hcju/proofwikips huggingface dataset:
Definition:Simultaneous Equations/Consistency
A system of simultaneous equations: :$\forall i \in \set {1, 2, \ldots m} : \map {f_i} {x_1, x_2, \ldots x_n} = \beta_i$ that has at least one solution is '''consistent'''. If a system has no solutions, it is '''inconsistent'''.
Definition:Echelon Matrix/Echelon Form/Non-Unity Variant
{{:Definition:Echelon Matrix/Echelon Form/Non-Unity Variant/Definition 1}}
Definition:Echelon Matrix/Reduced Echelon Form
The matrix $\mathbf A$ is in '''reduced echelon form''' , in addition to being in echelon form, the leading $1$ in any non-zero row is the only non-zero element in the column in which that $1$ occurs. Such a matrix is called a '''reduced echelon matrix'''.
Definition:Generalized Sum/Net Convergence
Let $\left({g_n}\right)_{n \in \N}$ be a sequence in $G$. The series $\displaystyle \sum_{n \mathop = 1}^\infty g_n$ '''converges as a net''' or '''has net convergence'''  the generalized sum $\displaystyle \sum \left\{{g_n: n \in \N}\right\}$ converges.
Definition:Generalized Sum/Absolute Net Convergence
Let $V$ be a Banach space. Let $\family {v_i}_{i \mathop \in I}$ be an indexed subset of $V$. Then $\displaystyle \sum \set {v_i: i \in I}$ '''converges absolutely'''  $\displaystyle \sum \set {\norm {v_i}: i \mathop \in I}$ converges. This nomenclature is appropriate as we have Absolutely Convergent Generalized Sum Converges.
Definition:Finite Cyclic Group
Let $\struct {G, \circ}$ be a cyclic group. Then $\struct {G, \circ}$ is a '''finite cyclic group'''  it is a finite group.
Definition:Sequential Continuity/Point
Let $x \in X$. Then $f$ is '''sequentially continuous at $x$''' iff: :For every sequence $\left\langle{x_n}\right\rangle_{n \ge 1}$ in $X$ which converges to $x$, the sequence $\left\langle{f \left({x_n}\right)}\right\rangle_{n \ge 1}$ in $Y$ converges to $f \left({x}\right)$.
Definition:Complex Number/Definition 1
A '''complex number''' is a number in the form $a + b i$ or $a + i b$ where: :$a$ and $b$ are real numbers :$i$ is a square root of $-1$, that is, $i = \sqrt {-1}$.
Definition:Complex Number/Definition 2
A '''complex number''' is an ordered pair $\tuple {x, y}$ where $x, y \in \R$ are real numbers, on which the operations of addition and multiplication are defined as follows: === Complex Addition === {{:Definition:Complex Number/Definition 2/Addition}} === Complex Multiplication === {{:Definition:Complex Number/Definition 2/Multiplication}} === Scalar Product === {{:Definition:Complex Number/Definition 2/Scalar Product}}
Definition:Complex Number/Imaginary Unit
The entity $i := 0 + 1 i$ is known as the '''imaginary unit'''.
Definition:Complex Number/Wholly Real
A complex number $z = a + i b$ is '''wholly real'''  $b = 0$.
Definition:Complex Number/Wholly Imaginary
A complex number $z = a + i b$ is '''wholly imaginary'''  $a = 0$.
Definition:Complex Number/Complex Plane
Because a complex number can be expressed as an ordered pair, we can plot the number $x + i y$ on the real number plane $\R^2$: :400px This representation is known as the '''complex plane'''. === Real Axis === {{:Definition:Complex Number/Complex Plane/Real Axis}} === Imaginary Axis === {{:Definition:Complex Number/Complex Plane/Imaginary Axis}}
Definition:Complex Number/Polar Form
For any complex number $z = x + i y \ne 0$, let: {{begin-eqn}} {{eqn | l = r       | r = \cmod z = \sqrt {x^2 + y^2}       | c = the modulus of $z$, and }} {{eqn | l = \theta       | r = \arg z       | c = the argument of $z$ (the angle which $z$ yields with the real line) }} {{end-eqn}} where $x, y \in \R$. From the definition of $\arg z$: :$(1): \quad \dfrac x r = \cos \theta$ :$(2): \quad \dfrac y r = \sin \theta$ which implies that: :$x = r \cos \theta$ :$y = r \sin \theta$ which in turn means that any number $z = x + i y \ne 0$ can be written as: :$z = x + i y = r \paren {\cos \theta + i \sin \theta}$ The pair $\polar {r, \theta}$ is called the '''polar form''' of the complex number $z \ne 0$.  The number $z = 0 + 0 i$ is defined as $\polar {0, 0}$.
Definition:Complex Number/Complex Plane/Real Axis
Complex numbers of the form $\tuple {x, 0}$, being wholly real, appear as points on the $x$-axis.
Definition:Complex Number/Complex Plane/Imaginary Axis
Complex numbers of the form $\tuple {0, y}$, being wholly imaginary, appear as points on the points on the $y$-axis. This line is known as the '''imaginary axis'''.
Definition:Complex Number/Definition 2/Addition
Let $\tuple {x_1, y_1}$ and $\tuple {x_2, y_2}$ be complex numbers. Then $\tuple {x_1, y_1} + \tuple {x_2, y_2}$ is defined as: :$\tuple {x_1, y_1} + \tuple {x_2, y_2}:= \tuple {x_1 + x_2, y_1 + y_2}$
Definition:Complex Number/Definition 2/Multiplication
Let $\tuple {x_1, y_1}$ and $\tuple {x_2, y_2}$ be complex numbers. Then $\tuple {x_1, y_1} \tuple {x_2, y_2}$ is defined as: :$\tuple {x_1, y_1} \tuple {x_2, y_2} := \tuple {x_1 x_2 - y_1 y_2, x_1 y_2 + y_1 x_2}$
Definition:Argument of Complex Number/Principal Argument
Let $R$ be the principal range of the complex numbers $\C$. The unique value of $\theta$ in $R$ is known as the '''principal value of the argument''', or just '''principal argument''', of $z$. This is denoted $\Arg z$. Note the capital $A$. The standard practice is for $R$ to be $\hointl {-\pi} \pi$. This ensures that the '''principal argument''' is continuous on the real axis for positive numbers. Thus, if $z$ is represented in the complex plane, the '''principal argument''' $\Arg z$ is intuitively defined as the angle which $z$ yields with the real ($y = 0$) axis.
Definition:Pairwise Disjoint Events
Let $\family {A_i}_{i \mathop \in I}$ be an indexed family of events in a probability space. Then $\family {A_i}$ is '''pairwise disjoint''' : :$\forall i, j \in I, i \ne j: A_i \cap A_j = \O$ That is, a set of events is '''pairwise disjoint'''  every pair of them is disjoint.
Definition:Complex Number/Polar Form/Exponential Form
From Euler's Formula: :$e^{i \theta} = \cos \theta + i \sin \theta$ so $z$ can also be written in the form: :$z = r e^{i \theta}$
Definition:Hyperbolic Cotangent
{{:Definition:Hyperbolic Cotangent/Definition 1}}
Definition:Hyperbolic Secant
{{:Definition:Hyperbolic Secant/Definition 1}}
Definition:Hyperbolic Cosecant
{{:Definition:Hyperbolic Cosecant/Definition 1}}
Definition:Hyperbolic Tangent/Definition 1
The '''hyperbolic tangent''' function is defined on the complex numbers as: :$\tanh: X \to \C$: :$\forall z \in X: \tanh z := \dfrac {e^z - e^{-z} } {e^z + e^{-z} }$ where: : $X = \set {z : z \in \C, \ e^z + e^{-z} \ne 0}$
Definition:Hyperbolic Tangent/Definition 2
The '''hyperbolic tangent''' function is defined on the complex numbers as: :$\tanh: X \to \C$: :$\forall z \in X: \tanh z := \dfrac {\sinh z} {\cosh z}$ where: :$\sinh$ is the hyperbolic sine :$\cosh$ is the hyperbolic cosine :$X = \set {z : z \in \C, \ \cosh z \ne 0}$
Definition:Hyperbolic Tangent/Definition 3
The '''hyperbolic tangent''' function is defined on the complex numbers as: :$\tanh: X \to \C$: :$\forall z \in X: \tanh z := \dfrac {e^{2 z} - 1} {e^{2 z} + 1}$ where: : $X = \set {z: z \in \C, \ e^{2 z} + 1 \ne 0}$
Definition:Hyperbolic Cotangent/Definition 1
The '''hyperbolic cotangent''' function is defined on the complex numbers as: :$\coth: X \to \C$: :$\forall z \in X: \coth z := \dfrac {e^z + e^{-z} } {e^z - e^{-z}}$ where: : $X = \set {z : z \in \C, \ e^z - e^{-z} \ne 0}$
Definition:Hyperbolic Cotangent/Definition 2
The '''hyperbolic cotangent''' function is defined on the complex numbers as: :$\coth: X \to \C$: :$\forall z \in X: \coth z := \dfrac {\cosh z} {\sinh z}$ where: :$\sinh$ is the hyperbolic sine :$\cosh$ is the hyperbolic cosine :$X = \set {z : z \in \C, \ \sinh z \ne 0}$
Definition:Hyperbolic Cotangent/Definition 3
The '''hyperbolic cotangent''' function is defined on the complex numbers as: :$\coth: X \to \C$: :$\forall z \in X: \coth z := \dfrac {e^{2 z} + 1} {e^{2 z} - 1}$ where: :$X = \set {z : z \in \C, \ e^{2 z} - 1 \ne 0}$
Definition:Hyperbolic Secant/Definition 1
The '''hyperbolic secant''' function is defined on the complex numbers as: :$\sech: X \to \C$: :$\forall z \in X: \sech z := \dfrac 2 {e^z + e^{-z} }$ where: : $X = \set {z: z \in \C, \ e^z + e^{-z} \ne 0}$
Definition:Hyperbolic Secant/Definition 2
The '''hyperbolic secant''' function is defined on the complex numbers as: :$\sech: X \to \C$: :$\forall z \in X: \sech z := \dfrac 1 {\cosh z}$ where: :$\cosh$ is the hyperbolic cosine :$X = \set {z: z \in \C, \ \cosh z \ne 0}$
Definition:Hyperbolic Cosecant/Definition 1
The '''hyperbolic cosecant''' function is defined on the complex numbers as: :$\csch: X \to \C$: :$\forall z \in X: \csch z := \dfrac 2 {e^z - e^{-z} }$ where: : $X = \set {z: z \in \C, \ e^z - e^{-z} \ne 0}$
Definition:Hyperbolic Cosecant/Definition 2
The '''hyperbolic cosecant''' function is defined on the complex numbers as: :$\csch: X \to \C$: :$\forall z \in X: \csch z := \dfrac 1 {\sinh z}$ where: :$\sinh$ is the hyperbolic sine :$X = \set {z: z \in \C, \ \sinh z \ne 0}$
Definition:Inverse Hyperbolic Function
Let $f: \C \to \C$ be one of the hyperbolic functions on the set of complex numbers. The '''inverse hyperbolic function''' $f^{-1} \subseteq \C \times \C$ is actually a multifunction, as in general for a given $y \in \C$ there is more than one $x \in \C$ such that $y = f \left({x}\right)$. As with the inverse trigonometric functions, it is usual to restrict the codomain of the multifunction so as to allow $f^{-1}$ to be single-valued.
Definition:Inverse Hyperbolic Sine/Real
=== Definition 1 === {{:Definition:Inverse Hyperbolic Sine/Real/Definition 1}} === Definition 2 === {{:Definition:Inverse Hyperbolic Sine/Real/Definition 2}}
Definition:Inverse Hyperbolic Sine/Complex/Definition 1
The '''inverse hyperbolic sine''' is a multifunction defined as: :$\forall z \in \C: \sinh^{-1} \left({z}\right) := \left\{{w \in \C: z = \sinh \left({w}\right)}\right\}$ where $\sinh \left({w}\right)$ is the hyperbolic sine function.
Definition:Inverse Hyperbolic Sine/Complex/Definition 2
The '''inverse hyperbolic sine''' is a multifunction defined as: :$\forall z \in \C: \sinh^{-1} \left({z}\right) := \left\{{\ln \left({z + \sqrt{\left|{z^2 + 1}\right|} e^{\left({i /  2}\right) \arg \left({z^2 + 1}\right)} }\right) + 2 k \pi i: k \in \Z}\right\}$ where: : $\sqrt{\left|{z^2 + 1}\right|}$ denotes the positive square root of the complex modulus of $z^2 + 1$ : $\arg \left({z^2 + 1}\right)$ denotes the argument of $z^2 + 1$ : $\ln$ denotes the complex natural logarithm considered as a  multifunction.
Definition:Inverse Hyperbolic Cosine/Complex
=== Definition 1 === {{:Definition:Inverse Hyperbolic Cosine/Complex/Definition 1}} === Definition 2 === {{:Definition:Inverse Hyperbolic Cosine/Complex/Definition 2}}
Definition:Inverse Hyperbolic Tangent/Complex
Let $S$ be the subset of the complex plane: :$S = \C \setminus \left\{{-1 + 0 i, 1 + 0 i}\right\}$ === Definition 1 === {{:Definition:Inverse Hyperbolic Tangent/Definition 1}} === Definition 2 === {{:Definition:Inverse Hyperbolic Tangent/Definition 2}}
Definition:Inverse Hyperbolic Cotangent/Complex
Let $S$ be the subset of the complex plane: :$S = \C \setminus \left\{{-1 + 0 i, 1 + 0 i}\right\}$ === Definition 1 === {{:Definition:Inverse Hyperbolic Cotangent/Complex/Definition 1}} === Definition 2 === {{:Definition:Inverse Hyperbolic Cotangent/Complex/Definition 2}}
Definition:Inverse Hyperbolic Secant/Complex
=== Definition 1 === {{:Definition:Inverse Hyperbolic Secant/Definition 1}} === Definition 2 === {{:Definition:Inverse Hyperbolic Secant/Definition 2}}
Definition:Inverse Hyperbolic Cosecant/Complex
=== Definition 1 === {{:Definition:Inverse Hyperbolic Cosecant/Definition 1}} === Definition 2 === {{:Definition:Inverse Hyperbolic Cosecant/Definition 2}}
Definition:Inverse Hyperbolic Cosine/Complex/Definition 1
The '''inverse hyperbolic cosine''' is a multifunction defined as: :$\forall z \in \C: \cosh^{-1} \left({z}\right) := \left\{{w \in \C: z = \cosh \left({w}\right)}\right\}$ where $\cosh \left({w}\right)$ is the hyperbolic cosine function.
Definition:Inverse Hyperbolic Tangent/Complex/Definition 1
The '''inverse hyperbolic tangent''' is a multifunction defined on $S$ as: :$\forall z \in S: \tanh^{-1} \paren z := \set {w \in \C: z = \tanh \paren w}$ where $\tanh \paren w$ is the hyperbolic tangent function.
Definition:Inverse Hyperbolic Cotangent/Complex/Definition 1
The '''inverse hyperbolic cotangent''' is a multifunction defined on $S$ as: :$\forall z \in S: \coth^{-1} \left({z}\right) := \left\{{w \in \C: z = \coth \left({w}\right)}\right\}$ where $\coth \left({w}\right)$ is the hyperbolic cotangent function.
Definition:Inverse Hyperbolic Secant/Complex/Definition 1
The '''inverse hyperbolic secant''' is a multifunction defined as: :$\forall z \in \C_{\ne 0}: \operatorname{sech}^{-1} \left({z}\right) := \left\{{w \in \C: z = \operatorname{sech} \left({w}\right)}\right\}$ where $\operatorname{sech} \left({w}\right)$ is the hyperbolic secant function.
Definition:Inverse Hyperbolic Cosecant/Complex/Definition 1
The '''inverse hyperbolic cosecant''' is a multifunction defined as: :$\forall z \in \C_{\ne 0}: \map {\csch^{-1} } z := \set {w \in \C: z = \map \csch w}$ where $\map \csch w$ is the hyperbolic cosecant function.
Definition:Inverse Hyperbolic Cosine/Complex/Definition 2
The '''inverse hyperbolic cosine''' is a multifunction defined as: :$\forall z \in \C: \cosh^{-1} \left({z}\right) := \left\{{\ln \left({z + \sqrt{\left|{z^2 - 1}\right|} e^{\left({i / 2}\right) \arg \left({z^2 - 1}\right)} }\right) + 2 k \pi i: k \in \Z}\right\}$ where: : $\sqrt{\left|{z^2 - 1}\right|}$ denotes the positive square root of the complex modulus of $z^2 - 1$ : $\arg \left({z^2 - 1}\right)$ denotes the argument of $z^2 - 1$ : $\ln$ denotes the complex natural logarithm considered as a  multifunction.
Definition:Inverse Hyperbolic Tangent/Complex/Definition 2
The '''inverse hyperbolic tangent''' is a multifunction defined on $S$ as: :$\forall z \in S: \tanh^{-1} \left({z}\right) := \left\{{\dfrac 1 2 \ln \left({\dfrac {1 + z} {1 - z} }\right) + k \pi i: k \in \Z}\right\}$ where $\ln$ denotes the complex natural logarithm considered as a  multifunction.
Definition:Inverse Hyperbolic Cotangent/Complex/Definition 2
The '''inverse hyperbolic cotangent''' is a multifunction defined  on $S$ as: :$\forall z \in S: \coth^{-1} \left({z}\right) := \left\{{\dfrac 1 2 \ln \left({\dfrac {z + 1} {z - 1} }\right) + k \pi i: k \in \Z}\right\}$ where $\ln$ denotes the complex natural logarithm considered as a  multifunction.
Definition:Inverse Hyperbolic Secant/Complex/Definition 2
The '''inverse hyperbolic secant''' is a multifunction defined as: :$\forall z \in \C_{\ne 0}: \map {\sech^{-1} } z := \set {\map \ln {\dfrac {1 + \sqrt {\size {1 - z^2} } e^{\paren {i / 2} \map \arg {1 - z^2} } } z} + 2 k \pi i: k \in \Z}$ where: :$\sqrt {\size {1 - z^2} }$ denotes the positive square root of the complex modulus of $1 - z^2$ :$\map \arg {1 - z^2}$ denotes the argument of $1 - z^2$ :$\ln$ denotes the complex natural logarithm as a multifunction.
Definition:Inverse Hyperbolic Cosecant/Complex/Definition 2
The '''inverse hyperbolic cosecant''' is a multifunction defined as: :$\forall z \in \C_{\ne 0}: \map {\csch^{-1} } z := \set {\map \ln {\dfrac {1 + \sqrt {\size {z^2 + 1} } e^{\paren {i / 2} \, \map \arg {z^2 + 1} } } z} + 2 k \pi i: k \in \Z}$ where: :$\sqrt {\size {z^2 + 1} }$ denotes the positive square root of the complex modulus of $z^2 + 1$ :$\map \arg {z^2 + 1}$ denotes the argument of $z^2 + 1$ :$\ln$ denotes the complex natural logarithm considered as a multifunction.
Definition:Periodic Function/Period
The '''period''' of $f$ is the smallest value $\cmod L \in \R_{\ne 0}$ such that: : $\forall x \in X: \map f x = \map f {x + L}$ where $\cmod L$ is the modulus of $L$.
Definition:Periodic Function/Real
Let $f: \R \to \R$ be a real function. Then $f$ is '''periodic''' : :$\exists L \in \R_{\ne 0}: \forall x \in \R: \map f x = \map f {x + L}$
Definition:Periodic Function/Complex
Let $f: \C \to \C$ be a complex function. Then $f$ is '''periodic''' : :$\exists L \in \C_{\ne 0}: \forall x \in \C: f \left({x}\right) = f \left({x + L}\right)$
Definition:Inverse Sine
=== Real Numbers === {{:Definition:Inverse Sine/Real}} === Complex Plane === {{:Definition:Inverse Sine/Complex}}
Definition:Inverse Cosine
=== Real Numbers === {{:Definition:Inverse Cosine/Real}} === Complex Plane === {{:Definition:Inverse Cosine/Complex}}
Definition:Inverse Tangent
=== Real Numbers === {{:Definition:Inverse Tangent/Real}} === Complex Plane === {{:Definition:Inverse Tangent/Complex}}
Definition:Inverse Cosecant
=== Real Numbers === {{:Definition:Inverse Cosecant/Real}} === Complex Plane === {{:Definition:Inverse Cosecant/Complex}}
Definition:Inverse Secant
=== Real Numbers === {{:Definition:Inverse Secant/Real}} === Complex Plane === {{:Definition:Inverse Secant/Complex}}
Definition:Inverse Cotangent
=== Real Numbers === {{:Definition:Inverse Cotangent/Real}} === Complex Plane === {{:Definition:Inverse Cotangent/Complex}}
Definition:Quadratic Equation
A '''quadratic equation''' is a polynomial equation of the form: :$a x^2 + b x + c = 0$ such that $a \ne 0$.
Definition:Cubic Equation/Resolvent
Let: : $y = x + \dfrac b {3 a}$ : $Q = \dfrac {3 a c - b^2} {9 a^2}$ : $R = \dfrac {9 a b c - 27 a^2 d - 2 b^3} {54 a^3}$ Let $y = u + v$ where $u v = -Q$. The '''resolvent equation''' of the cubic is given by: : $u^6 - 2 R u^3 - Q^3$
Definition:Division/Notation
The operation of division can be denoted as: :$a / b$, which is probably the most common in the general informal context :$\dfrac a b$, which is the preferred style on  :$a \div b$, which is rarely seen outside grade school.
Definition:Division/Rational Numbers
Let $\struct {\Q, +, \times}$ be the field of rational numbers. The operation of '''division''' is defined on $\Q$ as: :$\forall a, b \in \Q \setminus \set 0: a / b := a \times b^{-1}$ where $b^{-1}$ is the multiplicative inverse of $b$ in $\Q$.
Definition:Division/Real Numbers
Let $\struct {\R, +, \times}$ be the field of real numbers. The operation of '''division''' is defined on $\R$ as: :$\forall a, b \in \R \setminus \set 0: a / b := a \times b^{-1}$ where $b^{-1}$ is the multiplicative inverse of $b$ in $\R$.
Definition:Division/Complex Numbers
Let $\struct {\C, +, \times}$ be the field of complex numbers. The operation of '''division''' is defined on $\C$ as: :$\forall a, b \in \C \setminus \set 0: \dfrac a b := a \times b^{-1}$ where $b^{-1}$ is the multiplicative inverse of $b$ in $\C$.
Definition:Division/Divisor
Let $c = a / b$ denote the division operation on two elements $a$ and $b$ of a field. The element $b$ is the '''divisor''' of $a$.
Definition:Division/Dividend
Let $c = a / b$ denote the division operation on two elements $a$ and $b$ of a field. The element $a$ is the '''dividend''' of $b$.
Definition:Division/Quotient
Let $c = a / b$ denote the division operation on two elements $a$ and $b$ of a field. The element $c$ is the '''quotient of $a$ (divided) by $b$'''.
Definition:Division
Let $\struct {F, +, \times}$ be a field. Let the zero of $F$ be $0_F$. The operation of '''division''' is defined as: :$\forall a, b \in F \setminus \set {0_F}: a / b := a \times b^{-1}$ where $b^{-1}$ is the multiplicative inverse of $b$.
Definition:Irrational Number/Approximation
From its definition, it is not possible to express an irrational number precisely in terms of a fraction. From Decimal Expansion of Irrational Number neither Terminates nor Recurs, it is not possible to express it precisely by a decimal expansion either. However, it is possible to express it to an arbitrary level of precision. Let $x$ be an irrational number whose decimal expansion is $\sqbrk {n.d_1 d_2 d_3 \ldots}_{10}$. Then: :$\displaystyle n + \sum_{j \mathop = 1}^k \frac {d_j} {10^j} \le x < n + \sum_{j \mathop = 1}^k \frac {d_j} {10^j} + \frac 1 {10^k}$ for all $k \in \Z: k \ge 1$. Then all one needs to do is state that $x$ is expressed as '''accurate to $k$ decimal places'''.
Definition:Rational Number/Fraction
By definition, a rational number is a number which can be expressed in the form: :$\dfrac a b$ where $a$ and $b$ are integers. A '''fraction''' is a rational number such that, when expressed in canonical form $\dfrac a b$ (that is, such that $a$ and $b$ are coprime), the denominator $b$ is not $1$.
Definition:Subtraction/Complex Numbers
Let $\struct {\C, +, \times}$ be the field of complex numbers. The operation of '''subtraction''' is defined on $\C$ as: :$\forall a, b \in \C: a - b := a + \paren {-b}$ where $-b$ is the negative of $b$ in $\C$.
Definition:Negative/Complex Number
As the Complex Numbers cannot be Ordered Compatibly with Ring Structure, the concept of a '''negative complex number''', relative to a specified zero, is not defined. However, the '''negative''' of a complex number is defined as follows: Let $z = a + i b$ be a complex number. Then the '''negative of $z$''' is defined as: :$-z = -a - i b$
Definition:Cartesian Product/Coordinate
Let $\displaystyle \prod_{i \mathop \in I} S_i$ be a cartesian product. Let $j \in I$, and let $s = \sequence {s_i}_{i \mathop \in I} \in \displaystyle \prod_{i \mathop \in I} S_i$. Then $s_j$ is called the '''$j$th coordinate of $s$'''. If the indexing set $I$ consists of ordinary numbers $1, 2, \ldots, n$, one speaks about, for example, the '''first''', '''second''', or '''$n$th coordinate'''. For an element $\tuple {s, t} \in S \times T$ of a binary cartesian product, $s$ is the '''first coordinate''', and $t$ is the '''second coordinate'''.
Definition:Distance/Points/Complex Numbers
Let $x, y \in \C$ be complex numbers. Let $\cmod {x - y}$ be the complex modulus of $x - y$. Then the function $d: \C^2 \to \R$: :$\map d {x, y} = \cmod {x - y}$ is called the '''distance between $x$ and $y$'''.
Definition:Argument of Complex Number/Principal Range
It is understood that the argument of a complex number $z$ is unique only up to multiples of $2 k \pi$. With this understanding, we can limit the choice of what $\theta$ can be for any given $z$ by requiring that $\theta$ lie in some half open interval of length $2 \pi$. The most usual of these are: :$\hointr 0 {2 \pi}$ :$\hointl {-\pi} \pi$ but in theory any such interval may be used. This interval is known as the '''principal range'''.
Definition:Substitution (Formal Systems)/Well-Formed Part
Let $\FF$ be a formal language with alphabet $\AA$. Let $\mathbf B$ be a well-formed formula of $\FF$. Let $\mathbf A$ be a well-formed part of $\mathbf B$. Let $\mathbf A'$ be another well-formed formula. Then the '''substitution of $\mathbf A'$ for $\mathbf A$ in $\mathbf B$''' is the collation resulting from $\mathbf B$ by replacing all occurrences of $\mathbf A$ in $\mathbf B$ by $\mathbf A'$. It is denoted as $\map {\mathbf B} {\mathbf A' \mathbin {//} \mathbf A}$.
Definition:Complex Number as Vector
Let $z = x + i y$ be a complex number. Then $z$ can be considered as a vector $OP$ in the complex plane such that: : its initial point is the origin : its terminal point $P$ is the point $\tuple {x, y}$. Two vectors which have the same magnitude and direction, but different initial points, are considered '''equal'''.
Definition:Spherical Representation of Complex Number
Let $\PP$ be the complex plane. Let $\mathbb S$ be the unit sphere which is tangent to $\PP$ at $\tuple {0, 0}$ (that is, where $z = 0$). Let the diameter of $\mathbb S$ perpendicular to $\PP$ through $\tuple {0, 0}$ be $NS$ where $S$ is the point $\tuple {0, 0}$. Let the point $N$ be referred to as the '''north pole''' of $\mathbb S$ and $S$ be referred to as the '''south pole''' of $\mathbb S$. Let $A$ be a point on $P$. Let the line $NA$ be constructed. :900px Then $NA$ passes through a point of $\mathbb S$. Thus any complex number can be represented by a point on the surface of the unit sphere. The point $N$ on $\mathbb S$ corresponds to the point at infinity. Thus any point on the surface of the unit sphere corresponds to a point on the extended complex plane.