nyuuzyou/wikis huggingface dataset:
Complex-Valued Function Dominated by Function Vanishing at Infinity also Vanishes at Infinity

== Theorem ==
Let $X$ be a topological space. 
Let $g : X \to \C$ be a complex-valued function vanishing at infinity.
Let $f : X \to \C$ be such that $\cmod f \le \cmod g$.
Then $f$ vanishes at infinity.
== Proof ==
Let $\epsilon > 0$.
Since $g$ vanishes at infinity, there exists a compact set $F \subseteq X$ such that:
:$\cmod {\map g x} < \epsilon$ for $x \in X \setminus F$. 
We then have:
:$\cmod {\map f x} \le \cmod {\map g x} < \epsilon$ for $x \in X \setminus F$.
Since $\epsilon$ was arbitrary, we have that $f$ vanishes at infinity.
Category:Complex-Valued Functions Vanishing at Infinity
Complex-Valued Function Vanishing in Neighborhood of Infinity Vanishes at Infinity

== Theorem ==
Let $X$ be a locally compact Hausdorff space. 
Let $f : X \to \C$ be a function vanishing in a neighborhood of infinity. 
Then $f$ vanishes at infinity.
== Proof ==
Let $\epsilon > 0$. 
Since $f$ vanishes in a neighborhood of infinity, there exists a compact $F \subseteq X$ such that:
:$\map f x = 0$ for $x \in X \setminus F$
In particular:
:$\cmod {\map f x} = 0 < \epsilon$ for each $x \in X \setminus F$. 
Hence $f$ vanishes at infinity.
Category:Complex-Valued Functions Vanishing at Infinity
Category:Complex-Valued Functions Vanishing in Neighborhood of Infinity
Complex-Valued Function is Measurable iff Real and Imaginary Part Measurable

== Theorem ==
Let $\struct {X, \Sigma}$ be a measurable space.
Let $\struct {\C, \map \BB \C}$ be the complex numbers made into a measurable space with its Borel $\sigma$-algebra.
Let $\struct {\R, \map \BB \R}$ be the real numbers made into a measurable space with its Borel $\sigma$-algebra.
Let $f : X \to \C$ be a function.
Let $\map \Re f : X \to \R$ and $\map \Im f : X \to \R$ be the pointwise real part and imaginary part of $f$ respectively.
Then $f$ is $\Sigma/\map \BB \C$-measurable $\map \Re f$ and $\map \Im f$ are $\Sigma/\map \BB \R$-measurable.
== Proof ==
=== Necessary Condition ===
Suppose that $f$ is $\struct {X, \Sigma}$/$\struct {\C, \map \BB \C}$-measurable.
From Real and Imaginary Part Projections are Continuous and Continuous Mapping is Measurable, the real part and imaginary part are $\map \BB \C/\map \BB \R$-measurable as mappings $\C \to \R$.
Hence from Composition of Measurable Mappings is Measurable, $\map \Re f$ and $\map \Im f$ are $\Sigma/\map \BB \R$-measurable.
=== Sufficient Condition ===
Suppose that $\map \Re f$ and $\map \Im f$ are $\Sigma/\map \BB \R$-measurable.
Then from Measurability in Trace Sigma-Algebra, we have that $\map \Re f$ and $\map \Im f$ are $\Sigma/\map \BB \C$-measurable.
From Linear Combination of Measurable Functions valued in Topological Vector Space is Measurable, $f = \map \Re f + i \map \Im f$ is $\Sigma/\map \BB \C$-measurable.
Category:Measurable Functions
Complex-Valued Function on Compact Hausdorff Space Vanishes at Infinity

== Theorem ==
Let $X$ be a compact Hausdorff space. 
Let $f : X \to \C$ be a function.
Then $f$ vanishes at infinity.
== Proof ==
Let $\epsilon > 0$.
Then:
:$\cmod {\map f x} < \epsilon$ for all $x \in \O$.
By hypothesis:
:$X$ is compact.
Hence we in particular have:
:$\cmod {\map f x} < \epsilon$ for all $x \in X \setminus X$
with $X$ compact.
Hence $f$ vanishes at infinity.
Category:Complex-Valued Functions Vanishing at Infinity
Complex Addition/Examples

== Examples of Complex Addition ==
=== Example: $\paren {3 + 2 i} + \paren {5 + 6 i}$ ===
=== Example: $\paren {-1 + 4 i} + \paren {2 + \paren {-7} i}$ ===
=== Example: $\paren {3 + 4 i} + \paren {5 + 2 i}$ ===
=== Example: $\paren {4 - 3 i} + \paren {2 i - 8}$ ===
=== Example: $\paren {-3 + 5 i} + \paren {4 + 2 i} + \paren {5 - 3 i} + \paren {-4 - 6 i}$ ===
=== Example: $\paren {2 + 3 i} + \paren {4 - 5 i}$ ===
=== Example: $\paren {2 + 3 i} + \paren {5 - 2 i}$ ===
=== Example of Travel: $1$ ===
=== Example of Travel: $2$ ===
Category:Examples of Complex Addition
Category:Complex Arithmetic
Complex Addition/Examples/((5 + 3i) + (-1 + 2i)) + (7 - 5i)

== Example of Complex Addition ==
:$\paren {\paren {5 + 3 i} + \paren {-1 + 2 i} } + \paren {7 - 5 i} = 11$
== Proof ==
== Sources ==
* : $1$: Complex Numbers: Solved Problems: Fundamental Operations with Complex Numbers: $1 \ \text {(e)}$
Category:Examples of Complex Addition
Complex Addition/Examples/(-1 + 4i) + (2 - 7i)

== Example of Complex Addition ==
:$\paren {-1 + 4 i} + \paren {2 + \paren {-7} i} = 1 + \paren {-3} i$
== Proof ==
== Sources ==
* : $\S 1.2$. The Algebraic Theory
Category:Examples of Complex Addition
Complex Addition/Examples/(-3 + 5i) + (4 + 2i) + (5 - 3i) + (-4 - 6i)

== Example of Complex Addition ==
:$\paren {-3 + 5 i} + \paren {4 + 2 i} + \paren {5 - 3 i} + \paren {-4 - 6 i} = 2 - 2 i$
== Proof 1 ==
== Proof 2 ==
Category:Examples of Complex Addition
Complex Addition/Examples/(-3 + 5i) + (4 + 2i) + (5 - 3i) + (-4 - 6i)/Proof 1

== Example of Complex Addition ==
== Proof ==
== Sources ==
* : $1$: Complex Numbers: Solved Problems: Graphical Representation of Complex Numbers. Vectors: $5 \ \text{(c)}$
Category:Examples of Complex Addition
Complex Addition/Examples/(-3 + 5i) + (4 + 2i) + (5 - 3i) + (-4 - 6i)/Proof 2

== Example of Complex Addition ==
== Proof ==
Let:
These can be depicted in the complex plane as follows:
:
To find the required sum, proceed as in the following diagram:
:
Construct $z_2$ with its initial point placed at the terminal point of $z_1$.
Construct $z_3$ with its initial point placed at the terminal point of this instance of $z_2$.
Construct $z_4$ with its initial point placed at the terminal point of this instance of $z_3$.
The required resultant $z_1 + z_2 + z_3 + z_4$ of $z_1$ to $z_4$ is therefore represented by the terminal point of $z_4$.
== Sources ==
* : $1$: Complex Numbers: Solved Problems: Graphical Representation of Complex Numbers. Vectors: $5 \ \text{(c)}$
Category:Examples of Complex Addition
Complex Addition/Examples/(-7 - i) + (3 + 2i)

== Example of Complex Addition ==
:$\paren {-7 - i} + \paren {3 + 2 i} = -4 + i$
== Proof ==
== Sources ==
* : $1$: Complex Numbers: Solved Problems: Fundamental Operations with Complex Numbers: $1 \ \text{(b)}$
Category:Examples of Complex Addition
Complex Addition/Examples/(2 + 3i) + (4 - 5i)/Proof 2

== Example of Complex Addition ==
== Proof ==
:
Let the complex numbers $2 + 3 i$ and $4 - 5 i$ be represented by the points $P_1$ and $P_2$ respectively in the complex plane.
Complete the parallelogram with $OP_1$ and $OP_2$ as the adjacent sides.
By the Geometrical Interpretation of Complex Addition, the point $P$ represents the complex number $6 - 2 i$, which is the sum of $2 + 3 i$ and $4 - 5 i$.
== Sources ==
* : $1$: Complex Numbers: Supplementary Problems: Graphical Representation of Complex Numbers. Vectors: $61 \ \text {(a)}$
Category:Examples of Complex Addition
Complex Addition/Examples/(3 + 2i) + (-7 - i)

== Example of Use of Complex Addition is Commutative ==
:$\paren {3 + 2 i} + \paren {-7 - i} = -4 + i$
== Proof ==
== Sources ==
* : $1$: Complex Numbers: Solved Problems: Fundamental Operations with Complex Numbers: $1 \ \text{(a)}$
Category:Examples of Complex Addition
Complex Addition/Examples/(3 + 4i) + (5 + 2i)/Proof 2

== Example of Complex Addition ==
== Proof ==
:
Let the complex numbers $3 + 4 i$ and $5 + 2 i$ be represented by the points $P_1$ and $P_2$ respectively in the complex plane.
Complete the parallelogram with $OP_1$ and $OP_2$ as the adjacent sides.
By the Geometrical Interpretation of Complex Addition, the point $P$ represents the complex number $8 + 6 i$, which is the sum of $3 + 4 i$ and $5 + 2 i$.
== Sources ==
* : $1$: Complex Numbers: Solved Problems: Graphical Representation of Complex Numbers. Vectors: $5 \ \text{(a)}$
Category:Examples of Complex Addition
Complex Addition/Examples/(4 - 3i) + (2i - 8)

== Example of Complex Addition ==
:$\paren {4 - 3 i} + \paren {2 i - 8} = -4 - i$
== Proof ==
== Sources ==
* : $1$: Complex Numbers: Supplementary Problems: Fundamental Operations with Complex Numbers: $53 \ \text {(a)}$
Category:Examples of Complex Addition
Complex Addition/Examples/(5 + 3i) + ((-1 + 2i) + (7 - 5i))

== Example of Complex Addition ==
:$\paren {5 + 3 i} + \paren {\paren {-1 + 2 i} + \paren {7 - 5 i} } = 11$
== Proof ==
== Sources ==
* : $1$: Complex Numbers: Solved Problems: Fundamental Operations with Complex Numbers: $1 \ \text {(d)}$
Category:Examples of Complex Addition
Complex Addition/Examples/Travel 1

== Example of Complex Addition ==
A man travels:
:$12$ kilometres northeast
then:
:$20$ kilometres $30 \degrees$ west of north
then:
:$18$ kilometres $60 \degrees$ south of west.
Assuming the curvature of the Earth to be negligible at this scale, at the end of this travel, he is $14.7$ kilometres in a direction $45 \degrees 49'$ west of north from his starting point.
== Proof 1 ==
== Proof 2 ==
Category:Examples of Complex Addition
Complex Addition/Examples/Travel 1/Proof 1

== Example of Complex Addition ==
== Proof ==
Let the route of the traveller be embedded in the complex plane.
:
Let $P$ be the final location of the traveller.
The given directions can be translated into absolute arguments thus:
:Northeast is $45 \degrees$ north of east, and therefore an argument of $45 \degrees$
:$30 \degrees$ west of north is $30 + 90 \degrees$ north of east, and therefore an argument of $120 \degrees$
:$60 \degrees$ south of west is $60 + 180 \degrees$ north of east, and therefore an argument of $240 \degrees$.
Thus the problem reduces to finding the sum:
Then:
== Sources ==
* : $1$: Complex Numbers: Solved Problems: Polar Form of Complex Numbers: $18 \ \text {(a)}$
Category:Examples of Complex Addition
Complex Addition/Examples/Travel 2

== Example of Complex Addition ==
An airplane travels:
:$150$ kilometres southeast
then:
:$100$ kilometres due west
then:
:$225$ kilometres $30 \degrees$ north of east
then:
:$323$ kilometres northeast.
Assuming the curvature of Earth to be negligible at this scale, at the end of this travel, the plane is $490$ kilometres in a direction $28.7 \degrees$ north of east from its starting point.
== Proof 1 ==
== Proof 2 ==
Category:Examples of Complex Addition
Complex Addition/Examples/Travel 2/Mistake

== Source Work ==
:Chapter $1$: Complex Numbers
::Supplementary Problems: Polar Form of Complex Numbers: $85$
== Mistake ==
:An airplane travels $150 \, \mathrm {km}$ southeast, $100 \, \mathrm {km}$ due west, $225 \, \mathrm {km}$ $30 \degrees$ north of east, and then $323 \, \mathrm {km}$ northeast. Determine ... how far and in what direction it is from its starting point.
:Ans. $\ 375 \, \mathrm {km}$, $23 \degrees$ north of east (approx.)
== Correction ==
The correct solution is:
:$490$ kilometres in a direction $28.7 \degrees$ north of east from its starting point.
== Sources ==
* : $1$: Complex Numbers: Supplementary Problems: Polar Form of Complex Numbers: $85 \ \text {(b)}$
Category:Mistakes/Theory and Problems of Complex Variables
Complex Addition/Examples/Travel 2/Proof 1

== Example of Complex Addition ==
== Proof ==
Let the route of the traveller be embedded in the complex plane.
:
Let $P$ be the final location of the traveller.
The given directions can be translated into absolute arguments thus:
:Southeast is $45 \degrees$ south of east, and therefore an argument of $315 \degrees$
:Due west is an argument of $180 \degrees$
:$30 \degrees$ north of east is an argument of $30 \degrees$
:Northeast is $45 \degrees$ north of east, and therefore an argument of $45 \degrees$.
Thus the problem reduces to finding the sum:
Then:
== Sources ==
* : $1$: Complex Numbers: Supplementary Problems: Polar Form of Complex Numbers: $85 \ \text {(a)}$
Category:Examples of Complex Addition
Complex Addition Identity is Zero

== Theorem ==
Let $\C$ be the set of complex numbers.
The identity element of $\struct {\C, +}$ is the complex number $0 + 0 i$.
== Proof ==
We have:
:$\paren {x + i y} + \paren {0 + 0 i} = \paren {x + 0} + i \paren {y + 0} = x + i y$
:$\paren {0 + 0 i} + \paren {x + i y} = \paren {0 + x} + i \paren {0 + y} = x + i y$
== Sources ==
* : $\S 1.2$. The Algebraic Theory
* : Chapter $\text{II}$: Groups: Direct Products
* : $1$: Complex Numbers: Axiomatic Foundations of the Complex Number System: $7$
* : $1$ The complex plane: Complex numbers $\S 1.2$ The algebraic structure of the complex numbers
Category:Complex Addition
Category:Examples of Identity Elements
Complex Addition is Associative

== Theorem ==
The operation of addition on the set of complex numbers $\C$ is associative:
:$\forall z_1, z_2, z_3 \in \C: z_1 + \paren {z_2 + z_3} = \paren {z_1 + z_2} + z_3$
== Proof ==
From the definition of complex numbers, we define the following:
where $x_1, x_2, x_3, y_1, y_2, y_3 \in \R$.
Thus:
== Examples ==
== Also see ==
* Complex Addition is Commutative
== Sources ==
* : Chapter $\text I$: Functions of a Complex Variable: $\S 1$. Complex Numbers: $\text {(iii)}$ The fundamental operations $\text {(a)}$
* : Chapter $\text{II}$: Groups: Direct Products
* : $1$: Complex Numbers: Axiomatic Foundations of the Complex Number System: $2$
* : $1$: Complex Numbers: Supplementary Problems: Axiomatic Foundations of Complex Numbers: $77 \ \text{(a)}$
* : $1$ The complex plane: Complex numbers $\S 1.2$ The algebraic structure of the complex numbers
Category:Complex Addition is Associative
Category:Complex Addition
Category:Associative Law of Addition
Complex Addition is Associative/Examples

== Examples of Use of Complex Addition is Associative ==
=== Example: $\paren {5 + 3 i} + \paren {\paren {-1 + 2 i} + \paren {7 - 5 i} } = \paren {\paren {5 + 3 i} + \paren {-1 + 2 i} } + \paren {7 - 5 i}$ ===
Category:Examples of Use of Complex Addition is Associative
Category:Complex Addition is Associative
Complex Addition is Associative/Examples/(5 + 3i) + ((-1 + 2i) + (7 - 5i))

== Examples of Use of Complex Addition is Associative ==
==== Example: $\paren {5 + 3 i} + \paren {\paren {-1 + 2 i} + \paren {7 - 5 i} }$ ====
==== Example: $\paren {\paren {5 + 3 i} + \paren {-1 + 2 i} } + \paren {7 - 5 i}$ ====
As can be seen:
:$\paren {5 + 3 i} + \paren {\paren {-1 + 2 i} + \paren {7 - 5 i} } = \paren {\paren {5 + 3 i} + \paren {-1 + 2 i} } + \paren {7 - 5 i}$
== Sources ==
* : $1$: Complex Numbers: Solved Problems: Fundamental Operations with Complex Numbers: $1$
Category:Examples of Use of Complex Addition is Associative
Complex Addition is Closed

== Theorem ==
The set of complex numbers $\C$ is closed under addition:
:$\forall z, w \in \C: z + w \in \C$
== Proof 1 ==
== Proof 2 ==
== Sources ==
* : Chapter $1$: Integral Domains: $\S 2$. Operations: Example $1$
* : $1$: Complex Numbers: Axiomatic Foundations of the Complex Number System: $1$
Category:Complex Addition
Category:Algebraic Closure
Category:Complex Addition is Closed
Complex Addition is Closed/Proof 1

== Theorem ==
The set of complex numbers $\C$ is closed under addition:
:$\forall z, w \in \C: z + w \in \C$
== Proof ==
From the informal definition of complex numbers, we define the following:
:$z = x_1 + i y_1$
:$w = x_2 + i y_2$
where $i = \sqrt {-1}$ and $x_1, x_2, y_1, y_2 \in \R$.
Then from the definition of complex addition:
:$z + w = \paren {x_1 + x_2} + i \paren {y_1 + y_2}$
From Real Numbers under Addition form Group, real addition is closed.
So:
:$\paren {x_1 + x_2} \in \R$ and $\paren {y_1 + y_2} \in \R$
Hence the result.
Category:Complex Addition is Closed
Complex Addition is Closed/Proof 2

== Theorem ==
The set of complex numbers $\C$ is closed under addition:
:$\forall z, w \in \C: z + w \in \C$
== Proof ==
From the formal definition of complex numbers, we have:
:$z = \tuple {x_1, y_1}$
:$w = \tuple {x_2, y_2}$
where $x_1, x_2, y_1, y_2 \in \R$.
Then from the definition of complex addition:
:$z + w = \tuple {x_1 + x_2, y_1 + y_2}$
From Real Numbers under Addition form Group, real addition is closed.
So:
:$\paren {x_1 + x_2} \in \R$ and $\paren {y_1 + y_2} \in \R$
and hence the result.
Category:Complex Addition is Closed
Complex Addition is Commutative

== Theorem ==
The operation of addition on the set of complex numbers is commutative:
:$\forall z, w \in \C: z + w = w + z$
== Proof ==
From the definition of complex numbers, we define the following:
where $x_1, x_2, y_1, y_2 \in \R$.
Then:
== Examples ==
== Also see ==
* Complex Addition is Associative
== Sources ==
* : Chapter $\text I$: Functions of a Complex Variable: $\S 1$. Complex Numbers: $\text {(iii)}$ The fundamental operations $\text {(a)}$
* : Chapter $\text{II}$: Groups: Direct Products
* : $1$: Complex Numbers: Axiomatic Foundations of the Complex Number System: $2$
* : $1$: Complex Numbers: Supplementary Problems: Axiomatic Foundations of Complex Numbers: $76 \ \text{(a)}$
* : $1$ The complex plane: Complex numbers $\S 1.2$ The algebraic structure of the complex numbers
Category:Complex Addition
Category:Examples of Commutative Operations
Category:Commutative Law of Addition
Complex Addition is Commutative/Examples/(3 + 2i) + (-7 - i)

== Example of Use of Complex Addition is Commutative ==
==== Example: $\paren {3 + 2 i} + \paren {-7 - i}$ ====
==== Example: $\paren {-7 - i} + \paren {3 + 2 i}$ ====
As can be seen:
:$\paren {3 + 2 i} + \paren {-7 - i} = \paren {-7 - i} + \paren {3 + 2 i}$
== Sources ==
* : $1$: Complex Numbers: Solved Problems: Fundamental Operations with Complex Numbers: $1$
Category:Examples of Use of Complex Addition is Commutative
Complex Algebra/Examples

== Examples of Complex Algebra ==
=== Example: $\left({2 + i}\right) z + i = 3$ ===
=== Example: $\dfrac {z - 1} {z - i} = \dfrac 2 3$ ===
=== Example: $z^5 + 1$ ===
=== Example: $z^6 + z^3 + 1$ ===
=== Example: $z^8 + 1$ ===
=== Example: $\paren {1 + z}^5 = \paren {1 - z}^5$ ===
=== Example: $\paren {z - 1}^6 + \paren {z + 1}^6 = 0$ ===
</onlyinclude>
=== Example: $z^4 - 3 z^2 + 1 = 0$ ===
=== Example: $z^2 \paren {1 - z^2} = 16$ ===
=== Example: $3 x + 2 i y - i x + 5 y = 7 + 5 i$ ===
=== Example: $2 x - 3 i y + 4 i x - 2 y - 5 - 10 i = \paren {x + y + 2} - \paren {y - x + 3} i$ ===
Let $z_1$ and $z_2$ be arbitrary complex numbers represented in the complex plane as follows:
:
=== Example: $3 z_1 - 2 z_2$ ===
=== Example: $\dfrac 1 2 z_2 + \dfrac 5 3 z_1$ ===
Let $z_1$, $z_2$ and $z_3$ be arbitrary complex numbers represented in the complex plane as follows:
:
=== Example: $2 z_1 + z_3$ ===
=== Example: $\paren {z_1 + z_2} + z_3$ ===
=== Example: $z_1 + \paren {z_2 + z_3}$ ===
=== Example: $3 z_1 - 2 z_2 + 5 z_3$ ===
=== Example: $\dfrac 1 3 z_2 - \dfrac 3 4 z_1 + \dfrac 2 3 z_3$ ===
Category:Examples of Complex Algebra
Complex Algebra/Examples/(1+z)^5 = (1-z)^5

== Example of Complex Algebra ==
The roots of the equation:
:$\paren {1 + z}^5 = \paren {1 - z}^5$
are:
:$z = \set {\dfrac {\omega^k - 1} {\omega^k + 1}: k = 0, 1, 2, 3, 4}$
That is:
:$z = \set {0, \dfrac {\omega - 1} {\omega + 1}, \dfrac {\omega^2 - 1} {\omega^2 + 1}, \dfrac {\omega^3 - 1} {\omega^3 + 1} , \dfrac {\omega^4 - 1} {\omega^4 + 1} }$
where:
:$\omega = \cis \dfrac {2 \pi i} 5$
== Proof ==
We have:
Let $\cis \dfrac {2 \pi i} 5 = \omega$ as suggested.
Thus:
So:
When $n = 0$:
:$\omega^n = 1$
and thus:
:$z = \paren {1 - 1} / \paren {1 + 1}$
So:
:$z = 0, \dfrac {\omega - 1} {\omega + 1}, \dfrac {\omega^2 - 1} {\omega^2 + 1}, \dfrac {\omega^3 - 1} {\omega^3 + 1}, \dfrac {\omega^4 - 1} {\omega^4 + 1}$
== Sources ==
* : $1$: Complex Numbers: Supplementary Problems: The $n$th Roots of Unity: $109$
Category:Examples of Complex Algebra
Complex Algebra/Examples/(2+i)z + i = 3

== Example of Complex Algebra ==
Let $z \in \C$ be a complex number such that:
:$\paren {2 + i} z + i = 3$
Then:
:$z = 1 - i$
== Proof ==
== Sources ==
* : $\S 1$. Algebraic Theory of Complex Numbers: Exercise $2 \ \text{(i)}$
Category:Examples of Complex Algebra
Complex Algebra/Examples/(z-1)^6 + (z+1)^6 = 0

== Example of Complex Algebra ==
The roots of the equation:
:$\paren {z - 1}^6 + \paren {z + 1}^6 = 0$
are:
:$\pm i \cot \dfrac \pi {12}, \pm i \cot \dfrac {5 \pi} {12}, \pm i$
== Proof ==
By substituting $z = i$ in $(1)$ above, it is seen that $i$ is a root of $(1)$:
From Complex Roots of Polynomial with Real Coefficients occur in Conjugate Pairs it follows that $-i$ is also a root of $(1)$.
So it is now established $i$ and $-i$ are indeed roots of $(1)$.
Substituting $x = z^2$ into the second factor of $(2)$ gives the quadratic equation:
:$x^2 + 14 x + 1 = 0$
which, from Solution to Quadratic Equation, has roots:
Since:
:$7 + 4 \sqrt 3 = \paren {2 + \sqrt 3}^2$
and:
:$7 - 4 \sqrt 3 = \paren {2 - \sqrt 3}^2$
it follows we have roots:
:$z = \pm i \paren {2 \pm \sqrt 3}$
From Particular Values of Cotangent Function:
:$\cot \dfrac {\pi} {12} = 2 + \sqrt 3$
and:
:$\cot \dfrac {5 \pi} {12} = 2 - \sqrt 3$
Therefore we can simplify the remaining roots as:
:$\pm i \cot \dfrac \pi {12}, \pm i \cot \dfrac {5 \pi} {12}$
as required.
== Sources ==
* : $\S 3$. Roots of Unity: Exercise $10$
Category:Examples of Complex Algebra
Complex Algebra/Examples/(z-1) (z-i)^-1 = 2 over 3

== Example of Complex Algebra ==
Let $z \in \C$ be a complex number such that:
:$\dfrac {z - 1} {z - i} = \dfrac 2 3$
Then:
:$z = 3 - 2 i$
== Proof ==
== Sources ==
* : $\S 1$. Algebraic Theory of Complex Numbers: Exercise $2 \ \text{(ii)}$
Category:Examples of Complex Algebra
Complex Algebra/Examples/(z1 + z2) + z3

== Example of Complex Algebra ==
Let $z_1$, $z_2$ and $z_3$ be arbitrary complex numbers represented in the complex plane as follows:
:
The quantity $\paren {z_1 + z_2} + z_3$ can be depicted graphically as follows:
:
== Sources ==
* : $1$: Complex Numbers: Supplementary Problems: Graphical Representation of Complex Numbers. Vectors: $62 \ \text {(b)}$
Category:Examples of Complex Algebra
Complex Algebra/Examples/1 2 z2 + 5 3 z1

== Example of Complex Algebra ==
Let $z_1$ and $z_2$ be arbitrary complex numbers represented in the complex plane as follows:
:
The quantity $\dfrac 1 2 z_2 + \dfrac 5 3 z_1$ can be depicted graphically as follows:
:
== Sources ==
* : $1$: Complex Numbers: Solved Problems: Graphical Representation of Complex Numbers. Vectors: $6 \ \text{(b)}$
Category:Examples of Complex Algebra
Complex Algebra/Examples/1 3 z2 - 3 4 z1 + 2 3 z3

== Example of Complex Algebra ==
Let $z_1$, $z_2$ and $z_3$ be arbitrary complex numbers represented in the complex plane as follows:
:
The quantity $\dfrac 1 3 z_2 - \dfrac 3 4 z_1 + \dfrac 2 3 z_3$ can be depicted graphically as follows:
:
== Sources ==
* : $1$: Complex Numbers: Supplementary Problems: Graphical Representation of Complex Numbers. Vectors: $62 \ \text {(e)}$
Category:Examples of Complex Algebra
Complex Algebra/Examples/2x - 3iy + 4ix - 2y - 5 - 10i = (x + y + 2) - (y - x + 3)i

== Example of Complex Algebra ==
Let:
:$2 x - 3 i y + 4 i x - 2 y - 5 - 10 i = \paren {x + y + 2} - \paren {y - x + 3} i$
Then:
 
== Proof ==
Separating the real parts from the imaginary parts:
 
Substituting for $x$ from $(1)$ into $(2)$:
 
== Sources ==
* : $1$: Complex Numbers: Supplementary Problems: Fundamental Operations with Complex Numbers: $57$
Category:Examples of Complex Algebra
Complex Algebra/Examples/2z1 + z3

== Example of Complex Algebra ==
Let $z_1$, $z_2$ and $z_3$ be arbitrary complex numbers represented in the complex plane as follows:
:
The quantity $2 z_1 + z_3$ can be depicted graphically as follows:
:
== Sources ==
* : $1$: Complex Numbers: Supplementary Problems: Graphical Representation of Complex Numbers. Vectors: $62 \ \text {(a)}$
Category:Examples of Complex Algebra
Complex Algebra/Examples/3x + 2iy - ix + 5y = 7 + 5i

== Example of Complex Algebra ==
Let $3 x + 2 i y - i x + 5 y = 7 + 5 i$.
Then:
== Proof ==
== Sources ==
* : $1$: Complex Numbers: Solved Problems: Fundamental Operations with Complex Numbers: $3$
Category:Examples of Complex Algebra
Complex Algebra/Examples/3z1 - 2z2

== Example of Complex Algebra ==
Let $z_1$ and $z_2$ be arbitrary complex numbers represented in the complex plane as follows:
:
The quantity $3 z_1 - 2 z_2$ can be depicted graphically as follows:
:
== Sources ==
* : $1$: Complex Numbers: Solved Problems: Graphical Representation of Complex Numbers. Vectors: $6 \ \text{(a)}$
Category:Examples of Complex Algebra