nyuuzyou/wikis huggingface dataset:
Complex Algebra/Examples/3z1 - 2z2 + 5z3

== 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 $3 z_1 - 2 z_2 + 5 z_3$ can be depicted graphically as follows:
:
== Sources ==
* : $1$: Complex Numbers: Supplementary Problems: Graphical Representation of Complex Numbers. Vectors: $62 \ \text {(d)}$
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: $62 \ \text {(c)}$
Category:Examples of Complex Algebra
Complex Algebra/Examples/z^2 (1 - z^2) = 16

== Example of Complex Algebra ==
The roots of the equation:
:$z^2 \paren {1 - z^2} = 16$
are:
:$\pm \dfrac 3 2 \pm \dfrac {\sqrt 7} 2 i$
== Proof 1 ==
== Proof 2 ==
Category:Examples of Complex Algebra
Complex Algebra/Examples/z^2 (1 - z^2) = 16/Proof 2

== Example of Complex Algebra ==
== Proof ==
Let $w := z^2$.
Then:
Let $\paren {p + i q}^2 = \dfrac 1 2 \pm \dfrac 3 2 \sqrt 7 i$
Then:
which leads after unpleasant algebra to:
:$p + i q = \pm \dfrac 3 2 \pm \dfrac {\sqrt 7} 2 i$
== Sources ==
* : $1$: Complex Numbers: Solved Problems: Miscellaneous Problems: $50$
Category:Examples of Complex Algebra
Complex Algebra/Examples/z^4 - 3z^2 + 1 = 0

== Example of Complex Algebra ==
The roots of the equation:
:$z^4 - 3z^2 + 1 = 0$
are:
:$2 \cos 36 \degrees, 2 \cos 72 \degrees, 2 \cos 216 \degrees, 2 \cos 252 \degrees$
== Proof ==
From the Quadratic Formula applied to each of the above quadratic factors we can easily see that the four roots are:
:$\dfrac {\pm 1 \pm \sqrt 5} 2$
$360 \degrees = 2 \pi \radians$, so $72 \degrees = 2 \pi / 5 \radians$
From De Moivre's Formula, the roots of $x^5 - 1 = 0$ are:
:$(\cos(2 n \pi) + i \sin(2 n \pi))^{1/5}= \cos \dfrac {2 n \pi} 5 + i \sin \dfrac {2 n \pi} 5$
However, the coefficient of $x^4$ is $0$ and therefore, by Viète's Formulas, the sum of the roots of $x^5 - 1 = 0$ are also $0$ which means that the sum of the real parts of the roots are also $0$:
Now:
Hence:
and
We can now simplify the sum of the real parts of the roots of $x^5 - 1 = 0$:
From the Quadratic Formula we then have two potential values:
:$\cos \dfrac {2 \pi} 5 = \dfrac {-1 \pm \sqrt 5} 4$
$0 < 2 \pi / 5 < \pi / 2$, so we know that $\cos \dfrac {2 \pi} 5 > 0$, hence:
:$2 \cos 72 \degrees = 2 \cos \dfrac {2 \pi} 5 = \dfrac {-1 + \sqrt 5} 2$
From Cosine of Angle plus Straight Angle, $\map \cos {x + 180 \degrees} = -\cos x$, hence:
:$2 \cos 252 \degrees = \dfrac {1 - \sqrt 5} 2$
Now:
From Cosine of Angle plus Straight Angle, $\map \cos {x + 180 \degrees} = -\cos x$, hence:
:$2 \cos 216 \degrees = \dfrac {- 1 - \sqrt 5} 2$
We have therefore shown that the four roots of $z^4 - 3z^2 + 1 = 0$ are $\dfrac {\pm 1 \pm \sqrt 5} 2$ and that these four values are also equal to $2 \cos 36 \degrees, 2 \cos 72 \degrees, 2 \cos 216 \degrees, 2 \cos 252 \degrees$
== Sources ==
* : $1$: Complex Numbers: Supplementary Problems: De Moivre's Theorem: $91$
Category:Examples of Complex Algebra
Complex Algebra/Examples/z^5 + 1

== Example of Complex Algebra ==
:$z^5 + 1 = \paren {z + 1} \paren {z^2 - 2 z \cos \dfrac \pi 5 + 1} \paren {z^2 - 2 z \cos \dfrac {3 \pi} 5 + 1}$
== Proof ==
From Factorisation of $z^{2 n + 1} + 1$ in Real Domain:
:$z^5 + 1 = \ds \prod_{k \mathop = 0}^1 \paren {z + 1} \paren {z^2 - 2 z \cos \dfrac {\paren {2 k + 1} \pi} 5 + 1}$
Hence the result.
== Sources ==
* : $\S 3$. Roots of Unity: Exercise $9$
Category:Complex Roots
Category:Examples of Complex Algebra
Complex Algebra/Examples/z^6 + z^3 + 1

== Example of Complex Algebra ==
:$z^6 + z^3 + 1 = \paren {z^2 - 2 z \cos \dfrac {2 \pi} 9 + 1} \paren {z^2 - 2 z \cos \dfrac {4 \pi} 9 + 1} \paren {z^2 - 2 z \cos \dfrac {8 \pi} 9 + 1}$
== Proof ==
From Sum of Geometric Sequence or Difference of Two Cubes:
:$z^6 + z^3 + 1 = \dfrac {z^9 - 1} {z^3 - 1}$
Then from Factorisation of $x^{2 n + 1} - 1$ in Real Domain:
:$z^9 - 1 = \paren {z - 1} \ds \prod_{k \mathop = 1}^4 \paren {z^2 - 2 \cos \dfrac {2 \pi k} 9 + 1}$
and:
Thus:
:$\paren {z^3 - 1} \paren {z^6 + z^3 + 1} = \paren {\paren {z - 1} \paren {z^2 - 2 \cos \dfrac {6 \pi} 9 + 1} } \paren {\paren {z^2 - 2 \cos \dfrac {2 \pi} 9 + 1} \paren {z^2 - 2 \cos \dfrac {4 \pi} 9 + 1} \paren {z^2 - 2 \cos \dfrac {8 \pi} 9 + 1} }$
from which:
:$\paren {z^6 + z^3 + 1} = \paren {z^2 - 2 \cos \dfrac {2 \pi} 9 + 1} \paren {z^2 - 2 \cos \dfrac {4 \pi} 9 + 1} \paren {z^2 - 2 \cos \dfrac {8 \pi} 9 + 1}$
== Sources ==
* : $\S 3$. Roots of Unity: Example $6$: $(3.14)$
Category:Complex Roots
Category:Examples of Complex Algebra
Complex Algebra/Examples/z^8 + 1

== Example of Complex Algebra ==
:$z^8 + 1 = \paren {z^2 - 2 z \cos \dfrac \pi 8 + 1} \paren {z^2 - 2 z \cos \dfrac {3 \pi} 8 + 1} \paren {z^2 - 2 z \cos \dfrac {5 \pi} 8 + 1} \paren {z^2 - 2 z \cos \dfrac {7 \pi} 8 + 1}$
== Proof ==
From Roots of $z^8 + 1 = 0$ and the corollary to the Polynomial Factor Theorem:
:$z^8 + 1 = \ds \prod_{k \mathop = 0}^7 \paren {z - \paren {\cos \dfrac {\paren {2 k + 1} \pi} 8 + i \sin \dfrac {\paren {2 k + 1} \pi} 8} }$
Hence:
Hence the result.
== Sources ==
* : $\S 3$. Roots of Unity: Exercise $8$
Category:Complex Roots
Category:Examples of Complex Algebra
Complex Arithmetic/Examples

== Examples of Complex Arithmetic ==
=== Example: $\dfrac {\paren {1 + 2 i}^2} {1 - i}$ ===
=== Example: $\dfrac 1 {1 + i} + \dfrac 1 {1 - 2 i}$ ===
=== Example: Sum of Powers of $i$ from $0$ to $7$ ===
=== Example: $\dfrac 1 {\paren {4 + 2 i} \paren {2 - 3 i} }$ ===
=== Example: $\dfrac {5 + 5 i} {3 - 4 i} + \dfrac {20} {4 + 3 i}$ ===
=== Example: $\dfrac {3 i^{30} - i^{19} } {2 i - 1}$ ===
=== Example: $\paren {\dfrac {1 + \sqrt 3 i} {1 - \sqrt 3 i} }^{10}$ ===
=== Example: $\paren {\dfrac {\sqrt 3 - i} {\sqrt 3 + i} }^4 \paren {\dfrac {1 + i} {1 - i} }^5$ ===
=== Example: $3 \paren {-1 + 4 i} - 2 \paren {7 - i}$ ===
=== Example: $\paren {i - 2} \paren {2 \paren {1 + i} - 3 \paren {i - 1} }$ ===
=== Example: $\dfrac {\paren {2 + i} \paren {3 - 2 i} \paren {1 + 2 i} } {\paren {1 - i}^2}$ ===
=== Example: $\paren {2 i - 1}^2 \paren {\dfrac 4 {1 - i} + \dfrac {2 - i} {1 + i} }$ ===
=== Example: $\dfrac {i^4 + i^9 + i^{16} } {2 - i^5 + i^{10} - i^{15} }$ ===
=== Example: $3 \paren {\dfrac {1 + i} {1 - i} }^2 - 2 \paren {\dfrac {1 - i} {1 + i} }^3$ ===
=== Example: $3 \paren {1 + 2 i} - 2 \paren {2 - 3 i}$ ===
=== Example: $3 \paren {1 + i} + 2 \paren {4 - 3 i} - \paren {2 + 5 i}$ ===
=== Example: $\dfrac 1 2 \paren {4 - 3 i} + \dfrac 3 2 \paren {5 + 2 i}$ ===
Let $z_1 = 2 + i$, $z_2 = 3 - 2 i$ and $z^3 = -\dfrac 1 2 + \dfrac {\sqrt 3} 2 i$.
=== Example: $\cmod {3 z_1 - 4 z_2}$ ===
=== Example: $z_1^3 - 3 z_1^2 + 4 z_1 - 8$ ===
=== Example: $\paren {\overline {z_3} }^4$ ===
=== Example: $\cmod {\dfrac {2 z_2 + z_1 - 5 - 1} {2 z_1 - z_2 + 3 - 1} }^2$ ===
Let $z_1 = 1 - i$, $z_2 = -2 + 4 i$ and $z_3 = \sqrt 3 - 2 i$.
=== Example: ${z_1}^2 + 2 z_1 - 3$ ===
=== Example: $\cmod {2 z_2 - 3 z_1}^2$ ===
=== Example: $\paren {z_3 - \overline {z_3} }^5$ ===
=== Example: $\cmod {z_1 \overline {z_2} + z_2 \overline {z_1} }$ ===
=== Example: $\cmod {\dfrac {z_1 + z_2 + 1} {z_1 - z_2 + i} }$ ===
=== Example: $\dfrac 1 2 \paren {\dfrac {z_3} {\overline z_3} + \dfrac {\overline z_3} {z_3} }$ ===
=== Example: $\overline {\paren {z_2 + z_3} \paren {z_1 - z_3} }$ ===
=== Example: $\cmod { {z_1}^2 + \overline {z_2}^2}^2 + \cmod {\overline {z_3}^2 - {z_2}^2}^2$ ===
=== Example: $\map \Re {2 {z_1}^3 + 3 {z_2}^2 - 5 {z_3}^2}$ ===
=== Example: $\map \Im {\dfrac {z_1 z_2} {z_3} }$ ===
Category:Examples of Complex Arithmetic
Complex Arithmetic/Examples/((1 + root 3 i)(1 - root 3 i)^-1)^10

== Example of Complex Arithmetic ==
:$\paren {\dfrac {1 + \sqrt 3 i} {1 - \sqrt 3 i} }^{10} = -\dfrac 1 2 + \dfrac {\sqrt 3} 2 i$
== Proof 1 ==
== Proof 2 ==
Category:Examples of Complex Arithmetic
Complex Arithmetic/Examples/((root 3 - i)(root 3 + i)^-1)^4 ((1 + i)(1 - i)^-1)^5

== Example of Complex Arithmetic ==
:$\paren {\dfrac {\sqrt 3 - i} {\sqrt 3 + i} }^4 \paren {\dfrac {1 + i} {1 - i} }^5 = -\dfrac {\sqrt 3} 2 - \dfrac 1 2 i$
== Proof ==
== Sources ==
* : $1$: Complex Numbers: Supplementary Problems: De Moivre's Theorem: $89 \ \text{(e)}$
Category:Examples of Complex Arithmetic
Complex Arithmetic/Examples/(-1 + 2i) ((7 - 5i) + (-3 + 4i))

== Example of Complex Arithmetic ==
:$\paren {-1 + 2 i} \paren {\paren {7 - 5 i} + \paren {-3 + 4 i} } = -2 + 9 i$
== Proof ==
== Sources ==
* : $1$: Complex Numbers: Solved Problems: Fundamental Operations with Complex Numbers: $1 \ \text{(j)}$
Category:Examples of Complex Arithmetic
Complex Arithmetic/Examples/(-1 + 2i) (7 - 5i) + (-1 + 2i) (-3 + 4i)

== Example of Complex Arithmetic ==
:$\paren {-1 + 2 i} \paren {7 - 5 i} + \paren {-1 + 2 i} \paren {-3 + 4 i} = -2 + 9 i$
== Proof ==
== Sources ==
* : $1$: Complex Numbers: Solved Problems: Fundamental Operations with Complex Numbers: $1 \ \text{(j)}$
Category:Examples of Complex Arithmetic
Complex Arithmetic/Examples/(1+2i)^2 over 1-i

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

== Example of Complex Arithmetic ==
:$\dfrac {\paren {2 + i} \paren {3 - 2 i} \paren {1 + 2 i} } {\paren {1 - i}^2} = -\dfrac {15} 2 + 5 i$
== Proof ==
== Sources ==
* : $1$: Complex Numbers: Supplementary Problems: Fundamental Operations with Complex Numbers: $53 \ \text {(g)}$
Category:Examples of Complex Arithmetic
Complex Arithmetic/Examples/(2i-1)^2 (4(1-i)^-1 + (2-i) (1+i)^1)

== Example of Complex Arithmetic ==
:$\paren {2 i - 1}^2 \paren {\dfrac 4 {1 - i} + \dfrac {2 - i} {1 + i} } = -\dfrac {11} 2 - \dfrac {23} 2 i$
== Proof ==
== Sources ==
* : $1$: Complex Numbers: Supplementary Problems: Fundamental Operations with Complex Numbers: $53 \ \text {(h)}$
Category:Examples of Complex Arithmetic
Complex Arithmetic/Examples/(3i^30 - i^19) (2i - 1)^-1

== Example of Complex Arithmetic ==
:$\dfrac {3 i^{30} - i^{19} } {2 i - 1} = 1 + i$
== Proof ==
== Sources ==
* : $1$: Complex Numbers: Solved Problems: Fundamental Operations with Complex Numbers: $1 \ \text{(m)}$
Category:Examples of Complex Arithmetic
Complex Arithmetic/Examples/(4+2i)^-1 (2-3i)^-1

== Example of Complex Arithmetic ==
:$\dfrac 1 {\paren {4 + 2 i} \paren {2 - 3 i} } = \dfrac 7 {130} + \dfrac {2 i} {65}$
== Proof ==
== Sources ==
* : $\S 1$. Algebraic Theory of Complex Numbers: Exercise $1 \ \text{(iv)}$
Category:Examples of Complex Arithmetic
Complex Arithmetic/Examples/(5 + 5i) (3 - 4i)^-1 + 20 (4 + 3i)^-1

== Example of Complex Arithmetic ==
:$\dfrac {5 + 5 i} {3 - 4 i} + \dfrac {20} {4 + 3 i} = 3 - i$
== Proof ==
Then:
Finally:
== Sources ==
* : $1$: Complex Numbers: Solved Problems: Fundamental Operations with Complex Numbers: $1 \ \text{(l)}$
Category:Examples of Complex Arithmetic
Complex Arithmetic/Examples/(Conjugate of z 3)^4

== Example of Complex Arithmetic ==
Let $z^3 = -\dfrac 1 2 + \dfrac {\sqrt 3} 2 i$.
Then:
:$\paren {\overline {z_3} }^4 = -\dfrac 1 2 - \dfrac {\sqrt 3} 2 i$
== Proof ==
But from Cube Roots of Unity:
:$\paren {-\dfrac 1 2 - \dfrac {\sqrt 3} 2 i}^3 = 1$
Hence the result.
== Sources ==
* : $1$: Complex Numbers: Solved Problems: Fundamental Operations with Complex Numbers: $2 \ \text{(c)}$
Category:Examples of Complex Arithmetic
Complex Arithmetic/Examples/(Modulus of (z 1^2 + conj z 2^2))^2 + (Modulus of (conj z 3^2 - z 2^2))^2

== Example of Complex Arithmetic ==
Let $z_1 = 1 - i$, $z_2 = -2 + 4 i$ and $z_3 = \sqrt 3 - 2 i$.
Then:
:$\cmod { {z_1}^2 + \overline {z_2}^2}^2 + \cmod {\overline {z_3}^2 - {z_2}^2}^2 = 765 + 128 \sqrt 3$
== Proof ==
== Sources ==
* : $1$: Complex Numbers: Supplementary Problems: Fundamental Operations with Complex Numbers: $54 \ \text {(h)}$
Category:Examples of Complex Arithmetic
Complex Arithmetic/Examples/(Modulus of 2 z 2 - 3 z 1)^2

== Example of Complex Arithmetic ==
Let $z_1 = 1 - i$ and $z_2 = -2 + 4 i$.
Then:
:$\cmod {2 z_2 - 3 z_1}^2 = 170$
== Proof ==
== Sources ==
* : $1$: Complex Numbers: Supplementary Problems: Fundamental Operations with Complex Numbers: $54 \ \text {(b)}$
Category:Examples of Complex Arithmetic
Complex Arithmetic/Examples/(i-2)(2(1+i) - 3(i-1))

== Example of Complex Arithmetic ==
:$\paren {i - 2} \paren {2 \paren {1 + i} - 3 \paren {1 - i} } = -9 + 7 i$
== Proof ==
== Sources ==
* : $1$: Complex Numbers: Supplementary Problems: Fundamental Operations with Complex Numbers: $53 \ \text {(d)}$
Category:Examples of Complex Arithmetic
Complex Arithmetic/Examples/(i^4 + i^9 + i^16) (2 - i^5 + i^10 - i^15)^1

== Example of Complex Arithmetic ==
:$\dfrac {i^4 + i^9 + i^{16} } {2 - i^5 + i^{10} - i^{15} } = 2 + i$
== Proof ==
== Sources ==
* : $1$: Complex Numbers: Supplementary Problems: Fundamental Operations with Complex Numbers: $53 \ \text {(i)}$
Category:Examples of Complex Arithmetic
Complex Arithmetic/Examples/(z 3 - conj z 3)^5

== Example of Complex Arithmetic ==
Let $z_3 = \sqrt 3 - 2 i$.
Then:
:$\paren {z_3 - \overline {z_3} }^5 = -1024 i$
== Proof ==
== Sources ==
* : $1$: Complex Numbers: Supplementary Problems: Fundamental Operations with Complex Numbers: $54 \ \text {(c)}$
Category:Examples of Complex Arithmetic
Complex Arithmetic/Examples/(z 3 - conj z 3)^5/Mistake

== Source Work ==
:Chapter $1$: Complex Numbers
::Supplementary Problems: Fundamental Operations with Complex Numbers: $54 \ \text {(c)}$
== Mistake ==
:If ... $z_3 = \sqrt 3 - 2 i$, evaluate ...:
::$\paren {z_3 - \overline {z_3} }^5$
:Ans. $1024 i$
== Correction ==
This result actually works out to $-1024 i$:
== Sources ==
* : $1$: Complex Numbers: Supplementary Problems: Fundamental Operations with Complex Numbers: $54 \ \text {(c)}$
Category:Mistakes/Theory and Problems of Complex Variables
Complex Arithmetic/Examples/1 2^-1 (4-3i) + 3 2^-1 (5+2i)

== Example of Complex Arithmetic ==
:$\dfrac 1 2 \paren {4 - 3 i} + \dfrac 3 2 \paren {5 + 2 i} = \dfrac {19} 2 + \dfrac 3 2 i$
== Proof 1 ==
== Proof 2 ==
Category:Examples of Complex Arithmetic
Complex Arithmetic/Examples/1 2^-1 (4-3i) + 3 2^-1 (5+2i)/Proof 1

== Example of Complex Arithmetic ==
== Proof ==
== Sources ==
* : $1$: Complex Numbers: Supplementary Problems: Graphical Representation of Complex Numbers. Vectors: $61 \ \text {(e)}$
Category:Examples of Complex Arithmetic
Complex Arithmetic/Examples/1 2^-1 (4-3i) + 3 2^-1 (5+2i)/Proof 2

== Example of Complex Arithmetic ==
== Proof ==
:
Let the complex numbers $\dfrac 1 2 \paren {4 - 3 i}$ and $\dfrac 3 2 \paren {5 + 2 i}$ 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.
Using Geometrical Interpretation of Complex Addition, the point $P$ represents the complex number $\dfrac {19} 2 + \dfrac 3 2 i$, which is the sum of $\dfrac 1 2 \paren {4 - 3 i}$ and $\dfrac 3 2 \paren {5 + 2 i}$.
Hence, $\dfrac {19} 2 + \dfrac 3 2 i$ is the sum of $\dfrac 1 2 \paren {4 - 3 i}$ and $\dfrac 3 2 \paren {5 + 2 i}$.
== Sources ==
* : $1$: Complex Numbers: Supplementary Problems: Graphical Representation of Complex Numbers. Vectors: $61 \ \text {(e)}$
Category:Examples of Complex Arithmetic
Complex Arithmetic/Examples/1 over 1+i Plus 1 over 1-2i

== Example of Complex Arithmetic ==
:$\dfrac 1 {1 + i} + \dfrac 1 {1 - 2 i} = \dfrac 7 {10} - \dfrac 1 {10} i$
== Proof ==
== Sources ==
* : $\S 1.2$. The Algebraic Theory: Example $2$.
Category:Examples of Complex Arithmetic
Complex Arithmetic/Examples/3((1+i)(1-i)^1)^2 - 2((1-i)(1+i)^1)^3

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

== Example of Complex Arithmetic ==
:$3 \paren {-1 + 4 i} - 2 \paren {7 - i} = -17 + 14 i$
== Proof ==
== Sources ==
* : $1$: Complex Numbers: Supplementary Problems: Fundamental Operations with Complex Numbers: $53 \ \text {(b)}$
Category:Examples of Complex Arithmetic
Complex Arithmetic/Examples/3(1+2i) - 2(2-3i)/Proof 2

== Example of Complex Arithmetic ==
== Proof ==
:
By definition of complex subtraction:
:$3 \paren {1 + 2 i} - 2 \paren {2 - 3 i} = \paren {7 + i} + 2 \paren {-2 + 3 i}$
Let the complex numbers $3 \paren {1 + 2 i}$ and $2 \paren {-2 + 3 i}$ 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.
Using Geometrical Interpretation of Complex Addition, the point $P$ represents the complex number $-1 + 12 i$, which is the sum of $3 \paren {1 + 2 i}$ and $2 \paren {-2 + 3 i}$.
Hence, $-1 + 12 i$ is the difference of $3 \paren {1 + 2 i}$ and $2 \paren {2 - 3 i}$.
== Sources ==
* : $1$: Complex Numbers: Supplementary Problems: Graphical Representation of Complex Numbers. Vectors: $61 \ \text {(c)}$
Category:Examples of Complex Arithmetic
Complex Arithmetic/Examples/3(1+i) + 2(4-3i) - (2+5i)

== Example of Complex Arithmetic ==
:$3 \paren {1 + i} + 2 \paren {4 - 3 i} - \paren {2 + 5 i} = 9 - 8 i$
== Proof 1 ==
== Proof 2 ==
Category:Examples of Complex Arithmetic
Complex Arithmetic/Examples/3(1+i) + 2(4-3i) - (2+5i)/Proof 2

== Example of Complex Arithmetic ==
== Proof ==
We have:
These can be depicted in the complex plane as follows:
:
To find the required sum, proceed as in the following diagram:
:
Construct $8 - 6 i$ with its initial point placed at the terminal point of $3 + 3 i$.
Construct $-2 - 5 i$ with its initial point placed at the terminal point of this instance of $8 - 6 i$.
The required resultant $3 \paren {1 + i} + 2 \paren {4 - 3 i} - \paren {2 + 5 i}$ is therefore represented by the terminal point of $-2 - 5 i$.
== Sources ==
* : $1$: Complex Numbers: Supplementary Problems: Graphical Representation of Complex Numbers. Vectors: $61 \ \text {(d)}$
Category:Examples of Complex Arithmetic
Complex Arithmetic/Examples/Half (z 3 over conj z 3 + conj z 3 over z 3)

== Example of Complex Arithmetic ==
Let $z_3 = \sqrt 3 - 2 i$.
Then:
:$\dfrac 1 2 \paren {\dfrac {z_3} {\overline z_3} + \dfrac {\overline z_3} {z_3} } = -\dfrac 1 7$
== Proof ==
== Sources ==
* : $1$: Complex Numbers: Supplementary Problems: Fundamental Operations with Complex Numbers: $54 \ \text {(f)}$
Category:Examples of Complex Arithmetic
Complex Arithmetic/Examples/Imaginary (z 1 z 2 over z 3)

== Example of Complex Arithmetic ==
Let $z_1 = 1 - i$, $z_2 = -2 + 4 i$ and $z_3 = \sqrt 3 - 2 i$.
Then:
:$\map \Im {\dfrac {z_1 z_2} {z_3} } = \dfrac {6 \sqrt 3 + 4} 7$
== Proof ==
== Sources ==
* : $1$: Complex Numbers: Supplementary Problems: Fundamental Operations with Complex Numbers: $54 \ \text {(j)}$
Category:Examples of Complex Arithmetic
Complex Arithmetic/Examples/Modulus of ((2 z 2 + z 1 - 5 - 1) (2 z 1 - z 2 + 3 - 1)^-1)^2

== Example of Complex Arithmetic ==
Let $z^3 = -\dfrac 1 2 + \dfrac {\sqrt 3} 2 i$.
Then:
:$\paren {\overline {z_3} }^4 = -\dfrac 1 2 - \dfrac {\sqrt 3} 2 i$
== Proof ==
But from Cube Roots of Unity:
:$\paren {-\dfrac 1 2 - \dfrac {\sqrt 3} 2 i}^3 = 1$
Hence the result.
== Sources ==
* : $1$: Complex Numbers: Solved Problems: Fundamental Operations with Complex Numbers: $2 \ \text{(d)}$
Category:Examples of Complex Arithmetic
Complex Arithmetic/Examples/Modulus of ((z 1 + z 2 + 1) (z 1 - z 2 + i)^-1)

== Example of Complex Arithmetic ==
Let $z_1 = 1 - i$ and $z_2 = -2 + 4 i$.
Then:
:$\cmod {\dfrac {z_1 + z_2 + 1} {z_1 - z_2 + i} } = \dfrac 3 5$
== Proof ==
== Sources ==
* : $1$: Complex Numbers: Supplementary Problems: Fundamental Operations with Complex Numbers: $54 \ \text {(e)}$
Category:Examples of Complex Arithmetic
Complex Arithmetic/Examples/Modulus of (3 z 1 - 4 z 2)

== Example of Complex Arithmetic ==
Let $z_1 = 2 + i$ and $z_2 = 3 - 2 i$.
Then:
:$\cmod {3 z_1 - 4 z_2} = \sqrt {157}$
== Proof ==
== Sources ==
* : $1$: Complex Numbers: Solved Problems: Fundamental Operations with Complex Numbers: $2 \ \text{(a)}$
Category:Examples of Complex Arithmetic
Complex Arithmetic/Examples/Modulus of (z 1 conj z 2 + z 2 conj z 1)

== Example of Complex Arithmetic ==
Let $z_1 = 1 - i$ and $z_2 = -2 + 4 i$.
Then:
:$\cmod {z_1 \overline {z_2} + z_2 \overline {z_1} } = 12$
== Proof ==
== Sources ==
* : $1$: Complex Numbers: Supplementary Problems: Fundamental Operations with Complex Numbers: $54 \ \text {(d)}$
Category:Examples of Complex Arithmetic
Complex Arithmetic/Examples/Real (2 z 1^3 + 3 z 2^2 - 5 z 3^2)

== Example of Complex Arithmetic ==
Let $z_1 = 1 - i$, $z_2 = -2 + 4 i$ and $z_3 = \sqrt 3 - 2 i$.
Then:
:$\map \Re {2 {z_1}^3 + 3 {z_2}^2 - 5 {z_3}^2} = -35$
== Proof ==
== Sources ==
* : $1$: Complex Numbers: Supplementary Problems: Fundamental Operations with Complex Numbers: $54 \ \text {(i)}$
Category:Examples of Complex Arithmetic
Complex Arithmetic/Examples/Sum of Powers of i from 0 to 7

== Example of Complex Arithmetic ==
:$1 + i + i^2 + i^3 + i^4 + i^5 + i^6 + i^7 = 0$
== Proof ==
== Sources ==
* : $\S 1.2$. The Algebraic Theory: Example $3$.
Category:Examples of Complex Arithmetic
Complex Arithmetic/Examples/conj ((z 2 + z 3) (z 1 - z 3))

== Example of Complex Arithmetic ==
Let $z_1 = 1 - i$, $z_2 = -2 + 4 i$ and $z_3 = \sqrt 3 - 2 i$.
Then:
:$\overline {\paren {z_2 + z_3} \paren {z_1 - z_3} } = -7 + 3 \sqrt 3 + \sqrt 3 i$
== Proof ==
== Sources ==
* : $1$: Complex Numbers: Supplementary Problems: Fundamental Operations with Complex Numbers: $54 \ \text {(g)}$
Category:Examples of Complex Arithmetic
Complex Arithmetic/Examples/z 1^2 + 2 z 1 - 3

== Example of Complex Arithmetic ==
Let $z_1 = 1 - i$.
Then:
:${z_1}^2 + 2 z_1 - 3 = -1 - 4 i$
== Proof ==
== Sources ==
* : $1$: Complex Numbers: Supplementary Problems: Fundamental Operations with Complex Numbers: $54 \ \text {(a)}$
Category:Examples of Complex Arithmetic
Complex Arithmetic/Examples/z 1^3 - 3 z 1^2 + 4 z 1 - 8

== Example of Complex Arithmetic ==
Let $z_1 = 2 + i$.
Then:
:$z_1^3 - 3 z_1^2 + 4 z_1 - 8 = -7 + 3 i$
== Proof ==
So:
== Sources ==
* : $1$: Complex Numbers: Solved Problems: Fundamental Operations with Complex Numbers: $2 \ \text{(b)}$
Category:Examples of Complex Arithmetic
Complex Conjugate/Examples/conj z1 - conj z2

== Example of Complex Conjugates ==
Let $z_1 = 4 - 3 i$ and $z_2 = -1 + 2 i$.
Then:
:$\overline {z_1} - \overline {z_2} = 5 + 5 i$
== Proof 1 ==
== Proof 2 ==
Category:Examples of Complex Conjugates
Complex Conjugate/Examples/conj z1 - conj z2/Proof 1

== Example of Complex Conjugates ==
== Proof ==
An illustration of the modulus of the difference of the complex conjugates of the complex numbers:
:$z_1 = 4 - 3 i$
:$z_2 = -1 + 2 i$
is given below:
:
== Sources ==
* : $1$: Complex Numbers: Supplementary Problems: Graphical Representation of Complex Numbers. Vectors: $63 \ \text {(c)}$
Category:Examples of Complex Conjugates
Complex Conjugate Coordinates/Examples

== Examples of Complex Conjugate Coordinates ==
=== Example: $2 x + y = 5$ ===
=== Example: $x^2 + y^2 = 36$ ===
=== Example: $\paren {x - 3}^2 + y^2 = 9$ ===
=== Example: $2 x - 3 y = 5$ ===
=== Example: $4 x^2 + 16 y^2 = 25$ ===
Category:Examples of Complex Conjugate Coordinates
Complex Conjugate Coordinates/Examples/(x-3)^2 + y^2 = 9

== Example of Complex Conjugate Coordinates ==
The equation of the circle:
:$\paren {x - 3}^2 + y^2 = 9$
can be expressed in complex conjugate coordinates as:
:$\paren {z - 3} \paren {\overline z - 3} = 9$
== Proof ==
== Sources ==
* : $1$: Complex Numbers: Supplementary Problems: Conjugate Coordinates: $117 \ \text{(a)}$
Category:Examples of Complex Conjugate Coordinates