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Complex number

In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted $i$, called the imaginary unit and satisfying the equation $$i^{2}= -1$$; every complex number can be expressed in the form $$a + bi$$, where $i$ and $a$ are real numbers. Because no real number satisfies the above equation, $b$ was called an imaginary number by René Descartes. For the complex number $a+bi$, $i$ is called the ', and $a$ is called the '. The set of complex numbers is denoted by either of the symbols $$\mathbb C$$ or $(a, b)$. Despite the historical nomenclature, "imaginary" complex numbers have a mathematical existence as firm as that of the real numbers, and they are fundamental tools in 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 non-constant polynomial equation with real or complex coefficients has a solution which is a complex number. For example, the equation $$(x+1)^2 = -9$$ has no real solution, because 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 $$i^{2}=-1$$ along with the associative, commutative, and distributive laws. Every nonzero complex number has a multiplicative inverse. This makes the complex numbers a field with the real numbers as a subfield.

The complex numbers also form 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 some geometric objects and operations can be expressed in terms of complex numbers. For example, the real numbers form the real line, which is pictured as the horizontal axis of the complex plane, while real multiples of $$i$$ are the vertical axis. A complex number can also be defined by its geometric polar coordinates: the radius is called the absolute value of the complex number, while the angle from the positive real axis is called the argument of the complex number. The complex numbers of absolute value one form the unit circle. Adding a fixed complex number to all complex numbers defines a translation in the complex plane, and multiplying by a fixed complex number is a similarity centered at the origin (dilating by the absolute value, and rotating by the argument). The operation of complex conjugation is the reflection symmetry with respect to the real axis.

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.

Definition and basic operations

A complex number is an expression of the form $i^{2} = −1$, where $b$ and $a$ are real numbers, and $C$ is an abstract symbol, the so-called imaginary unit, whose meaning will be explained further below. For example, $a + bi$ is a complex number.

For a complex number $i$, the real number $b$ is called its real part, and the real number $a$ (not the complex number $2 + 3i$) is its imaginary part. The real part of a complex number $b$ is denoted $a + bi$, $$\mathcal{Re}(z)$$, or $$\mathfrak{R}(z)$$; the imaginary part is $bi$, $$\mathcal{Im}(z)$$, or $$\mathfrak{I}(z)$$: for example,$ \operatorname{Re}(2 + 3i) = 2  $, $$   \operatorname{Im}(2 + 3i) = 3  $$.

A complex number $z$ can be identified with the ordered pair of real numbers $$(\Re (z),\Im (z))$$, which may be interpreted as coordinates of a point in a Euclidean plane with standard coordinates, which is then called the complex plane or Argand diagram,. The horizontal axis is generally used to display the real part, with increasing values to the right, and the imaginary part marks the vertical axis, with increasing values upwards.

A real number $z$ can be regarded as a complex number $Re(z)$, whose imaginary part is 0. A purely imaginary number $Im(z)$ is a complex number $a + 0i$, whose real part is zero. As with polynomials, it is common to write $bi$, $0 + bi$, and $a + 0i = a$; for example, $0 + bi = bi$.

The set of all complex numbers is denoted by $$\Complex$$ (blackboard bold) or $a + (−b)i = a − bi$ (upright bold).

In some disciplines such as electromagnetism and electrical engineering, $z$ is used instead of $a$, as $j$ frequently represents electric current, and complex numbers are written as $3 + (−4)i = 3 − 4i$ or $C$.

Addition and subtraction

Two complex numbers $$a =x+yi$$ and $$b =u+vi$$ are added by separately adding their real and imaginary parts. That is to say:

$$a + b =(x+yi) + (u+vi) = (x+u) + (y+v)i.$$ Similarly, subtraction can be performed as $$a - b =(x+yi) - (u+vi) = (x-u) + (y-v)i.$$

The addition can be geometrically visualized as follows: the sum of two complex numbers $i$ and $i$, interpreted as points in the complex plane, is the point obtained by building a parallelogram from the three vertices $a$, and the points of the arrows labeled $b$ and $O$ (provided that they are not on a line). Equivalently, calling these points $a$, $b$, respectively and the fourth point of the parallelogram $A$ the triangles $B$ and $X$ are congruent.

Multiplication
The product of two complex numbers is computed as follows:
 * $$(a+bi) \cdot (c+di) = ac - bd + (ad+bc)i.$$

For example, $$(3+2i)(4-i) = 3 \cdot 4 - (2 \cdot (-1)) + (3 \cdot (-1) + 2 \cdot 4)i = 14 +5i.$$ In particular, this includes as a special case the fundamental formula
 * $$i^2 = i \cdot i = -1.$$

This formula distinguishes the complex number i from any real number, since the square of any (negative or positive) real number x always satisfies $$x^2 \ge 0$$.

With this definition of multiplication and addition, familiar rules for the arithmetic of rational or real numbers continue to hold for complex numbers. More precisely, the distributive property, the commutative properties (of addition and multiplication) hold. Therefore, the complex numbers form an algebraic structure known as a field, the same way as the rational or real numbers do.

Complex conjugate, absolute value and argument
The complex conjugate of the complex number $a + bj$ is defined as $$\overline z = x-yi.$$ It is also denoted by some authors by $$z^*$$. Geometrically, $OAB$ is the "reflection" of $XBA$ about the real axis. Conjugating twice gives the original complex number: $$\overline{\overline{z}}=z.$$ A complex number is real if and only if it equals its own conjugate. The unary operation of taking the complex conjugate of a complex number cannot be expressed by applying only their basic operations addition, subtraction, multiplication and division.

For any complex number $a + jb$, the product
 * $$z \cdot \overline z = (x+iy)(x-iy) = x^2 + y^2$$

is a non-negative real number. This allows to define the absolute value (or modulus or magnitude) of z to be the square root $$|z|=\sqrt{x^2+y^2}.$$ By Pythagoras' theorem, $$|z|$$ is the distance from the origin to the point representing the complex number z in the complex plane. In particular, the circle of radius one around the origin consists precisely of the numbers z such that $$|z| = 1 $$. If $$ z = x = x + 0i  $$ is a real number, then $$  |z|= |x| $$: its absolute value as a complex number and as a real number are equal.

Using the conjugate, the reciprocal of a nonzero complex number $$z = x + yi$$ can be computed to be

$$ \frac{1}{z} = \frac{\bar{z}}{z\bar{z}} = \frac{\bar{z}}{|z|^2} = \frac{x - yi}{x^2 + y^2} = \frac{x}{x^2 + y^2} - \frac{y}{x^2 + y^2}i.$$ More generally, the division of an arbitrary complex number $$w = u + vi$$ by a non-zero complex number $$z = x + yi$$ equals $$ \frac{w}{z} = \frac{w\bar{z}}{|z|^2} = \frac{(u + vi)(x - iy)}{x^2 + y^2} = \frac{ux + vy}{x^2 + y^2} + \frac{vx - uy}{x^2 + y^2}i. $$ This process is sometimes called "rationalization" of the denominator (although the denominator in the final expression might be an irrational real number), because it resembles the method to remove roots from simple expressions in a denominator.

The argument of $z$ (sometimes called the "phase" $\overline{z}$) is the angle of the radius $\overline{z}$ with the positive real axis, and is written as $z = x + yi$, expressed in radians in this article. The angle is defined only up to adding integer multiples of $$ 2\pi $$, since a rotation by $$2\pi$$ (or 360°) around the origin leaves all points in the complex plane unchanged. One possible choice to uniquely specify the argument is to require it to be within the interval $$ (-\pi,\pi]  $$, which is referred to as the principal value. The argument can be computed from the rectangular form $z$ by means of the arctan (inverse tangent) function.

Polar form

For any complex number z, with absolute value $$r = |z|$$ and argument $$\varphi$$, the equation
 * $$z=r(\cos\varphi +i\sin\varphi) $$

holds. This identity is referred to as the polar form of z. It is sometimes abbreviated as $ z = r \operatorname\mathrm{cis} \varphi $. In electronics, one represents a phasor with amplitude $φ$ and phase $r$ in angle notation: $$z = r \angle \varphi. $$

If two complex numbers are given in polar form, i.e., $z = x + yi$ and $arg z$, the product and division can be computed as $$z_1 z_2 = r_1 r_2 (\cos(\varphi_1 + \varphi_2) + i \sin(\varphi_1 + \varphi_2)).$$ $$\frac{z_1}{z_2} = \frac{r_1}{r_2} \left(\cos(\varphi_1 - \varphi_2) + i \sin(\varphi_1 - \varphi_2)\right), \text{if }z_2 \ne 0.$$ (These are a consequence of the trigonometric identities for the sine and cosine function.) In other words, the absolute values are multiplied and the arguments are added to yield the polar form of the product. The picture at the right illustrates the multiplication of $$(2+i)(3+i)=5+5i. $$ Because the real and imaginary part of $2 + i$ are equal, the argument of that number is 45 degrees, or $3 + i$ (in radian). On the other hand, it is also the sum of the angles at the origin of the red and blue triangles are arctan(1/3) and arctan(1/2), respectively. Thus, the formula $$\frac{\pi}{4} = \arctan\left(\frac{1}{2}\right) + \arctan\left(\frac{1}{3}\right) $$ holds. As the arctan function can be approximated highly efficiently, formulas like this – known as Machin-like formulas – are used for high-precision approximations of $\pi$.

Powers and roots
The n-th power of a complex number can be computed using de Moivre's formula, which is obtained by repeatedly applying the above formula for the product: $$ z^{n}=\underbrace{z \cdot \dots \cdot z}_{n \text{ factors}} = (r(\cos \varphi + i\sin \varphi ))^n = r^n \, (\cos n\varphi + i \sin n \varphi).$$ For example, the first few powers of the imaginary unit i are $$i, i^2 = -1, i^3 = -i, i^4 = 1, i^5 = i, \dots$$.

The $z$ $φ$th roots of a complex number $Oz$ are given by $$z^{1/n} = \sqrt[n]r \left( \cos \left(\frac{\varphi+2k\pi}{n}\right) + i \sin \left(\frac{\varphi+2k\pi}{n}\right)\right)$$ for $z_{1} = r_{1}(cos φ_{1} + i sin φ_{1})$. (Here $$\sqrt[n]r$$ is the usual (positive) $x + yi$th root of the positive real number $r$.) Because sine and cosine are periodic, other integer values of $φ$ do not give other values. For any $$z \ne 0$$, there are, in particular n distinct complex n-th roots. For example, there are 4 fourth roots of 1, namely
 * $$z_1 = 1, z_2 = i, z_3 = -1, z_4 = -i.$$

In general there is no natural way of distinguishing one particular complex $n$th root of a complex number. (This is in contrast to the roots of a positive real number x, which has a unique positive real n-th root, which is therefore commonly referred to as the n-th root of x.) One refers to this situation by saying that the $n$th root is a $z$-valued function of $n$.

Fundamental theorem of algebra
The fundamental theorem of algebra, of Carl Friedrich Gauss and Jean le Rond d'Alembert, states that for any complex numbers (called coefficients) $z_{2} = r_{2}(cos φ_{2} + i sin φ_{2})$, the equation $$a_n z^n + \dotsb + a_1 z + a_0 = 0$$ has at least one complex solution z, provided that at least one of the higher coefficients $5 + 5i$ is nonzero. This property does not hold for the field of rational numbers $$\Q$$ (the polynomial $π/4$ does not have a rational root, because $0 ≤ k ≤ n − 1$ is not a rational number) nor the real numbers $$\R$$ (the polynomial $a_{0}, ..., a_{n}$ does not have a real root, because the square of $r$ is positive for any real number $k$).

Because of this fact, $$\Complex$$ is called an algebraically closed field. It is a cornerstone of various applications of complex numbers, as is detailed further below. There are various proofs of this theorem, by either analytic methods such as Liouville's theorem, or topological ones such as the winding number, or a proof combining Galois theory and the fact that any real polynomial of odd degree has at least one real root.

History
The solution in radicals (without trigonometric functions) of a general cubic equation, when all three of its roots are real numbers, contains the square roots of negative numbers, a situation that cannot be rectified by factoring aided by the rational root test, if the cubic is irreducible; this is the so-called casus irreducibilis ("irreducible case"). This conundrum led Italian mathematician Gerolamo Cardano to conceive of complex numbers in around 1545 in his Ars Magna, though his understanding was rudimentary; moreover, he later described complex numbers as being "as subtle as they are useless". Cardano did use imaginary numbers, but described using them as "mental torture." This was prior to the use of the graphical complex plane. Cardano and other Italian mathematicians, notably Scipione del Ferro, in the 1500s created an algorithm for solving cubic equations which generally had one real solution and two solutions containing an imaginary number. Because they ignored the answers with the imaginary numbers, Cardano found them useless.

Work on the problem of general polynomials ultimately led to the fundamental theorem of algebra, which shows that with complex numbers, a solution exists to every polynomial equation of degree one or higher. Complex numbers thus form an algebraically closed field, where any polynomial equation has a root.

Many mathematicians contributed to the development of complex numbers. The rules for addition, subtraction, multiplication, and root extraction of complex numbers were developed by the Italian mathematician Rafael Bombelli. A more abstract formalism for the complex numbers was further developed by the Irish mathematician William Rowan Hamilton, who extended this abstraction to the theory of quaternions.

The earliest fleeting reference to square roots of negative numbers can perhaps be said to occur in the work of the Greek mathematician Hero of Alexandria in the 1st century AD, where in his Stereometrica he considered, apparently in error, the volume of an impossible frustum of a pyramid to arrive at the term $$\sqrt{81 - 144}$$ in his calculations, which today would simplify to $$\sqrt{-63} = 3i\sqrt{7}$$. Negative quantities were not conceived of in Hellenistic mathematics and Hero merely replaced it by its positive $$\sqrt{144 - 81} = 3\sqrt{7}.$$

The impetus to study complex numbers as a topic in itself first arose in the 16th century when algebraic solutions for the roots of cubic and quartic polynomials were discovered by Italian mathematicians (Niccolò Fontana Tartaglia and Gerolamo Cardano). It was soon realized (but proved much later) that these formulas, even if one were interested only in real solutions, sometimes required the manipulation of square roots of negative numbers. In fact, it was proved later that the use of complex numbers is unavoidable when all three roots are real and distinct. However, the general formula can still be used in this case, with some care to deal with the ambiguity resulting from the existence of three cubic roots for nonzero complex numbers. Rafael Bombelli was the first to address explicitly these seemingly paradoxical solutions of cubic equations and developed the rules for complex arithmetic, trying to resolve these issues.

The term "imaginary" for these quantities was coined by René Descartes in 1637, who was at pains to stress their unreal nature: "... sometimes only imaginary, that is one can imagine as many as I said in each equation, but sometimes there exists no quantity that matches that which we imagine. [... quelquefois seulement imaginaires c'est-à-dire que l'on peut toujours en imaginer autant que j'ai dit en chaque équation, mais qu'il n'y a quelquefois aucune quantité qui corresponde à celle qu'on imagine.]" A further source of confusion was that the equation $$\sqrt{-1}^2 = \sqrt{-1}\sqrt{-1} = -1$$ seemed to be capriciously inconsistent with the algebraic identity $$\sqrt{a}\sqrt{b} = \sqrt{ab}$$, which is valid for non-negative real numbers $n$ and $n$, and which was also used in complex number calculations with one of $n$, $z$ positive and the other negative. The incorrect use of this identity in the case when both $x$ and $x$ are negative, and the related identity $\frac{1}{\sqrt{a}} = \sqrt{\frac{1}{a}}$, even bedeviled Leonhard Euler. This difficulty eventually led to the convention of using the special symbol $a_{1}, ..., a_{n}$ in place of $$\sqrt{-1}$$ to guard against this mistake. Even so, Euler considered it natural to introduce students to complex numbers much earlier than we do today. In his elementary algebra text book, Elements of Algebra, he introduces these numbers almost at once and then uses them in a natural way throughout.

In the 18th century complex numbers gained wider use, as it was noticed that formal manipulation of complex expressions could be used to simplify calculations involving trigonometric functions. For instance, in 1730 Abraham de Moivre noted that the identities relating trigonometric functions of an integer multiple of an angle to powers of trigonometric functions of that angle could be re-expressed by the following de Moivre's formula:

$$(\cos \theta + i\sin \theta)^{n} = \cos n \theta + i\sin n \theta. $$

In 1748, Euler went further and obtained Euler's formula of complex analysis:

$$e ^{i\theta } = \cos \theta + i\sin \theta $$

by formally manipulating complex power series and observed that this formula could be used to reduce any trigonometric identity to much simpler exponential identities.

The idea of a complex number as a point in the complex plane (above) was first described by Danish–Norwegian mathematician Caspar Wessel in 1799, although it had been anticipated as early as 1685 in Wallis's A Treatise of Algebra.

Wessel's memoir appeared in the Proceedings of the Copenhagen Academy but went largely unnoticed. In 1806 Jean-Robert Argand independently issued a pamphlet on complex numbers and provided a rigorous proof of the fundamental theorem of algebra. Carl Friedrich Gauss had earlier published an essentially topological proof of the theorem in 1797 but expressed his doubts at the time about "the true metaphysics of the square root of −1". It was not until 1831 that he overcame these doubts and published his treatise on complex numbers as points in the plane, largely establishing modern notation and terminology:

"If one formerly contemplated this subject from a false point of view and therefore found a mysterious darkness, this is in large part attributable to clumsy terminology. Had one not called +1, −1, $\sqrt{-1}$ positive, negative, or imaginary (or even impossible) units, but instead, say, direct, inverse, or lateral units, then there could scarcely have been talk of such darkness."

In the beginning of the 19th century, other mathematicians discovered independently the geometrical representation of the complex numbers: Buée, Mourey, Warren,   Français and his brother, Bellavitis.

The English mathematician G.H. Hardy remarked that Gauss was the first mathematician to use complex numbers in "a really confident and scientific way" although mathematicians such as Norwegian Niels Henrik Abel and Carl Gustav Jacob Jacobi were necessarily using them routinely before Gauss published his 1831 treatise.

Augustin-Louis Cauchy and Bernhard Riemann together brought the fundamental ideas of complex analysis to a high state of completion, commencing around 1825 in Cauchy's case.

The common terms used in the theory are chiefly due to the founders. Argand called $x^{2} − 2$ the direction factor, and $$r = \sqrt{a^2 + b^2}$$ the modulus; Cauchy (1821) called $√2$ the reduced form (l'expression réduite) and apparently introduced the term argument; Gauss used $x^{2} + 4$ for $$\sqrt{-1}$$, introduced the term complex number for $i$, and called $cos φ + i sin φ$ the norm. The expression direction coefficient, often used for $cos φ + i sin φ$, is due to Hankel (1867), and absolute value, for modulus, is due to Weierstrass.

Later classical writers on the general theory include Richard Dedekind, Otto Hölder, Felix Klein, Henri Poincaré, Hermann Schwarz, Karl Weierstrass and many others. Important work (including a systematization) in complex multivariate calculus has been started at beginning of the 20th century. Important results have been achieved by Wilhelm Wirtinger in 1927.

Abstract algebraic aspects
While the above low-level definitions, including the addition and multiplication, accurately describes the complex numbers, there are other, equivalent approaches that reveal the abstract algebraic structure of the complex numbers more immediately.

Construction as a quotient field
One approach to $$\C$$ is via polynomials, i.e., expressions of the form $$p(X) = a_nX^n+\dotsb+a_1X+a_0,$$ where the coefficients $i$ are real numbers. The set of all such polynomials is denoted by $$\R[X]$$. Since sums and products of polynomials are again polynomials, this set $$\R[X]$$ forms a commutative ring, called the polynomial ring (over the reals). To every such polynomial p, one may assign the complex number $$p(i) = a_n i^n + \dotsb + a_1 i + a_0$$, i.e., the value obtained by setting $$X = i$$. This defines a function
 * $$\R[X] \to \C$$

This function is surjective since every complex number can be obtained in such a way: the evaluation of a linear polynomial $$a+bX$$ at $$X = i$$ is $$a+bi$$. However, the evaluation of polynomial $$X^2 + 1$$ at i is 0, since $$i^2 + 1 = 0.$$ This polynomial is irreducible, i.e., cannot be written as a product of two linear polynomials. Basic facts of abstract algebra then imply that the kernel of the above map is an ideal generated by this polynomial, and that the quotient by this ideal is a field, and that there is an isomorphism
 * $$\R[X] / (X^2 + 1) \stackrel \cong \to \C$$

between the quotient ring and $$\C$$. Some authors take this as the definition of $$\C$$.

Accepting that $$\Complex$$ is algebraically closed, because it is an algebraic extension of $$\mathbb{R}$$ in this approach, $$\Complex$$ is therefore the algebraic closure of $$\R.$$

Matrix representation of complex numbers
Complex numbers $a + bi$ can also be represented by $a^{2} + b^{2}$ matrices that have the form

$$ \begin{pmatrix} a &  -b  \\ b & \;\; a \end{pmatrix}. $$ Here the entries $a$ and $b$ are real numbers. As the sum and product of two such matrices is again of this form, these matrices form a subring of the ring of $cos φ + i sin φ$ matrices.

A simple computation shows that the map $$a+ib\mapsto \begin{pmatrix} a &  -b  \\ b & \;\; a \end{pmatrix}$$ is a ring isomorphism from the field of complex numbers to the ring of these matrices, proving that these matrices form a field. This isomorphism associates the square of the absolute value of a complex number with the determinant of the corresponding matrix, and the conjugate of a complex number with the transpose of the matrix.

The geometric description of the multiplication of complex numbers can also be expressed in terms of rotation matrices by using this correspondence between complex numbers and such matrices. The action of the matrix on a vector $a_{0}, ..., a_{n}$ corresponds to the multiplication of $a + bi$ by $2 × 2$. In particular, if the determinant is $2 × 2$, there is a real number $a$ such that the matrix has the form

$$\begin{pmatrix} \cos t & - \sin t  \\ \sin t & \;\; \cos t \end{pmatrix}.$$ In this case, the action of the matrix on vectors and the multiplication by the complex number $$\cos t+i\sin t$$ are both the rotation of the angle $b$.

Complex analysis
The study of functions of a complex variable is known as complex analysis and has enormous practical use in applied mathematics as well as in other branches of mathematics. Often, the most natural proofs for statements in real analysis or even number theory employ techniques from complex analysis (see prime number theorem for an example).

Unlike real functions, which are commonly represented as two-dimensional graphs, complex functions have four-dimensional graphs and may usefully be illustrated by color-coding a three-dimensional graph to suggest four dimensions, or by animating the complex function's dynamic transformation of the complex plane.

Convergence
The notions of convergent series and continuous functions in (real) analysis have natural analogs in complex analysis. A sequence of complex numbers is said to converge if and only if its real and imaginary parts do. This is equivalent to the (ε, δ)-definition of limits, where the absolute value of real numbers is replaced by the one of complex numbers. From a more abstract point of view, $$\mathbb{C}$$, endowed with the metric $$\operatorname{d}(z_1, z_2) = |z_1 - z_2|$$ is a complete metric space, which notably includes the triangle inequality $$|z_1 + z_2| \le |z_1| + |z_2|$$ for any two complex numbers $(x, y)$ and $x + iy$.

Complex exponential
Like in real analysis, this notion of convergence is used to construct a number of elementary functions: the exponential function $a + ib$, also written $1$, is defined as the infinite series, which can be shown to converge for any z: $$\exp z:= 1+z+\frac{z^2}{2\cdot 1}+\frac{z^3}{3\cdot 2\cdot 1}+\cdots = \sum_{n=0}^{\infty} \frac{z^n}{n!}. $$ For example, $$\exp (1)$$ is Euler's constant $$e \approx 2.718$$. Euler's formula states: $$\exp(i\varphi) = \cos \varphi + i\sin \varphi $$ for any real number $a$. This formula is a quick consequence of general basic facts about convergent power series and the definitions of the involved functions as power series. As a special case, this includes Euler's identity $$\exp(i \pi) = -1. $$

Complex logarithm
For any positive real number t, there is a unique real number x such that $$\exp(x) = t$$. This leads to the definition of the natural logarithm as the inverse $$\ln \colon \R^+ \to \R ; x \mapsto \ln x $$ of the exponential function. The situation is different for complex numbers, since
 * $$\exp(z+2\pi i) = \exp z \exp (2 \pi i) = \exp z$$

by the functional equation and Euler's identity. For example, $|1| = 1$, so both $b$ and $|z/w| = |z|/|w|$ are possible values for the complex logarithm of $d(z, w) = |z − w|$.

In general, given any non-zero complex number w, any number z solving the equation
 * $$\exp z = w$$

is called a complex logarithm of $a$, denoted $$\log w$$. It can be shown that these numbers satisfy $$z = \log w = \ln|w| + i\arg w, $$ where arg is the argument defined above, and ln the (real) natural logarithm. As arg is a multivalued function, unique only up to a multiple of $d(z, w) = |z − w|$, log is also multivalued. The principal value of log is often taken by restricting the imaginary part to the interval $b$. This leads to the complex logarithm being a bijective function taking values in the strip $$\R^+ + \; i \, \left(-\pi, \pi\right]$$ (that is denoted $$S_0$$ in the above illustration) $$\ln \colon \; \Complex^\times \; \to \; \; \; \R^+ + \; i \, \left(-\pi, \pi\right] .$$

If $$z \in \Complex \setminus \left( -\R_{\ge 0} \right)$$ is not a non-positive real number (a positive or a non-real number), the resulting principal value of the complex logarithm is obtained with $(z^{2} − 1)(z − 2 − i)^{2}⁄z^{2} + 2 + 2i$. It is an analytic function outside the negative real numbers, but it cannot be prolongated to a function that is continuous at any negative real number $$z \in -\R^+ $$, where the principal value is $±1, (2 + i)$.

Complex exponentiation $z_{1}$ is defined as $$z^\omega = \exp(\omega \ln z), $$ and is multi-valued, except when $t$ is an integer. For $z_{2}$, for some natural number $t$, this recovers the non-uniqueness of $z$th roots mentioned above. If $exp z$ is real (and $w$ an arbitrary complex number), one has a preferred choice of $$\ln x$$, the real logarithm, which can be used to define a preferred exponential function.

Complex numbers, unlike real numbers, do not in general satisfy the unmodified power and logarithm identities, particularly when naïvely treated as single-valued functions; see failure of power and logarithm identities. For example, they do not satisfy $$a^{bc} = \left(a^b\right)^c.$$ Both sides of the equation are multivalued by the definition of complex exponentiation given here, and the values on the left are a subset of those on the right.

Complex sine and cosine
The series defining the real trigonometric functions sine and cosine, as well as the hyperbolic functions sinh and cosh, also carry over to complex arguments without change. For the other trigonometric and hyperbolic functions, such as tangent, things are slightly more complicated, as the defining series do not converge for all complex values. Therefore, one must define them either in terms of sine, cosine and exponential, or, equivalently, by using the method of analytic continuation.

Holomorphic functions
A function $$f: \mathbb{C}$$ → $$\mathbb{C}$$ is called holomorphic or complex differentiable at a point $$z_0$$ if the limit
 * $$\lim_{z \to z_0} {f(z) - f(z_0) \over z - z_0 }$$

exists (in which case it is denoted by $$f'(z_0)$$). This mimics the definition for real differentiable functions, except that all quantities are complex numbers. Loosely speaking, the freedom of approaching $$z_0$$ in different directions imposes a much stronger condition than being (real) differentiable. For example, the function
 * $$f(z) = \overline z$$

is differentiable as a function $$\R^2 \to \R^2$$, but is not complex differentiable. A real differentiable function is complex differentiable if and only if it satisfies the Cauchy–Riemann equations, which are sometimes abbreviated as
 * $$\frac{\partial f}{\partial \overline z} = 0.$$

Complex analysis shows some features not apparent in real analysis. For example, the identity theorem asserts that two holomorphic functions $z$ and $w$ agree if they agree on an arbitrarily small open subset of $$\mathbb{C}$$. Meromorphic functions, functions that can locally be written as $e^{z}$ with a holomorphic function $φ$, still share some of the features of holomorphic functions. Other functions have essential singularities, such as $eiπ = e3iπ = −1$ at $3iπ$.

Applications
Complex numbers have applications in many scientific areas, including signal processing, control theory, electromagnetism, fluid dynamics, quantum mechanics, cartography, and vibration analysis. Some of these applications are described below.

Complex conjugation is also employed in inversive geometry, a branch of geometry studying reflections more general than ones about a line. In the network analysis of electrical circuits, the complex conjugate is used in finding the equivalent impedance when the maximum power transfer theorem is looked for.

Shapes
Three non-collinear points $$u, v, w$$ in the plane determine the shape of the triangle $$\{u, v, w\}$$. Locating the points in the complex plane, this shape of a triangle may be expressed by complex arithmetic as $$S(u, v, w) = \frac {u - w}{u - v}. $$ The shape $$S$$ of a triangle will remain the same, when the complex plane is transformed by translation or dilation (by an affine transformation), corresponding to the intuitive notion of shape, and describing similarity. Thus each triangle $$\{u, v, w\}$$ is in a similarity class of triangles with the same shape.

Fractal geometry
The Mandelbrot set is a popular example of a fractal formed on the complex plane. It is defined by plotting every location $$c$$ where iterating the sequence $$f_c(z)=z^2+c$$ does not diverge when iterated infinitely. Similarly, Julia sets have the same rules, except where $$c$$ remains constant.

Triangles
Every triangle has a unique Steiner inellipse – an ellipse inside the triangle and tangent to the midpoints of the three sides of the triangle. The foci of a triangle's Steiner inellipse can be found as follows, according to Marden's theorem: Denote the triangle's vertices in the complex plane as $−1$, $2π$, and $−π < φ < π$. Write the cubic equation $$(x-a)(x-b)(x-c)=0$$, take its derivative, and equate the (quadratic) derivative to zero. Marden's theorem says that the solutions of this equation are the complex numbers denoting the locations of the two foci of the Steiner inellipse.

Algebraic number theory
As mentioned above, any nonconstant polynomial equation (in complex coefficients) has a solution in $$\mathbb{C}$$. A fortiori, the same is true if the equation has rational coefficients. The roots of such equations are called algebraic numbers – they are a principal object of study in algebraic number theory. Compared to $$\overline{\mathbb{Q}}$$, the algebraic closure of $$\mathbb{Q}$$, which also contains all algebraic numbers, $$\mathbb{C}$$ has the advantage of being easily understandable in geometric terms. In this way, algebraic methods can be used to study geometric questions and vice versa. With algebraic methods, more specifically applying the machinery of field theory to the number field containing roots of unity, it can be shown that it is not possible to construct a regular nonagon using only compass and straightedge – a purely geometric problem.

Another example is the Gaussian integers; that is, numbers of the form $ln z = ln(−z) + iπ$, where $iπ$ and $w$ are integers, which can be used to classify sums of squares.

Analytic number theory
Analytic number theory studies numbers, often integers or rationals, by taking advantage of the fact that they can be regarded as complex numbers, in which analytic methods can be used. This is done by encoding number-theoretic information in complex-valued functions. For example, the Riemann zeta function $z^{ω}$ is related to the distribution of prime numbers.

Improper integrals
In applied fields, complex numbers are often used to compute certain real-valued improper integrals, by means of complex-valued functions. Several methods exist to do this; see methods of contour integration.

Dynamic equations
In differential equations, it is common to first find all complex roots $(−π, π]$ of the characteristic equation of a linear differential equation or equation system and then attempt to solve the system in terms of base functions of the form $ω = 1 / n$. Likewise, in difference equations, the complex roots $ω$ of the characteristic equation of the difference equation system are used, to attempt to solve the system in terms of base functions of the form $z > 0$.

Linear algebra
Since $$\C$$ is algebraically closed, any non-empty complex square matrix has at least one (complex) eigenvalue. By comparison, real matrices do not always have real eigenvalues, for example rotation matrices (for rotations of the plane for angles other than 0° or 180°) leave no direction fixed, and therefore do not have any real eigenvalue. The existence of (complex) eigenvalues, and the ensuing existence of eigendecomposition is a useful tool for computing matrix powers and matrix exponentials.

Complex numbers often generalize concepts originally conceived in the real numbers. For example, the conjugate transpose generalizes the transpose, hermitian matrices generalize symmetric matrices, and unitary matrices generalize orthogonal matrices.

Control theory
In control theory, systems are often transformed from the time domain to the complex frequency domain using the Laplace transform. The system's zeros and poles are then analyzed in the complex plane. The root locus, Nyquist plot, and Nichols plot techniques all make use of the complex plane.

In the root locus method, it is important whether zeros and poles are in the left or right half planes, that is, have real part greater than or less than zero. If a linear, time-invariant (LTI) system has poles that are

 * in the right half plane, it will be unstable,
 * all in the left half plane, it will be stable,
 * on the imaginary axis, it will have marginal stability.

If a system has zeros in the right half plane, it is a nonminimum phase system.

Signal analysis
Complex numbers are used in signal analysis and other fields for a convenient description for periodically varying signals. For given real functions representing actual physical quantities, often in terms of sines and cosines, corresponding complex functions are considered of which the real parts are the original quantities. For a sine wave of a given frequency, the absolute value $sin(1/z)$ of the corresponding $n$ is the amplitude and the argument $f(z)/(z − z_{0})^{n}$ is the phase.

If Fourier analysis is employed to write a given real-valued signal as a sum of periodic functions, these periodic functions are often written as complex-valued functions of the form

$$x(t) = \operatorname{Re} \{X( t ) \} $$

and

$$X( t ) = A e^{i\omega t} = a e^{ i \phi } e^{i\omega t} = a e^{i (\omega t + \phi) } $$

where ω represents the angular frequency and the complex number A encodes the phase and amplitude as explained above.

This use is also extended into digital signal processing and digital image processing, which use digital versions of Fourier analysis (and wavelet analysis) to transmit, compress, restore, and otherwise process digital audio signals, still images, and video signals.

Another example, relevant to the two side bands of amplitude modulation of AM radio, is:

$$\begin{align} \cos((\omega + \alpha)t) + \cos\left((\omega - \alpha)t\right) & = \operatorname{Re}\left(e^{i(\omega + \alpha)t} + e^{i(\omega - \alpha)t}\right) \\ & = \operatorname{Re}\left(\left(e^{i\alpha t} + e^{-i\alpha t}\right) \cdot e^{i\omega t}\right) \\ & = \operatorname{Re}\left(2\cos(\alpha t) \cdot e^{i\omega t}\right) \\ & = 2 \cos(\alpha t) \cdot \operatorname{Re}\left(e^{i\omega t}\right) \\ & = 2 \cos(\alpha t) \cdot \cos\left(\omega t\right). \end{align}$$

Electromagnetism and electrical engineering
In electrical engineering, the Fourier transform is used to analyze varying voltages and currents. The treatment of resistors, capacitors, and inductors can then be unified by introducing imaginary, frequency-dependent resistances for the latter two and combining all three in a single complex number called the impedance. This approach is called phasor calculus.

In electrical engineering, the imaginary unit is denoted by $n$, to avoid confusion with $ω$, which is generally in use to denote electric current, or, more particularly, $f$, which is generally in use to denote instantaneous electric current.

Because the voltage in an AC circuit is oscillating, it can be represented as

$$ V(t) = V_0 e^{j \omega t} = V_0 \left (\cos\omega t + j \sin\omega t \right ),$$

To obtain the measurable quantity, the real part is taken:

$$ v(t) = \operatorname{Re}(V) = \operatorname{Re}\left [ V_0 e^{j \omega t} \right ] = V_0 \cos \omega t.$$

The complex-valued signal $sin(1/z)$ is called the analytic representation of the real-valued, measurable signal $z = 0$.

Fluid dynamics
In fluid dynamics, complex functions are used to describe potential flow in two dimensions.

Quantum mechanics
The complex number field is intrinsic to the mathematical formulations of quantum mechanics, where complex Hilbert spaces provide the context for one such formulation that is convenient and perhaps most standard. The original foundation formulas of quantum mechanics – the Schrödinger equation and Heisenberg's matrix mechanics – make use of complex numbers.

Relativity
In special and general relativity, some formulas for the metric on spacetime become simpler if one takes the time component of the spacetime continuum to be imaginary. (This approach is no longer standard in classical relativity, but is used in an essential way in quantum field theory.) Complex numbers are essential to spinors, which are a generalization of the tensors used in relativity.

Algebraic characterization
The field $$\Complex$$ has the following three properties: It can be shown that any field having these properties is isomorphic (as a field) to $$\Complex.$$ For example, the algebraic closure of the field $$\Q_p$$ of the $g$-adic number also satisfies these three properties, so these two fields are isomorphic (as fields, but not as topological fields). Also, $$\Complex$$ is isomorphic to the field of complex Puiseux series. However, specifying an isomorphism requires the axiom of choice. Another consequence of this algebraic characterization is that $$\Complex$$ contains many proper subfields that are isomorphic to $$\Complex$$.
 * First, it has characteristic 0. This means that $a = x_{A} + y_{A}i$ for any number of summands (all of which equal one).
 * Second, its transcendence degree over $$\Q$$, the prime field of $$\Complex,$$ is the cardinality of the continuum.
 * Third, it is algebraically closed (see above).

Characterization as a topological field
The preceding characterization of $$\Complex$$ describes only the algebraic aspects of $$\Complex.$$ That is to say, the properties of nearness and continuity, which matter in areas such as analysis and topology, are not dealt with. The following description of $$\Complex$$ as a topological field (that is, a field that is equipped with a topology, which allows the notion of convergence) does take into account the topological properties. $$\Complex$$ contains a subset $b = x_{B} + y_{B}i$ (namely the set of positive real numbers) of nonzero elements satisfying the following three conditions: Moreover, $$\Complex$$ has a nontrivial involutive automorphism $c = x_{C} + y_{C}i$ (namely the complex conjugation), such that $x + iy$ is in $ζ(s)$ for any nonzero $f$ in $$\Complex.$$
 * $f(t) = e^{rt}$ is closed under addition, multiplication and taking inverses.
 * If $x$ and $y$ are distinct elements of $f(t) = r^{t}$, then either $|z|$ or $arg z$ is in $V(t)$.
 * If $r$ is any nonempty subset of $v(t)$, then $1 + 1 + ⋯ + 1 ≠ 0$ for some $r$ in $$\Complex.$$

Any field $z$ with these properties can be endowed with a topology by taking the sets $P$ as a base, where $j$ ranges over the field and $I$ ranges over $P$. With this topology $i$ is isomorphic as a topological field to $$\Complex.$$

The only connected locally compact topological fields are $$\R$$ and $$\Complex.$$ This gives another characterization of $$\Complex$$ as a topological field, because $$\Complex$$ can be distinguished from $$\R$$ because the nonzero complex numbers are connected, while the nonzero real numbers are not.

Other number systems
The process of extending the field $$\mathbb R$$ of reals to $$\mathbb C$$ is an instance of the Cayley–Dickson construction. Applying this construction iteratively to $$\C$$ then yields the quaternions, the octonions and the sedenions. This construction turns out to diminish the structural properties of the involved number systems.

Unlike the reals, $$\Complex$$ is not an ordered field, that is to say, it is not possible to define a relation $P$ that is compatible with the addition and multiplication. In fact, in any ordered field, the square of any element is necessarily positive, so $x − y$ precludes the existence of an ordering on $$\Complex.$$ Passing from $$\C$$ to the quaternions $$\mathbb H$$ loses commutativity, while the octonions (additionally to not being commutative) fail to be associative. The reals, complex numbers, quaternions and octonions are all normed division algebras over $$\mathbb R$$. By Hurwitz's theorem they are the only ones; the sedenions, the next step in the Cayley–Dickson construction, fail to have this structure.

The Cayley–Dickson construction is closely related to the regular representation of $$\mathbb C,$$ thought of as an $$\mathbb R$$-algebra (an $$\mathbb{R}$$-vector space with a multiplication), with respect to the basis $y − x$. This means the following: the $$\mathbb R$$-linear map $$\begin{align} \mathbb{C} &\rightarrow \mathbb{C} \\ z &\mapsto wz \end{align}$$ for some fixed complex number $p$ can be represented by a $P$ matrix (once a basis has been chosen). With respect to the basis $P$, this matrix is $$\begin{pmatrix} \operatorname{Re}(w) & -\operatorname{Im}(w) \\ \operatorname{Im}(w) & \operatorname{Re}(w) \end{pmatrix},$$ that is, the one mentioned in the section on matrix representation of complex numbers above. While this is a linear representation of $$\mathbb C$$ in the 2 × 2 real matrices, it is not the only one. Any matrix $$J = \begin{pmatrix}p & q \\ r & -p \end{pmatrix}, \quad p^2 + qr + 1 = 0$$ has the property that its square is the negative of the identity matrix: $S + P = x + P$. Then $$\{ z = a I + b J : a,b \in \mathbb{R} \}$$ is also isomorphic to the field $$\mathbb C,$$ and gives an alternative complex structure on $$\mathbb R^2.$$ This is generalized by the notion of a linear complex structure.

Hypercomplex numbers also generalize $$\mathbb R,$$ $$\mathbb C,$$ $$\mathbb H,$$ and $$\mathbb{O}.$$ For example, this notion contains the split-complex numbers, which are elements of the ring $$\mathbb R[x]/(x^2-1)$$ (as opposed to $$\mathbb R[x]/(x^2+1)$$ for complex numbers). In this ring, the equation $x ↦ x*$ has four solutions.

The field $$\mathbb R$$ is the completion of $$\mathbb Q,$$ the field of rational numbers, with respect to the usual absolute value metric. Other choices of metrics on $$\mathbb Q$$ lead to the fields $$\mathbb Q_p$$ of $x$-adic numbers (for any prime number $y$), which are thereby analogous to $$\mathbb{R}$$. There are no other nontrivial ways of completing $$\mathbb Q$$ than $$\mathbb R$$ and $$\mathbb Q_p,$$ by Ostrowski's theorem. The algebraic closures $$\overline {\mathbb{Q}_p}$$ of $$\mathbb Q_p$$ still carry a norm, but (unlike $$\mathbb C$$) are not complete with respect to it. The completion $$\mathbb{C}_p$$ of $$\overline {\mathbb{Q}_p}$$ turns out to be algebraically closed. By analogy, the field is called $S$-adic complex numbers.

The fields $$\mathbb R,$$ $$\mathbb Q_p,$$ and their finite field extensions, including $$\mathbb C,$$ are called local fields.

Historical references

 * — A gentle introduction to the history of complex numbers and the beginnings of complex analysis.
 * — An advanced perspective on the historical development of the concept of number.
 * — A gentle introduction to the history of complex numbers and the beginnings of complex analysis.
 * — An advanced perspective on the historical development of the concept of number.
 * — A gentle introduction to the history of complex numbers and the beginnings of complex analysis.
 * — An advanced perspective on the historical development of the concept of number.
 * — An advanced perspective on the historical development of the concept of number.

title: Complex number


text:
Talk:Chinese language

Change "dialects" wording?
The list of Chinese languages on the info card is labeled "Dialects:" but they're not dialects, really not even close. They're mutually unintelligible languages that happen to share a writing system. Mandarin speakers and Cantonese speakers can't understand each other any better than speakers of English and French do. LaymansLinguist (talk) 07:55, 1 August 2023 (UTC)
 * The analogy of English and French sounds a little extreme – the relationship between these two languages is only remote, and if it weren't for the numerous loanwords they would be hard to recognise as related at all, while even the most divergent varieties of Chinese or Sinitic are still relatively closely related. A closer analogy, with respect to time depth and similarity (from what I've seen), would be West Germanic or Western Romance for non-Min Chinese, with Min being more analogous to outliers like North Germanic or Sardinian in their respective families respectively.
 * Ultimately, I understand what we have in the case of Chinese or Sinitic is more like a strongly differentiated group (like Germanic or Romance) of several moderately differentiated groups (like Frisian or Oïl) of dialect groups (like North Frisian or Walloon), which elsewhere would be classified as a language family composed of several branches each consisting of several individual languages. Just as a very rough idea. --Florian Blaschke (talk) 17:44, 17 August 2023 (UTC)
 * Just as an illustration, our article Southern Min says:
 * Southern Min is not mutually intelligible with other branches of Min Chinese nor with non-Min varieties of Chinese, such as Mandarin, and the principal varieties of Southern Min are not intelligible with each other.
 * So Southern Min is a merely a branch of Min Chinese, yet itself like a language family.
 * The tables here give a rough impression of just how diverse Min Chinese is internally. --Florian Blaschke (talk) 20:56, 31 August 2023 (UTC)
 * A couple hundred languages, so a larger family than Romance or Slavic or Western Germanic, but maybe about that time depth / degree of similarity. — kwami (talk) 08:57, 1 September 2023 (UTC)
 * Yeah, I'm not talking about the number of languages, but only the linguistic distance within the family. Some of the most divergent members, such as Standard Chinese and Hokkien, are comparatively distantly related – certainly there's no question their divergence would typically get them classed as distinct languages, that is, Hokkien is an Abstandsprache with respect to Standard Chinese. --Florian Blaschke (talk) 22:30, 7 September 2023 (UTC)
 * Just a point of emphasis that these linguistic situations do not map neatly onto each other: the European sprachbund, as it were, has had a very different linguistic history than the Chinese one, though obviously you can make various points of comparison. Comparisons to how European language families are structured on the site shouldn't be quite enough to make decisions here.
 * Ultimately, I've been trying to read more of the Sino-Tibetan/Sinitic academic literature to see if there are better structures to organize these articles into. Unfortunately, bahaha, a lot of what I want to read is in Chinese, and mine isn't good enough yet. Remsense (talk) 17:47, 4 October 2023 (UTC)
 * As we all know a language is a dialect with an army and navy, so calling them languages isn't ideal. But they are clearly not just dialects either. That's why "variety" is used, as in the above quote and more generally when discussing Chinese language(s). That would be the obvious thing to change it to I think.--2A04:4A43:907F:FEB1:E804:215E:5C25:7100 (talk) 23:30, 7 September 2023 (UTC)
 * However, that would mean that we'd have to change the infobox Template:Infobox language to Template:Infobox language family, which uses "Subdivisions" instead of "Dialects", and finally get around to merging Sinitic languages and Varieties of Chinese into here. Standard Chinese is a single language, Chinese is not.
 * By the way, Mandarin Chinese has an analogous problem: even Mandarin is a language family rather than a single language, by linguistic distance of its varieties. --Florian Blaschke (talk) 01:07, 8 September 2023 (UTC)
 * I'd be in favor of merging Varieties of Chinese into Sinitic languages, but not into Chinese language. This article is about Chinese as conceived on as a single language, contrary to what the others cover. I could see merging Modern Standard Chinese here, and restricting the scope to that. — kwami (talk) 02:28, 8 September 2023 (UTC)
 * Double sharp (talk) 05:56, 8 September 2023 (UTC)
 * That would be a reasonable solution. --Florian Blaschke (talk) 22:03, 10 September 2023 (UTC)
 * Okay, I tagged all four articles to merge into two (Chinese language and Sinitic languages). — kwami (talk) 23:07, 1 October 2023 (UTC)

 * I'm not sure I understand exactly what is being proposed here, but I see Standard Chinese has been tagged for merging to Chinese language, which I strongly oppose. Standard Chinese is just one variety of Chinese; the subject of the Chinese language article is much broader. I don't see what would be gained by merging these two together. —Mx. Granger (talk · contribs) 03:51, 2 October 2023 (UTC)
 * The idea was to make "chinese language" about the standard language, with everything else in the other article.
 * This was because people thought 4 articles was redundant for 2 topics (Chinese in the broad sense and in the narrow sense.) — kwami (talk) 22:16, 2 October 2023 (UTC)
 * I would assume that meant merging Chinese language into Varieties of Chinese, as they both cover Chinese in the broad sense, and moving Standard Chinese to "Chinese language". That seems more reasonable, but I don't support it either, because I don't think "Chinese language" is the common name for Standard Chinese as distinguished from Chinese in general.
 * The other three articles could conceivably be merged together (leaving Standard Chinese about the standardized form and another article, presumably Chinese language, about Chinese as a whole), though I'm not sure whether that's a good idea. At first glance I think it makes some sense to have an article focused specifically on differences between varieties (currently Varieties of Chinese) separate from the main overview article (currently Chinese language). Sinitic languages does strike me as possibly redundant and maybe should be merged into one of the others. —Mx. Granger (talk · contribs) 02:55, 3 October 2023 (UTC)
 * IMO sinitic languages should be the article about the language family. Calling it "chinese language" would be like having an article on the "romance language" or the "slavic language", with french and russian as "dialects". IMO, an article on the chinese language should be about the language. We could have a hatnote that other languages called "chinese" have their own articles. — kwami (talk) 23:29, 3 October 2023 (UTC)
 * Well, no one considers the Romance languages or the Slavic languages to be varieties of one language, whereas that is how Chinese is usually seen, including by most of its speakers, and what it is called by most sources. Mutual intelligibility is not the only criterion out there for what constitutes a language. Also, making the Chinese language article only about Standard Chinese would give the impression that only that variety is the "real" Chinese language and other varieties aren't. —Mx. Granger (talk · contribs) 14:54, 4 October 2023 (UTC)
 * No, people don't generally consider it a single language. That's to some extent a translation issue, where 'topolect' is translated as 'dialect'. There's also common usage of 'dialect' in the sense of not an official language, both in Chinese and English, but that's not WP usage.
 * Same issue with only one variety being the "real" French or German or Italian language. Not unique to Chinese. — kwami (talk) 22:42, 5 October 2023 (UTC)
 * If people object to the word dialect, we can say topolect or variety instead. —Mx. Granger (talk · contribs) 14:55, 4 October 2023 (UTC)
 * First, I think there kinda has to be a Chinese language page. So, I understand why Sinitic languages is that, and not Chinese languages, because it would be odd to have with Chinese language, both having different scopes. I also strongly with merging Standard Chinese with Chinese language, it is not taxonomically appropriate. I've been working on Chinese characters lately, which has a lot of overlap with this discussion but is naturally not coterminous—the general issue being that spoken and written languages do not perfectly map onto each other, but they do map significantly onto each other–the relevant structure I've been mulling over (insofar as one should be constructed) is as follows, with representative examples of where pages should 'go' in the hierarchy:
 * Sinitic languages · Varieties of Chinese—it's really difficult to try to wedge orthography and the spoken languages apart, but maybe this tree should focus on the spoken languages specifically.
 * Mandarin
 * Standard Chinese
 * Yue (&c.)
 * Chinese characters · Chinese family of scripts—orthography tree, the first page should provide an overview of the main trunk of the logographic family from oracle bones → modern written chinese et al, while appropriately treating kanji etc.
 * 'Stylistic scripts'
 * Clerical script
 * Cursive script (&c.)
 * Written Chinese
 * Classical Chinese—I basically have been treating this as a wholly artificial written language, almost more akin to a writing system itself than a natural language that was spoken, but I know that's not quite right.
 * Simplified characters
 * Traditional characters
 * Written Cantonese
 * Kanji (&c.)
 * Jōyō kanji (&c.)
 * I think, regarding default terms, the best I've settled on is varieties for the spoken language, and forms for the written language, e.g. traditional versus simplified versus kanji, as opposed to the less-structural, stylistic 'scripts', like regular script vs. gothic script, etc.
 * this stuff is complicated! help! Remsense (talk) 17:19, 4 October 2023 (UTC)
 * So you're arguing for a dab page at Chinese language? If not, what scope would the Chinese language article be about? (t · c)  buidhe  07:50, 12 October 2023 (UTC)
 * No, no! Honestly, I'm not sure. It's tricky, I think some overlap in coverage is required, because there are many common, core aspects shared between Chinese understood monolithically, and as a phylum. Remsense  聊  08:04, 12 October 2023 (UTC)
 * This started with the idea that 'varieties of Chinese' was redundant with 'Sinitic languages'. We could merge those and not these, if ppl want. Or keep all 4, if the overhead and content-fork issues aren't a problem. — kwami (talk) 22:27, 12 October 2023 (UTC)
 * yes, certainly, i figured it was worth sharing some extended thoughts while i was at it. are there any languages in that phylum that aren't really considered Chinese per se? Bai?
 * i think it might be worth making the articles more distinct instead, i.e. with Sinitic languages for a comparative linguistics treatment of the grouping, and Varieties of Chinese with a historiographical, cultural treatment. Which is already mostly the case, but maybe it would be apropos to more clearly delineate. Remsense  聊  22:48, 12 October 2023 (UTC)
 * Yes, sometimes also Bai. Also perhaps Caijia, which may or may not be closest to Bai (if Bai is even Sinitic). So they're not exact synonyms according to some, but not enough of difference IMO to require separate articles just for that point, if there isn't additional reason to split them. — kwami (talk) 23:46, 13 October 2023 (UTC)
 * Agreed. I do think the academic/cultural distinction is enough so that they should be separate, is where I'm coming down at the moment. Remsense  聊  00:30, 14 October 2023 (UTC)
 * I agree that this article should not be merged with Standard Chinese. This article is about a different topic, encompassing all aspects of Chinese including its history, literary styles and modern standard and regional forms.
 * The key question is where this article should be placed in the hierarchy of language families and languages on the wiki. It is the worst kind of POVFORK to have separate articles about the same thing just so that one can have a singular title and the other a plural.
 * It is generally agreed that Chinese is as diverse as a language family, though we outrun the literature if we try to enumerate languages. So this article should use infobox language family instead of infobox language, and have a plural title. To return to the original poster's complaint, the reason the infobox says "dialects" is that it uses infobox language. Very little of the content would need to change. Kanguole 15:00, 15 October 2023 (UTC)
 * I support a family box.
 * Yeah, no-one even knows what the Chinese languages are, we only have isoglosses. — kwami (talk) 03:59, 16 October 2023 (UTC)
 * ✅ with infobox language family proposal. now not so sure, after reading 209.2.226.172's insights below. Remsense  聊  16:52, 16 October 2023 (UTC)
 * I would say the Sinitic languages article is a POV fork. It should exist because the Chinese areal features do not extend to Bai, etc. (Bai, etc. do share a lot of areal features with Chinese but those are typical interlanguage features, usually in phonological level, few codevelopement) Yet unfortunately it developed into an article that pretends it's always possible to scientifically separate out different strata in each spoken Chinese variety. 209.2.226.172 (talk) 21:26, 20 October 2023 (UTC)
 * I disagree with the merging and disagree with a proposal. The concept "Chinese language" is a loose translation of "漢語" which should really be translated as the Chinese  yuyan  . It is not simply a language, a language family, a written language or many languages, but a unified entity that incorporate areal features and codevelopement a lot of intertwined lects. Each sound change in one variety of Chinese may have various degrees of effects on all varieties of Chinese that is usually not borrowings. If an infobox other than "language" has to be used, it might be something like a linguistic area or sprachbund, but infobox language suits with the need of this article better. Using a language family infobox would misled the reader that a tree model works with Chinese, or that it's easy to separate strata in Chinese.  209.2.226.172 (talk) 20:31, 20 October 2023 (UTC)
 * For the merging issue, a good distinction is that other prestige varieties exist, and many are prestige not due to representing the corresponding subfamily but the Chinese language. For example, Guangzhou Cantonese is definitely not the best variety to represent the Yue Chinese (many Goulou varieties do a better job), but is prestiged in Hong Kong because the locals believe it represents Chinese better. Hangzhou Wu is definitely not as representative to Wu when compared with Shaoxing Wu, however many find it prestiged as a representative of Chinese yet none considers it a representative of Wu (on the other hand, a Gascon speaker would be proud of his language because he speaks perfect Gascon, not because he considers his Gascon is the best French, thus applying a French model to Chinese may be offensive to those speakers). A Chinese language article should has its phonology section focused at the common part of "epicenter Chinese" that are more likely to impose an areal effect on others (such as Beijing Mandarin, Eastern Henan Mandarin, Nanjing Mandarin, Hangzhou Wu, Guangzhou Cantonese and the literary layer of some Min), their relations to the Middle Chinese and the range of acceptable articulation, and the effect of these varieties toward each other and toward varieties less likely to impose an areal effect, rather than any one single specific spoken Chinese. Pure comparative linguistic information should be removed from this article to Sinitic languages since it is terribly off-topic. "Dialect" is a common established translation of 方言 so it doesn't need any change, but it's important to inform the reader that the "dialect"s in these articles should not be understood in the English way, and that it's used simply because there's no better English terms for it. 209.2.226.172 (talk) 20:44, 20 October 2023 (UTC)
 * I appreciate the deliberation! if I'm understanding your points correctly, you're saying that this article should focus on the areal overlap of the Chinese varieties, and a comparative lens should be reserved for Sinitic languages? Remsense  聊  21:56, 20 October 2023 (UTC)
 * Comparative lens should be kept and stated with the areal overlap in an intertwined way. Pure comparative lens (such as an obsolete layer in Waxiang) should be reserved for Sinitic languages. There should be focus when stating the comparative model: vernacular stratum of Min, although with great linguistic value, is less likely to become an important source of global areal effect, should not be elaborated in this article but instead included in Sinitic languages and Min Chinese. It is important to mention that Chinese is not a dialect continuum, but a chimera of various dialect continuums that connected with each other with complex structures, and that when such connection fails it's also common for Chinese to have apparent internal borrowings as if it were a family of independent languages. 209.2.226.172 (talk) 22:11, 20 October 2023 (UTC)
 * Grammatically speaking, there are certain Cantonese (and Wu) grammar that has somewhat entered "common Chinese" but not yet in MSC (yet well understood by all). Such features should be included (such as 張三大過李四). Features in Beijing Mandarin that has not entered common Chinese should not be included (though I cannot immediately get an example out of my head). In comparative linguistic, all features are equal (or unequal based on "linguistic value"), but in the natural development of a language not all features are equal. 209.2.226.172 (talk) 22:33, 20 October 2023 (UTC)
 * In English, a language usually refers to a smallest unit that develops relatively independently, subject to some sprachbund restraints (but not strong enough to be affect its evolvement) and to some sprachbund borrowing process (but not to a degree that one cannot tell whether a word is borrowed or evolved independently), usually characterized by mutually intelligible classes. Such a concept is successful when dealing most languages in the world, such as the Dungan language (even though this one is not characterize by mutually intelligible classes). However, it has some problem when dealing with some dialect continuums and has serious problem when dealing with many Asian languages with complex internal relations such as Chinese. Consequently, it would make better sense to move this article to the Chinese linguistic monster (which it is) than to move it to the "Chinese languages". --209.2.226.172 (talk) 22:20, 20 October 2023 (UTC)
 * I'd like to mention the following fact: "On the other hand, Dungan speakers like Iasyr Shivaza and others have reported that people who speak the Beijing Mandarin dialect can understand Dungan, but Dungans could not understand the Beijing Mandarin. " It may be explained with the assumption that Dungan contains strictly more information than Beijing Mandarin, but that's not the case. It makes more sense to say due to there's is no monolingual native speakers of any individual spoken Chinese variety (no matter it's Standard Mandarin or something else), a "native speaker" of either Beijing Mandarin or Standard Mandarin can grasp the meaning of Dungan sentence with his prior knowledge in some other varieties. If even Standard Mandarin doesn't not have a monolingual native speakers (with the exception of some earlyschoolkids near Beijing, but I wouldn't say schoolkids have complete knowledge on Beijing Mandarin either), the "Chinese languages" would be a family of dead languages that as a whole spoken by everyone. Since there are a good number of Dungan speakers without knowledge of any spoken Chinese and a good number of Chinese speakers without any knowledge of Dungan, these are two independent languages. Suppose one day every Chinese speaker speaks perfect Standard Mandarin and uses Standard Mandarin as the single source of absolute standard, like the case of French, one may say probably Chinese is a language family. 209.2.226.172 (talk) 00:54, 21 October 2023 (UTC)
 * Chinese publications, including the Encyclopedia of China, often use the formulation 汉语在语言系属分类中相当于一个语族的地位. 'In language family classification, Chinese is equivalent to a language family.' This argues for a language family infobox.
 * The aim of enwiki is to communicate information in English. Blanket translation of all uses of fāngyán 方言 into English as "dialect" has long been recognized as misleading, so we should avoid it. Using infobox language family would label Wu, Yue, etc as "subdivisions", which everyone agrees they are. There is no implication of a tree model, let alone separation of strata. Kanguole 11:46, 21 October 2023 (UTC)
 * When I mentioned "Chinese languages" would be a family of dead languages that as a whole spoken by everyone, I didn't mean to deny the diversity of Chinese, but to confirm it yet meanwhile point out that it forms only one small aspect of the topic. Take the article George Washington as an example: if we write "George Washington is an un-mummified symbiote of microbials, protists and protostomes" in the top paragraph introduction, the tone of the article would be terribly inappropriate even if the statement itself may be technically correct; however, if that sentence appears in a "Tomb" section and starts with "Physically speaking, the body of George Washington is equivalent to ...", it would be appropriate though of no importance to be mentioned. The situation with this article is similar: if the language family infobox appears in a section of this article there's no problem (though for such infobox qieyun-strata should not be mentioned at the same level with colloquial Min strata), but the infobox appears on the top paragraph should be a language infobox or a "linguistic monster" infobox instead of a language family one in that Chinese is a living yuyan with codeveloping varieties. The aim of enwiki is to communicate info in English and you are right that "dialect" is not a perfect translation, but that doesn't mean one should use a worse one that brings more misleading than clarification. I said the Sinitic languages is a POV-fork not because the mention of language family, but because the article failed to inform the reader that the topic of the article is only one small aspect of the Chinese language. It's like an article tomb of Washington failed to tell the more important fact fact Washington was once a human being that rewrote the track of the history, or an article about Macanese man'o'war failed to tell a Macanese man'o'war is a single living creature in ecology point of view. Otherwise, the development of that article apart from the top paragraph is okay. No only is it practical both infoboxes to be mentioned with the language, if we make the language family infobox in corresponding section concise, it is somewhat necessary because, taking Min Chinese as an example, it is a sprachbund of literary strata and a cognitive family of colloquial strara, but not quite a sprachbund of colloquial strata and probably not at all a valid family of their literary reading strata. What's more, the two main topics of Min only roughly coincide: some topolects near Shaojiang Min may be considered in Gan sprachbund but in Min family or reverse (since it's hard to tell which stratum is the main stratum and even unlikely to separate a stratum out from the intertwined system). That said, probably each large topodivision of Chinese should also be introduced both as an actively developing intralanguage sprachbund and a cognitive family of diverse "dead" languages, but currently even this Chinese language article doesn't manage to inform the fact that Chinese is actively developing as one integrated body (while Encyclopedia of China of course does).
 * The below section on Ba-Shu Chinese can also be solved this way because Ba-Shu seems to be well-accepted as an extinct linguistic area but not so clear as an extinct comparative family. 209.2.227.244 (talk) 02:17, 11 November 2023 (UTC)
 * This is an example of a two-infoboxes solution. In the language infobox, each variety is a linguistic area (intralanguage sprachbund, 区片), while the language family infobox lists only legit toplevel comparative entries. The Template:infobox language may be edited to allow a parameter to change "dialects" to "varieties", and maybe more. The Chinese designation 方言区 and 方言片 apparently refers to geographical area, which is the origin of the current classification, and are not comparative linguistic subdivisions. Chinese is a "language" jn the sense of, if you send a some Standard Mandarin speakers to 300 light years away, then even if they still remember their language and still use Chinese characters, what they speak is no longer Chinese because their language has lost all possible coevolutionary opportunity with Chinese (that's why Dungan is frequently not considered Chinese until/unless coevolutionary conditions from again). Similarily, a few "speakers of a variety of Chinese" once reach 300 l.y. far away, based on their full knowledge of their variety, partial (passive) knowledge of varieties of their neighbor area before migration, their choice of knowledge about Classical Chinese terms and so on, a new living language just formed from a not-individually-living variety of the living language Chinese and become a separate language at the same time of becoming a living language (just like what happened when a zooid is cut off from the colony-body and put into a growth medium). Simply being Overseas Chinese or simply being illiterate cannot guarantee a sever of codevelopement, only by cutting several ties could a sever of coevolution to be guaranteed. 209.2.227.244 (talk) 04:48, 11 November 2023 (UTC)
 * Suggestion. Why don't we just create a infobox? I don't think it's unwarranted given all the special considerations.  Remsense  聊  11:48, 11 November 2023 (UTC)
 * I don't believe it is warranted – though Chinese is an extreme case, dialect continua, dialect borrowing and diglossia also occur elsewhere in the world.
 * But in any case, it wouldn't touch on the question being discussed here, namely what infobox should go in this article. Kanguole 12:38, 11 November 2023 (UTC)
 * I agree with your stance that creating an Infobox Chinese variety infobox is not warranted, but I disagree with your reasoning. It is oversimplification to describe the Chinese language phenomenon as simply "dialect continua, dialect borrowing and diglossia" but an organic interaction among dialect continua, multi-directional dialect influencing (not just borrowing), bidirectional diglossia, social linguistic, etc. Some of these phenomenons are not common among western languages (espcially multi-directionality of dialect influencing and of diglossia), not to mention a combination of them all. However, it is a somewhat typical phenomenon in the Chinese yuyan, in the Tibetan skad, and in a number of Asian linguistic entities. That said, what we need is a Infobox cognate coevolving areal linguistic entity for this article and for a number of other articles (such as the Tibetan skad), but it's premature to do that. As for now, the best fit should be Infobox language.
 * Talking about the directionality of diglossia is beyond common logic. What I intend to say is that a speaker of A is likely a speaker of B may or may not imply a speaker of B is likely a speaker of A: in Chinese monodirectional diglossia exists only for a few varieties such as Standard Mandarin, most diglossia phenomenon are bidirectional. This is not very common in the west (espcially not common in nation-states with a titular ethnic group amd in the "Five Eyes" English-speaking world), but common in a number of eastern linguistic entities where indigenous cultures dominate. 209.2.231.4 (talk) 02:24, 20 December 2023 (UTC)
 * By "elsewhere in the world", I did not mean the West, or even the Anglosphere. Kanguole 23:53, 20 December 2023 (UTC)
 * I have reverted these changes. The second infobox is mainly prose, which belongs in article text (provided it can be supported by citations) – infoboxes are for summary information. A further problem is that the subdivision proposed here appears to be novel rather than taken from the literature.
 * I have reverted these changes. The second infobox is mainly prose, which belongs in article text (provided it can be supported by citations) – infoboxes are for summary information. A further problem is that the subdivision proposed here appears to be novel rather than taken from the literature.

 * I have also reverted the re-addition of Ba–Shu for the reasons given in below.

 * More minor points are:
 * The link to macrolanguage is unhelpful – it is not a linguistic concept.
 * The link to stratum (linguistics) refers to a different concept than the vocabulary layers intended here.
 * I'm afraid that phrases like "a sprachbund of literary strata and a cognitive family of colloquial strata" convey no meaning. The individual words, and even "literary strata" and "colloquial strata", have meanings, but the combination does not.

 * And I'll repeat that labelling Wu, Yue, etc as "subdivisions" (which infobox language family would do) is uncontroversial, even if people would disagree on the meaning of the subdivisions. Kanguole 12:38, 11 November 2023 (UTC)
 * I apologize for my mistake: all "cognitive" above should be "cognate". I was reading about something about psychology that day and when writing a comment at night my output experienced a disorder. So it really should be "a sprachbund of literary strata and a cognate family (with shared innovations) of colloquial strata". Now the combination should make sense. Talking about "stratum", it was a slight abuse of the term because the stratum usually refers to the source language but I identified it with its image on the object language, that is, the part of vocabulary, phonology, grammar, etc. of the object language that can be traced to a forementioned stratum. Under such a conceptual equivalence, the vocabulary layer is an instance of stratum (because all the information we can attest from the substratum language are from its image on modern Min Chinese). 209.2.231.4 (talk) 02:38, 20 December 2023 (UTC)
 * Macrolanguage, on the other hand, is helpful. The definition of macrolanguage is due to SIL, and under no clause did SIL say the concept is not linguistic. So your claim is unfounded. The official definition, clusters of closely-related language varieties that, based on certain criteria is clearly leaning toward a linguistic rather than non-linguistic definition. Social linguistic is an integral part of linguistic and even if it doesn't, the inept of the current linguistic is no excuse to deny the very existance of Chinese language as something beyond a language family. Like, that biologists hadn't developed a concept for Portuguese man'o'war by 18th century didn't mean Portuguese man'o'wars in 18th century should be physically severed into individual zooids and then wait for malnutrition-death. The conventional concept of Chinese and of Tibetan doesn't need linguistic to justify. If a ljnk to macrolanguage is not usable, the only way to achieve NPOV in the head paragraph is following the Chinese Wikipedia tone: "The Chinese language is both a language and a (paraphyletic) language family (due to the exclusion of Dungan and Taz)." (The Encyclopedia of China you mentioned above can be used as a citation for this.) The current leading paragraph development represents a strong western bias. 209.2.231.4 (talk) 03:00, 20 December 2023 (UTC)
 * Labelling Wu, Yue, etc as top-level "subdivisions" itself was never a problem because "subdivisions" could mean "areal subdivisions". The problem is:
 * The use of the infobox language family alone in the top paragraph is inappropriate (however it can be used in a section);
 * The "subdivisions" in infobox language family are language family subdivisions (generally based on shared innovations), not areal subdivisions. Labelling Wu, Yue, and other top level areal subdivisions other than Min as top-level family subdivisions (generally based on shared innovations) are inappropriate.
 * 209.2.231.4 (talk) 03:13, 20 December 2023 (UTC)
 * A "macrolanguage" is an entity that has an ISO code and is composed of smaller entities that also have ISO codes. It has no meaning outside of ISO. — kwami (talk) 07:57, 20 December 2023 (UTC)

Suggestion. It seems the bigger linguistic values a feature has, the less importance it has in "common Chinese" development. So IMHO the following arrangement may be a solution:
 * 1) The article Chinese language may cover any phonetical features equivalent to the various yunbai systems (incl. Pekingeese yunbai, Suzhou Wu yunbai, and Cantonese yunbai), to the Old National Pronunciation, to the literary reading systems in Cantonese and Hokkien, but with without inclusion of any specific one of them, together with the grammar mentioned above and the frequent incorporation (switching) of Classical Chinese grammar in formal speaking, how different varieties interact, etc., with 80% of its content. The remaining 20% covers various spoken varieties, with maybe one or two sentences mentioning vernacular Min and Waxiang.
 * 2) The article Varieties of Chinese may cover a thorough phonetical development from qieyun to all spoken Chinese varieties except for vernacular Min and Waxiang, briefly mentioned in the article without deep introduction.
 * 3) The article Sinitic languages have qieyun-derived phonetics occupying less than a third of its contents, and include the development from Old Chinese to qieyun (and then to modern varieties), Min, Waxiang and Bai; and briefly mention the fact that a good number of modern pronunciations may involve a mix of different strata thus cannot be referenced, and the fact that with such a limited info available the reconstruction of Old Chinese is highly inconclusive (unlike PIE). An article with such a name deserves highly technical contents. 209.2.226.172 (talk) 01:18, 21 October 2023 (UTC)
 * All of that seems to belong in History of the Chinese language and Historical Chinese phonology, both of which have much scope for improvement. Kanguole 11:46, 21 October 2023 (UTC)
 * That one piece of information can be included in one article doesn't mean it cannot be included in another. And by including some information I didn't mean only including that information. The map from Old Chinese to Sinitic languages can be concisely mentioned in a way that from Old Chinese reconstruction directly shooting into modern varieties while a history of a language has to deal with how it develops step through each dynasty (and Middle Chinese was never a legit unitary language as qieyun was based on two varieties of Chinese, a modern variety based on qieyun is based on one of the two varieties, or a thrid intermediate one, or many of the above mentioned. Historical linguistic is different from both comparative linguistic and areal one). Sinitic languages may deal with the influences between modern varieties due to unintelligibility such as internal loan words; while Varieties of Chinese may deal with influences from common core. There are a lot to be included that cannot be covered by historical linguistics. 209.2.227.244 (talk) 04:34, 11 November 2023 (UTC)

✅ with infobox language family proposal, despite insights from 209.2.226.172. None of the factors mentioned change the fact that the Chinese languages are, for the most part, extremely mutually unintelligible, and the subdivisions should be treated as separate languages. In any case, there should not be two separate pages for Sinitic languages and Chinese language [sic], as Chinese-Bai monophyly has not to my knowledge been shown definitively. If "Sinitic" is preserved as a term for Chinese-Bai separate from "Chinese", it is merely a hypothetical subgroup akin to Italo-Celtic, which its infobox does not reflect. The page also has less than 10 lines on Greater Bai, with the vast majority of its content on comparisons between Chinese varieties, so I support its merger. I would also suggest moving Chinese language to Chinese languages, with Chinese language redirecting to a disambiguation page linking to both Standard Chinese [sic] and the new Chinese languages. Vampyricon (talk) 19:40, 10 November 2023 (UTC)
 * I agree with you that the Chinese-Bai monophyly should probably be discussed separately, but my point is never to deny the diversity of Chinese, but to point out that Chinese should not be introduced in one aspect, when most readers would search Chinese for a more broad topic and usually its more important aspects. Even without Bai, the article may still be developed in a way that qieyun-derived phonetics occupying less than a third of its contents (so the above arrangement still works). For the development of the article Chinese language, Portuguese man o' war makes a perfect example: "Portuguese man o' war constitutes a single individual from an ecological perspective, but is made up of many individuals from an embryological perspective." with a top paragraph introduction "The Portuguese man o' war, is a marine hydrozoan" instead of "a colony of tiny marine hydrozoans". A redirect page cannot serve the purpose of introducing Chinese as a whole, but more like
 * Portuguese man o' war may refers to
 * An individual tiny Portuguese man o' war zooid
 * The Portuguese man o' war colony (like a group of ants)
 * The sail of a Portuguese man o' war, or Standard Manowar
 * 209.2.227.244 (talk) 02:36, 11 November 2023 (UTC)

 * While there are still subjects to speak about, are there any objections to closing the two proposed mergers with the consensus not merged? Remsense  留  09:23, 28 November 2023 (UTC)

 * No objections. Please go ahead. 2001:8003:9100:2C01:2C47:80BD:EAA6:D798 (talk) 15:12, 18 January 2024 (UTC)
 * Agreed. Also I believe it's time to create a infobox topolektbund as well as comparative family to meet the need of Chinese, Tibetan, and a number of other language that may need it. The concept Swiss German (contains both Swiss Standard German and Swiss Alemanian) may be a concept similar to Chinese language but for that case it is not a comparative family (strictly speaking not is Chinese a comparative family because it probably doesn't contain Dungan). 209.2.228.41 (talk) 01:54, 3 February 2024 (UTC)

NPOV

 * re: your recent edits: for shorthand/leading terminology like this, our choices really need to be backed up verbatim by plenty of sources, as well as being accurate. English language sources would not introduce Chinese as a "coevolutionary linguistic entity with highly mutually unintelligible varieties". It is not our domain to invent our own phraseology, even if we think it's better—we have to pick the most accurate, but adequately well-attested and representative, descriptors that we can from existing sources. Remsense  留  04:00, 20 December 2023 (UTC)
 * I agree that such phraseology should be avoided whenever avoidable. But the alternative should meet NPOV. Putting language family as the main topic of the article, espcially on the top sentence, doesn't meet the NPOV criteria, just like calling a Portuguese man-o-war a colony in the first sentence wouldn't be a NPOV introduction. One option would be following the article in Chinese Wikipedia "Chinese is a language and a synonymous language family". 209.2.228.41 (talk) 01:37, 3 February 2024 (UTC)

Ba–Shu
I have reverted the addition of Ba-Shu to the infobox, because it seems to have no coverage in any overviews of Chinese dialects or Chinese language history. There is a Wikipedia article Ba–Shu Chinese, but it appears to be a synthesis of one of the dialect areas mentioned in Fangyan and studies of poetry from Song-era Sichuan. Kanguole 23:52, 31 October 2023 (UTC)

 * Would you please tell me why you thought this BaShu issue couldn't be solved in the two infoboxes way above, Given that Ba-Shu seems to be well-accepted as an extinct linguistic area but not so clear as an extinct comparative family? Or you just reverted the re-addition of Ba–Shu because it is improper to put it into a single infobox and that the two infoboxes way was "prose"? 209.2.231.4 (talk) 03:06, 20 December 2023 (UTC)
 * As I mentioned, it seems to have no coverage in any overviews of Chinese dialects or Chinese language history. Kanguole 10:06, 20 December 2023 (UTC)
 * It's not covered primarily because its relative importance is not as high as those living topolects, but there is strong evidence of its existance as a historical topolect. IMHO it can be here with an asterisk. 209.2.228.41 (talk) 02:16, 3 February 2024 (UTC)
 * Wikipedia articles need to be based on coverage in reliable secondary sources, to avoid original research and undue weight. If those sources don't find it worth mentioning, nor should we. Kanguole 10:49, 3 February 2024 (UTC)

Ambiguous "Standard Cantonese"
In Hong Kong, Written Cantonese has basically no official status, and has never appeared on any official document historically. The Standard Written language is largely based on Modern Standard Chinese, with a few influences from Cantonese. The spoken variety has always been Cantonese (de facto official), together with a system reading Mandarin into Cantonese. When the local use the term "中文", one usually means this complex system. This is why I made an edit, later reverted by 123.195.224.196, then redid and undid by Remsense. The efn statement doesn't explain it well because the Hong Kong government never actively promotes literacy in Cantonese (it does promote Written Cantonese in the similar extent how mainland China promote dialects, though). 209.2.227.244 (talk) 04:06, 11 November 2023 (UTC)
 * "De facto—while no specific variety of Chinese is official in Hong Kong and Macau, Cantonese is the predominant spoken form and the de facto regional standard, written in Traditional characters. Standard Chinese and Simplified characters are only occasionally used in some official and educational settings. The Hong Kong government promotes biliteracy in Chinese and English, and trilingualism between Cantonese, Mandarin, and English; while the Macau government promotes triliteracy between Chinese, Portuguese, and English, and quadrilingualism between Cantonese, Mandarin, Portuguese, and English, especially in public education."
 * Sorry if I was rash on this: feel free to put it back for now, I don't want to appear possessive on things where I am not remotely an expert. Remsense  留  23:51, 27 November 2023 (UTC)
 * See Special:Diff/1190834510. 209.2.231.4 (talk) 03:25, 20 December 2023 (UTC)
 * Hongkongers usually emphasize that their Written Standard Chinese contains Classical Chinese, but that's true to all Written Standard Chinese around the world to various degrees, though the one in Hkng Kong contains slightly more. 209.2.231.4 (talk) 03:28, 20 December 2023 (UTC)
 * What about Special:Diff/1202574855 then? 209.2.228.41 (talk) 02:10, 3 February 2024 (UTC)

Lead NPOV
Creating a new heading for 209.2.228.41's NPOV concerns raised here, where they would like the article lead to initially define Chinese as something like

—which I feel is too novel a phraseology for an article with such a large scope. But I appreciate their concerns, and am not quite sure what to do about it. Remsense 诉  20:44, 20 February 2024 (UTC)

 * I would really like to get this banner off the top of this article. IP 209..41, is there any formulation where we use the term "group of languages" in the first sentence that would be acceptable? It's simply an issue of prose. Remsense  诉  07:44, 11 March 2024 (UTC)

Hokkien and Hakka
Both hokkien and hakka are official in taiwan. Both are sinitic languages. Should it be added into the "Official status" section of the template? Guotaian (talk) 15:59, 14 May 2024 (UTC)

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Talk:Chinese language