All I know about the equation is that there is a root with the real part of $1$.
My approach ...
7
votes
2answers
78 views
How to show that $\overline{zw}=\overline{z}\,\overline{w}$?
I thought about first multiplying two complex which aren't in the conjugate form:
$$zw=a c+i a d+i b c-b d$$
Then multiply two complex conjugates:
$$\overline{z}\,\overline{w}=a c\color{red}{-}i a ...
7
votes
5answers
159 views
If $A,B,C,D$ are complex numbers on the unit circle with $A+B+C+D=0$, then they form a rectangle
Let $A, B, C, D$ be points on a unit circle. Prove that if $A+B+C+D=0$, then $A,B,C,D$ make a rectangle. (Use complex numbers.)
How do I prove this? I tried to use the dot product of 2 adjacent ...
7
votes
2answers
276 views
Why isn't $\log(-1)=i\pi$?
Reading http://people.math.gatech.edu/~cain/winter99/ch3.pdf, $\log(z)$ is defined as $=\ln|z|+i\arg(z)$. Looking on the Wessel plane, isn't $\arg(-1)=\pi$ (more generally $\pi \pm 2 \pi n$)? And ...
7
votes
3answers
114 views
Connectedness of $\lbrace z\in\mathbb{C} : |z^2+az+b|<r\rbrace$
What are the values of $r$ for which the set $$\lbrace z\in\mathbb{C} : |z^2+az+b|<r\rbrace$$ is connected ? Here $a,b\in\mathbb{C}$ and $r\in\mathbb{R}$.
7
votes
3answers
856 views
Drawing $z^4 +16 = 0$
I need to draw $z^4 +16 = 0$ on the complex numbers plane.
By solving $z^4 +16 = 0$ I get:
$z = 2 (-1)^{3/4}$ or $z = -2 (-1)^{3/4}$ or $z = -2 (-1)^{1/4}$ or $z = 2 (-1)^{1/4}$
However, the ...
7
votes
2answers
94 views
Product of $ |z^k - 1| $
Problem: Prove the following identity about the product involving the nth roots of unity:
$$
\prod_{k=1}^{N-1}|z^k-1| = N
$$
where $ z^k $ is the primitive nth root of unity.
Attempt:
$$
...
7
votes
5answers
222 views
I don't understand $\sqrt{-9i}$.
I try to visualise it on a graph, where x is real numbers and y is the imaginary numbers.
$\sqrt{9} = (3,0)$ and $(-3,0)$.
$\sqrt{-9} = \sqrt{-1} \times \sqrt{9} = (0,3) $ and $(0,-3)$.
...
7
votes
2answers
228 views
How to show a complex number inequality
A classmate consulted me this problem, after a few moment's thought I found it was difficult, so I wish to try my luck here.
Let $z_1,z_2,z_3,z_4\in \mathbb{C}$ such that ...
7
votes
1answer
96 views
Is there a complex variant of Möbius' function?
When you're dealing with arithmetic functions, you might have come across the classical Möbius' function
$$
\mu(n)=\begin{cases} (-1)^{\omega(n)}=(-1)^{\Omega(n)} &\mbox{if }\; \omega(n) = ...
7
votes
1answer
261 views
Why is $i$ called “imaginary”?
I was reading this question, and, after reading the responses, I felt like I had a much better understanding about how they're just another type of number definition.
Why, then, are they called ...
6
votes
7answers
397 views
What is the value of $i+i^2+i^3+\cdots+i^{23}$? [duplicate]
Can anyone help me with this question and show me a step by step solution please?
The imaginary number is $i$ is defined such that $i^2=-1$. What is $i+i^2+i^3+\cdots+i^{23}$?
6
votes
2answers
885 views
What does $\mathrm{Re}(x)$ mean?
I see this all the time in Mathematica output as well as in text, such as near the top of the Wikipedia Beta function page.
6
votes
5answers
124 views
Strong characterization of $\mathbb C$ with respect to $\mathbb R$
$\mathbb R^2$ is not a field, but $2$-tuple arithmetic rules like $(a,b)(c,d)=(ac-bd,ad+bc)$ coupled with $\mathbb R^2$ make it a field, but are there other rules than $(a,b)(c,d)=(ac-db,ad+bc)$ ...
