text:
1892 in music

Events in the year 1892 in music.

Specific locations

 * 1892 in Norwegian music

Events

 * April 28 – Jean Sibelius conducts the world première of his Kullervo Symphony in Helsinki.
 * May 26 – A statue of Felix Mendelssohn, by Werner Stein, is dedicated at St. Thomas Church, Leipzig. Removed by the Nazis in the 1930s, it is re-dedicated in 2008.
 * September 24 – Opening of the Theater Unter den Linden, Berlin with Adolf Ferron's operetta Daphne and Gaul and Haßreiter's ballet Die Welt in Bild und Tanz.
 * September 26 – Antonín Dvořák arrives in the United States to take up his post as artistic director of the National Conservatory of Music.
 * December 18
 * Anton Bruckner's Symphony No. 8 in C minor is premiered by the Vienna Philharmonic with Hans Richter conducting at the Stadttheater, Leipzig.
 * December 6 O.S. – Pyotr Ilyich Tchaikovsky's ballet The Nutcracker and opera Iolanta are premiered in a double bill at the Imperial Mariinsky Theatre in Saint Petersburg, Russia.
 * "After the Ball" becomes the first sheet music to sell over 1 million copies (for a single publisher in a single year).
 * Erik Satie composes his first pieces in his own compositional system.
 * Sergei Diaghilev graduates from the Saint Petersburg Conservatory.
 * Ferdinand Praeger's controversial biography Wagner As I Knew Him is published posthumously in London.
 * Sudrophone patented by François Sudre in France.

Published popular music

 * "After the Ball"    w.m. Charles K. Harris
 * "The Bowery"    w. Charles H. Hoyt m. Percy Gaunt
 * "Daddy Wouldn't Buy Me a Bow Wow"    w.m. Joseph Tabrar
 * "Daisy Bell" (aka "A Bicycle Built For Two")    w.m. Harry Dacre
 * "Flanagan" w.m. C. W. Murphy & William Letters
 * "Future Mrs. 'Awkins" by Albert Chevalier
 * "The Holy City"    w. Frederick Edward Weatherly m. Stephen Adams
 * "La Sultana Turkish March" m. Fred Linden
 * "Liebestraum Nocturne" m. Virginia Field
 * "Molly And I And The Baby"    w.m. Harry Kennedy
 * "My Old Dutch"    w. Albert Chevalier m. Charles Ingle
 * "My Sweetheart's The Man In The Moon"    w.m. James Thornton
 * "The Sweetest Story Ever Told"    w.m. R. M. Stults
 * "The Virginia Skedaddle" w.m. Monroe H. Rosenfeld

Recorded popular music

 * "Daddy Wouldn't Buy Me a Bow-wow " – Dan W. Quinn
 * "Daddy Wouldn't Buy Me a Bow-wow " – Silas Leachman
 * "Esquimeau Dance" – William Tuson
 * "Grover Cleveland March" – Gilmore's Band
 * "Michael Casey At The Telephone" – Russell Hunting
 * "Michael Casey Taking The Census" – Russell Hunting
 * "Pat Brady as President" – Dan Kelly
 * "Parody of "We'll Never Turn His Picture Toward The Wall'" – Al Reeves
 * "Riding through the Glen" – Issler's Orchestra
 * "Take Your Time Gentlemen" – Press Eldridge
 * "The Blind Boy" – Richard Jose
 * "The Bowery" – Dan W. Quinn
 * "The Laughing Darkie" – George W. Johnson (singer)
 * "The Night Alarm" – Holding's Military Band
 * "The Old Folks at Home" – Len Spencer
 * "Uncle Ned's Dream" – George W. Johnson (singer)

Classical music

 * Johannes Brahms – Intermezzos Opus number 117
 * Edward Elgar – Serenade for Strings, Op. 20
 * Alexander Glazunov – String Quintet in A major (Opus 39)
 * Gustav Mahler – Das himmlische Leben (later incorporated into his fourth symphony)
 * Miguel Marqués – El Centinelo
 * Carl Nielsen – First Symphony
 * Joseph Parry – Saul of Tarsus (oratorio)
 * Sergei Rachmaninoff
 * Prelude in C-sharp Minor
 * Piano Concerto No. 1 in F-sharp minor, Op. 1
 * Max Reger
 * Cello Sonata No. 1, Op. 5
 * Twelve Waltz-Caprices, Op. 9, four-hand piano
 * 20 German Dances, Op. 10, four-hand piano
 * Alexander Scriabin – Piano Sonata No. 1 in F minor, opus 6
 * Jean Sibelius – Kullervo Symphony
 * Josef Suk – Serenade for Strings in E-flat major
 * Alexander von Zemlinsky – Symphony in D minor

Opera

 * Herman Bemberg – Elaine
 * Karel Bendl – Dite Tabora
 * Alfredo Catalani – La Wally
 * Gialdino Gialdini – I due soci premiered February 24 at the Teatro Brunetti, Bologna
 * Umberto Giordano – Mala Vita
 * Isidore de Lara – The Light of Asia
 * Ruggiero Leoncavallo – I Pagliacci
 * Jules Massenet – Werther (Composed 1887)
 * Adolf Neuendorff – The Minstrel
 * Pyotr Ilyich Tchaikovsky – Iolanta
 * The World's Fair Colored Opera Company, with featured singer, soprano Matilda Sissieretta Jones are the first African-American performers to appear at Carnegie Hall

Ballet

 * Pyotr Ilyich Tchaikovsky – The Nutcracker

Musical theater

 * Blue-Eyed Susan, libretto by George Robert Sims, on London stage
 * "Maid Marian" (retitled version of "Robin Hood"); London production
 * "Maid Marian" (sequel to "Robin Hood"); Broadway production
 * The Wicklow Postman; initial production in Boston, Massachusetts

Top hits

 * "Slide, Kelly, Slide" by George J. Gaskin
 * "Sally in Our Alley" by Manhansett Quartette

Births

 * January 1 – Artur Rodziński, Polish conductor (d. 1958)
 * January 31 – Eddie Cantor, born Isidore Itzkowitz, American singer and entertainer (d. 1964)
 * February 4 – Yrjö Kilpinen, Finnish composer known most for his lieder (d. 1959)
 * February 15 – Ján Valašťan Dolinský, Slovak composer (d. 1965)
 * March 10
 * Arthur Honegger, composer (d. 1955)
 * Eva Turner, operatic soprano (d. 1990)
 * March 27 – Ferde Grofé, composer (d. 1972)
 * April 1 – Renato Zanelli, Chilean baritone, later tenor (d. 1935)
 * April 2 – Roy Palmer, jazz trombonist (d. 1962)
 * April 10 – Victor de Sabata, conductor and composer (d. 1967)
 * April 12 – Johnny Dodds, jazz clarinetist (d. 1940)
 * April 15 – Manuel Quiroga, violinist (d. 1961)
 * April 19 – Germaine Tailleferre, composer (d. 1983)
 * April 21 – Jaroslav Kvapil, composer (d. 1958)
 * May 14 – Arthur Lourié, Russian-born expatriate composer (d. 1966)
 * May 18 – Ezio Pinza, Italian bass singer and actor (d. 1957)
 * May 19 – Pops Foster, jazz bass player (d. 1969)
 * June 6 – Ted Lewis, singer and bandleader (d. 1971)
 * June 18 – Eduard Steuermann, pianist (d. 1964)
 * June 21 – Hilding Rosenberg, Swedish composer (d. 1985)
 * June 23 – Mieczysław Horszowski, Polish pianist (d. 1993)
 * June 30 – László Lajtha, Hungarian symphonist (d. 1963)
 * July 2 – Jack Hylton, British bandleader (d. 1965)
 * July 8 – J. Russel Robinson, dixieland pianist-composer (d. 1963)
 * July 10 – Ján Móry, Slovak composer (d. 1978)
 * July 26 – Philipp Jarnach, composer of German-French origins (d. 1982)
 * August 14 – Kaikhosru Shapurji Sorabji, composer, music critic and pianist (d. 1988)
 * August 15 – Knud Jeppesen, musicologist (d. 1974)
 * September 4 – Darius Milhaud, composer (d. 1974)
 * September 5 – Joseph Szigeti, violinist (d. 1973)
 * September 17 – Hendrik Andriessen, Dutch composer and organist (d. 1981)
 * October 17 – Herbert Howells, church music composer (d. 1983)
 * October 19 – Ilmari Hannikainen, composer (d. 1955)
 * October 25 – Janszieka (Jennie) Deutsch and Roszicka (Rosie) Deutsch, Hungarian-born dancers, actresses and singers, billed as the Dolly sisters (d. 1941 and 1970 respectively)
 * November 11 – Isidor Achron, Polish-American pianist and composer (d. 1948)
 * November 28 – Thomas Wood, English composer (d. 1950)
 * December 9 – Beatrice Harrison, cellist (d. 1965)
 * December 11 – Giacomo Lauri-Volpi, operatic tenor (d. 1979)

Deaths

 * January 10 – Heinrich Dorn, German conductor, composer and journalist (born 1804)
 * February 13 – Lambert Massart, violinist (born 1811)
 * February 20 – Róza Csillag, opera singer (born 1832)
 * March 11 – Caroline Reinagle, pianist, composer and writer (born 1818)
 * March 20 – Arthur Goring Thomas, composer (born 1850) (suicide)
 * April 22 – Édouard Lalo, composer (born 1823)
 * May 2 – Wilhelm Rust, composer (born 1822)
 * May 6 – Ernest Guiraud, composer (born 1837)
 * June 5 – Robert Rees, Welsh tenor (born 1841)
 * August 18 – Jules Perrot, ballet dancer (born 1810)
 * August 19 – František Zdeněk Skuherský, composer, teacher and music theorist (born 1830)
 * September 5 – Henry Christian Timm, pianist, conductor and composer (born 1811)
 * September 24 – Patrick Gilmore, bandmaster and composer (born 1829)
 * October 24 – Robert Franz, composer (born 1815)
 * October 28 – Felix Otto Dessoff, conductor and composer (born 1835)
 * November 4 – Hervé, organist and composer (born 1825)
 * November 19 – Antonio Torres Jurado, guitar maker (born 1817)
 * date unknown – Adolf Rzepko, Polish composer, oboist, choral and orchestral conductor and pianist (born 1825)

title: 1892 in music


text:
Root of unity

In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power $n$. Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group characters, and the discrete Fourier transform.

Roots of unity can be defined in any field. If the characteristic of the field is zero, the roots are complex numbers that are also algebraic integers. For fields with a positive characteristic, the roots belong to a finite field, and, conversely, every nonzero element of a finite field is a root of unity. Any algebraically closed field contains exactly $n$ $n$th roots of unity, except when $n$ is a multiple of the (positive) characteristic of the field.

General definition

An $r$th root of unity, where $φ$ is a positive integer, is a number $n$ satisfying the equation $$z^n = 1. $$ Unless otherwise specified, the roots of unity may be taken to be complex numbers (including the number 1, and the number −1 if $n$ is even, which are complex with a zero imaginary part), and in this case, the $z$th roots of unity are $$\exp\left(\frac{2k\pi i}{n}\right)=\cos\frac{2k\pi}{n}+i\sin\frac{2k\pi}{n},\qquad k=0,1,\dots, n-1.$$

However, the defining equation of roots of unity is meaningful over any field (and even over any ring) $F$, and this allows considering roots of unity in $F$. Whichever is the field $F$, the roots of unity in $F$ are either complex numbers, if the characteristic of $F$ is 0, or, otherwise, belong to a finite field. Conversely, every nonzero element in a finite field is a root of unity in that field. See Root of unity modulo n and Finite field for further details.

An $n$th root of unity is said to be  if it is not an $n$th root of unity for some smaller $n$, that is if

 * $$z^n=1\quad \text{and} \quad z^m \ne 1 \text{ for } m = 1, 2, 3, \ldots, n-1. $$

If n is a prime number, then all $n$th roots of unity, except 1, are primitive.

In the above formula in terms of exponential and trigonometric functions, the primitive $m$th roots of unity are those for which $m$ and $n$ are coprime integers.

Subsequent sections of this article will comply with complex roots of unity. For the case of roots of unity in fields of nonzero characteristic, see. For the case of roots of unity in rings of modular integers, see Root of unity modulo n.

Elementary properties
Every $n$th root of unity $z$ is a primitive $a$th root of unity for some $a ≤ n$, which is the smallest positive integer such that $z^{a} = 1$.

Any integer power of an $n$th root of unity is also an $n$th root of unity, as
 * $$(z^k)^n = z^{kn} = (z^n)^k = 1^k = 1.$$

This is also true for negative exponents. In particular, the reciprocal of an $n$th root of unity is its complex conjugate, and is also an $n$th root of unity:
 * $$\frac{1}{z} = z^{-1} = z^{n-1} = \bar z.$$

If $z$ is an $n$th root of unity and $a ≡ b (mod n)$ then $z^{a} = z^{b}$. Indeed, by the definition of congruence modulo n, $a = b + kn$ for some integer $k$, and hence
 * $$ z^a = z^{b+kn} = z^b z^{kn} = z^b (z^n)^k = z^b 1^k = z^b.$$

Therefore, given a power $z^{a}$ of $z$, one has $z^{a} = z^{r}$, where $0 ≤ r < n$ is the remainder of the Euclidean division of $k$ by $n$.

Let $z$ be a primitive $n$th root of unity. Then the powers $z$, $z^{2}$, ..., $z^{n−1}$, $z^{n} = z^{0} = 1$ are $n$th roots of unity and are all distinct. (If $z^{a} = z^{b}$ where $1 ≤ a < b ≤ n$, then $z^{b−a} = 1$, which would imply that $z$ would not be primitive.) This implies that $z$, $z^{2}$, ..., $z^{n−1}$, $z^{n} = z^{0} = 1$ are all of the $n$th roots of unity, since an $n$th-degree polynomial equation over a field (in this case the field of complex numbers) has at most $n$ solutions.

From the preceding, it follows that, if $z$ is a primitive $n$th root of unity, then $$z^a = z^b$$ if and only if $$a\equiv b \pmod{ n}.$$ If $z$ is not primitive then $$a\equiv b \pmod{ n}$$ implies $$z^a = z^b,$$ but the converse may be false, as shown by the following example. If $n = 4$, a non-primitive $n$th root of unity is $z = –1$, and one has $$z^2 = z^4 = 1$$, although $$ 2 \not\equiv 4 \pmod{4}.$$

Let $z$ be a primitive $n$th root of unity. A power $w = z^{k}$ of $a$ is a primitive $a$th root of unity for
 * $$ a = \frac{n}{\gcd(k,n)},$$

where $$\gcd(k,n)$$ is the greatest common divisor of $n$ and $z$. This results from the fact that $ka$ is the smallest multiple of $n$ that is also a multiple of $k$. In other words, $ka$ is the least common multiple of $k$ and $n$. Thus
 * $$a =\frac{\operatorname{lcm}(k,n)}{k}=\frac{kn}{k\gcd(k,n)}=\frac{n}{\gcd(k,n)}.$$

Thus, if $k$ and $n$ are coprime, $z^{k}$ is also a primitive $n$th root of unity, and therefore there are $φ(n)$ distinct primitive $n$th roots of unity (where $φ$ is Euler's totient function). This implies that if $n$ is a prime number, all the roots except $+1$ are primitive.

In other words, if $R(n)$ is the set of all $n$th roots of unity and $P(n)$ is the set of primitive ones, $R(n)$ is a disjoint union of the $P(n)$:

 * $$\operatorname{R}(n) = \bigcup_{d \,|\, n}\operatorname{P}(d),$$

where the notation means that $d$ goes through all the positive divisors of $n$, including $1$ and $n$.

Since the cardinality of $R(n)$ is $n$, and that of $P(n)$ is $φ(n)$, this demonstrates the classical formula

 * $$\sum_{d \,|\, n}\varphi(d) = n.$$

Group of all roots of unity
The product and the multiplicative inverse of two roots of unity are also roots of unity. In fact, if $x^{m} = 1$ and $y^{n} = 1$, then $(x^{−1})m = 1$, and $(xy)k = 1$, where $k$ is the least common multiple of $m$ and $n$.

Therefore, the roots of unity form an abelian group under multiplication. This group is the torsion subgroup of the circle group.

Group of $n$th roots of unity
For an integer n, the product and the multiplicative inverse of two $n$th roots of unity are also $n$th roots of unity. Therefore, the $n$th roots of unity form an abelian group under multiplication.

Given a primitive $n$th root of unity $ω$, the other $n$th roots are powers of $ω$. This means that the group of the $n$th roots of unity is a cyclic group. It is worth remarking that the term of cyclic group originated from the fact that this group is a subgroup of the circle group.

Galois group of the primitive $n$th roots of unity
Let $$\Q(\omega)$$ be the field extension of the rational numbers generated over $$\Q$$ by a primitive $n$th root of unity $ω$. As every $n$th root of unity is a power of $ω$, the field $$\Q(\omega)$$ contains all $n$th roots of unity, and $$\Q(\omega)$$ is a Galois extension of $$\Q.$$

If $k$ is an integer, $ω^{k}$ is a primitive $n$th root of unity if and only if $k$ and $n$ are coprime. In this case, the map

 * $$\omega \mapsto \omega^k$$

induces an automorphism of $$\Q(\omega)$$, which maps every $n$th root of unity to its $k$th power. Every automorphism of $$\Q(\omega)$$ is obtained in this way, and these automorphisms form the Galois group of $$\Q(\omega)$$ over the field of the rationals.

The rules of exponentiation imply that the composition of two such automorphisms is obtained by multiplying the exponents. It follows that the map

 * $$k\mapsto \left(\omega \mapsto \omega^k\right)$$

defines a group isomorphism between the units of the ring of integers modulo $n$ and the Galois group of $$\Q(\omega).$$

This shows that this Galois group is abelian, and implies thus that the primitive roots of unity may be expressed in terms of radicals.

Galois group of the real part of the primitive roots of unity
The real part of the primitive roots of unity are related to one another as roots of the minimal polynomial of $$2\cos(2\pi/n).$$ The roots of the minimal polynomial are just twice the real part; these roots form a cyclic Galois group.

Trigonometric expression

De Moivre's formula, which is valid for all real $k$ and integers $n$, is

 * $$\left(\cos x + i \sin x\right)^n = \cos nx + i \sin nx.$$

Setting $x = 2π⁄n$ gives a primitive $x$th root of unity – one gets

 * $$\left(\cos\frac{2\pi}{n} + i \sin\frac{2\pi}{n}\right)^{\!n} = \cos 2\pi + i \sin 2\pi = 1,$$

but
 * $$\left(\cos\frac{2\pi}{n} + i \sin\frac{2\pi}{n}\right)^{\!k} = \cos\frac{2k\pi}{n} + i \sin\frac{2k\pi}{n} \neq 1$$

for $k = 1, 2, …, n − 1$. In other words,
 * $$\cos\frac{2\pi}{n} + i \sin\frac{2\pi}{n}$$

is a primitive $n$th root of unity.

This formula shows that in the complex plane the $n$th roots of unity are at the vertices of a regular $n$-sided polygon inscribed in the unit circle, with one vertex at 1 (see the plots for $n = 3$ and $n = 5$ on the right). This geometric fact accounts for the term "cyclotomic" in such phrases as cyclotomic field and cyclotomic polynomial; it is from the Greek roots "cyclo" (circle) plus "tomos" (cut, divide).

Euler's formula

 * $$e^{i x} = \cos x + i \sin x,$$

which is valid for all real $n$, can be used to put the formula for the $n$th roots of unity into the form

 * $$e^{2\pi i \frac{k}{n}}, \quad 0 \le k < n.$$

It follows from the discussion in the previous section that this is a primitive $x$th-root if and only if the fraction $k⁄n$ is in lowest terms; that is, that $n$ and $n$ are coprime. An irrational number that can be expressed as the real part of the root of unity; that is, as $$\cos(2\pi k/n)$$, is called a trigonometric number.

Algebraic expression
The $n$th roots of unity are, by definition, the roots of the polynomial $x^{n} − 1$, and are thus algebraic numbers. As this polynomial is not irreducible (except for $n = 1$), the primitive $n$th roots of unity are roots of an irreducible polynomial (over the integers) of lower degree, called the $n$th cyclotomic polynomial, and often denoted $Φ_{n}$. The degree of $Φ_{n}$ is given by Euler's totient function, which counts (among other things) the number of primitive $n$th roots of unity. The roots of $Φ_{n}$ are exactly the primitive $n$th roots of unity.

Galois theory can be used to show that the cyclotomic polynomials may be conveniently solved in terms of radicals. (The trivial form $$\sqrt[n]{1}$$ is not convenient, because it contains non-primitive roots, such as 1, which are not roots of the cyclotomic polynomial, and because it does not give the real and imaginary parts separately.) This means that, for each positive integer $k$, there exists an expression built from integers by root extractions, additions, subtractions, multiplications, and divisions (and nothing else), such that the primitive $n$th roots of unity are exactly the set of values that can be obtained by choosing values for the root extractions ($n$ possible values for a $n$th root). (For more details see, below.)

Gauss proved that a primitive $k$th root of unity can be expressed using only square roots, addition, subtraction, multiplication and division if and only if it is possible to construct with compass and straightedge the regular $k$-gon. This is the case if and only if $n$ is either a power of two or the product of a power of two and Fermat primes that are all different.

If $n$ is a primitive $n$th root of unity, the same is true for $1/z$, and $$r=z+\frac 1z$$ is twice the real part of $z$. In other words, $Φ_{n}$ is a reciprocal polynomial, the polynomial $$R_n$$ that has $n$ as a root may be deduced from $Φ_{n}$ by the standard manipulation on reciprocal polynomials, and the primitive $z$th roots of unity may be deduced from the roots of $$R_n$$ by solving the quadratic equation $$z^2-rz+1=0.$$ That is, the real part of the primitive root is $$\frac r2,$$ and its imaginary part is $$\pm i\sqrt{1-\left(\frac r2\right)^2}.$$

The polynomial $$R_n$$ is an irreducible polynomial whose roots are all real. Its degree is a power of two, if and only if $r$ is a product of a power of two by a product (possibly empty) of distinct Fermat primes, and the regular $n$-gon is constructible with compass and straightedge. Otherwise, it is solvable in radicals, but one are in the casus irreducibilis, that is, every expression of the roots in terms of radicals involves nonreal radicals.

Explicit expressions in low degrees

 * For $n = 1$, the cyclotomic polynomial is $Φ_{1}(x) = x − 1$ Therefore, the only primitive first root of unity is 1, which is a non-primitive $n$th root of unity for every n > 1.
 * As $Φ_{2}(x) = x + 1$, the only primitive second (square) root of unity is −1, which is also a non-primitive $n$th root of unity for every even $n > 2$. With the preceding case, this completes the list of real roots of unity.
 * As $Φ_{3}(x) = x^{2} + x + 1$, the primitive third (cube) roots of unity, which are the roots of this quadratic polynomial, are $$\frac{-1 + i \sqrt{3}}{2},\ \frac{-1 - i \sqrt{3}}{2} .$$
 * As $Φ_{4}(x) = x^{2} + 1$, the two primitive fourth roots of unity are $i$ and $−i$.
 * As $Φ_{5}(x) = x^{4} + x^{3} + x^{2} + x + 1$, the four primitive fifth roots of unity are the roots of this quartic polynomial, which may be explicitly solved in terms of radicals, giving the roots $$\frac{\varepsilon\sqrt 5 - 1}4 \pm i \frac{\sqrt{10 + 2\varepsilon\sqrt 5}}{4},$$ where $$\varepsilon$$ may take the two values 1 and −1 (the same value in the two occurrences).
 * As $Φ_{6}(x) = x^{2} − x + 1$, there are two primitive sixth roots of unity, which are the negatives (and also the square roots) of the two primitive cube roots: $$\frac{1 + i \sqrt{3}}{2},\ \frac{1 - i \sqrt{3}}{2}.$$
 * As 7 is not a Fermat prime, the seventh roots of unity are the first that require cube roots. There are 6 primitive seventh roots of unity, which are pairwise complex conjugate. The sum of a root and its conjugate is twice its real part. These three sums are the three real roots of the cubic polynomial $$r^3+r^2-2r-1,$$ and the primitive seventh roots of unity are $$\frac{r}{2}\pm i\sqrt{1-\frac{r^2}{4}},$$ where $n$ runs over the roots of the above polynomial. As for every cubic polynomial, these roots may be expressed in terms of square and cube roots. However, as these three roots are all real, this is casus irreducibilis, and any such expression involves non-real cube roots.
 * As $Φ_{8}(x) = x^{4} + 1$, the four primitive eighth roots of unity are the square roots of the primitive fourth roots, $± i$. They are thus $$ \pm\frac{\sqrt{2}}{2} \pm i\frac{\sqrt{2}}{2}.$$
 * See Heptadecagon for the real part of a 17th root of unity.

Periodicity
If $n$ is a primitive $r$th root of unity, then the sequence of powers

is $z$-periodic (because $… , z^{−1}, z^{0}, z^{1}, …$ for all values of $n$), and the $n$ sequences of powers

for $z^{ j + n} = z^{ j}z^{ n} = z^{ j}$ are all $j$-periodic (because $s_{k}: … , z^{ k⋅(−1)}, z^{ k⋅0}, z^{ k⋅1}, …$). Furthermore, the set $k = 1, … , n$ of these sequences is a basis of the linear space of all $n$-periodic sequences. This means that any $n$-periodic sequence of complex numbers

can be expressed as a linear combination of powers of a primitive $n$th root of unity:
 * $$ x_j = \sum_k X_k \cdot z^{k \cdot j} = X_1 z^{1\cdot j} + \cdots + X_n \cdot z^{n \cdot j}$$

for some complex numbers $z^{ k⋅(j + n)} = z^{ k⋅j}$ and every integer $n$.

This is a form of Fourier analysis. If $n$ is a (discrete) time variable, then $j$ is a frequency and ${s_{1}, … , s_{n}}$ is a complex amplitude.

Choosing for the primitive $j$th root of unity
 * $$z = e^\frac{2\pi i}{n} = \cos\frac{2\pi}{n} + i \sin\frac{2\pi}{n}$$

allows $… , x_{−1} , x_{0} , x_{1}, …$ to be expressed as a linear combination of $X_{1}, … , X_{n}$ and $X_{k}$:
 * $$x_j = \sum_k A_k \cos \frac{2\pi jk}{n} + \sum_k B_k \sin \frac{2\pi jk}{n}.$$

This is a discrete Fourier transform.

Summation
Let $x_{j}$ be the sum of all the $k$th roots of unity, primitive or not. Then

 * $$\operatorname{SR}(n) =

\begin{cases} 1, & n=1\\ 0, & n>1. \end{cases}$$

This is an immediate consequence of Vieta's formulas. In fact, the $n$th roots of unity being the roots of the polynomial $cos$, their sum is the coefficient of degree $sin$, which is either 1 or 0 according whether $SR(n)$ or $X^{ n} – 1$.

Alternatively, for $n – 1$ there is nothing to prove, and for $n = 1$ there exists a root $n > 1$ – since the set $n = 1$ of all the $n$th roots of unity is a group, $n > 1$, so the sum satisfies $z ≠ 1$, whence $S$.

Let $zS = S$ be the sum of all the primitive $n$th roots of unity. Then

 * $$\operatorname{SP}(n) = \mu(n),$$

where $z SR(n) = SR(n)$ is the Möbius function.

In the section Elementary properties, it was shown that if $SR(n) = 0$ is the set of all $n$th roots of unity and $SP(n)$ is the set of primitive ones, $μ(n)$ is a disjoint union of the $R(n)$:

 * $$\operatorname{R}(n) = \bigcup_{d \,|\, n}\operatorname{P}(d),$$

This implies

 * $$\operatorname{SR}(n) = \sum_{d \,|\, n}\operatorname{SP}(d).$$

Applying the Möbius inversion formula gives

 * $$\operatorname{SP}(n) = \sum_{d \,|\, n}\mu(d)\operatorname{SR}\left(\frac{n}{d}\right).$$

In this formula, if $P(n)$, then $R(n)$, and for $P(n)$: $d < n$. Therefore, $SR(n⁄d) = 0$.

This is the special case $d = n$ of Ramanujan's sum $SR(n⁄d) = 1$, defined as the sum of the $n$th powers of the primitive $n$th roots of unity:

 * $$c_n(s) = \sum_{a = 1 \atop \gcd(a, n) = 1}^n e^{2 \pi i \frac{a}{n} s}.$$

Orthogonality
From the summation formula follows an orthogonality relationship: for $SP(n) = μ(n)$ and $c_{n}(1)$

 * $$\sum_{k=1}^{n} \overline{z^{j\cdot k}} \cdot z^{j'\cdot k} = n \cdot\delta_{j,j'}$$

where $s$ is the Kronecker delta and $n$ is any primitive $δ$th root of unity.

The $c_{n}(s)$ matrix $z$ whose $j = 1, … , n$th entry is

 * $$U_{j,k} = n^{-\frac{1}{2}}\cdot z^{j\cdot k}$$

defines a discrete Fourier transform. Computing the inverse transformation using Gaussian elimination requires $j′ = 1, … , n$ operations. However, it follows from the orthogonality that $n$ is unitary. That is,

 * $$\sum_{k=1}^{n} \overline{U_{j,k}} \cdot U_{k,j'} = \delta_{j,j'},$$

and thus the inverse of $U$ is simply the complex conjugate. (This fact was first noted by Gauss when solving the problem of trigonometric interpolation.) The straightforward application of $U$ or its inverse to a given vector requires $n × n$ operations. The fast Fourier transform algorithms reduces the number of operations further to $(j, k)$.

Cyclotomic polynomials
The zeros of the polynomial
 * $$p(z) = z^n - 1$$

are precisely the $U$th roots of unity, each with multiplicity 1. The $U$th cyclotomic polynomial is defined by the fact that its zeros are precisely the primitive $n$th roots of unity, each with multiplicity 1.
 * $$\Phi_n(z) = \prod_{k=1}^{\varphi(n)}(z-z_k)$$

where $O(n^{3})$ are the primitive $n$th roots of unity, and $O(n^{2})$ is Euler's totient function. The polynomial $O(n log n)$ has integer coefficients and is an irreducible polynomial over the rational numbers (that is, it cannot be written as the product of two positive-degree polynomials with rational coefficients). The case of prime $n$, which is easier than the general assertion, follows by applying Eisenstein's criterion to the polynomial

 * $$\frac{(z+1)^n - 1}{(z+1) - 1},$$

and expanding via the binomial theorem.

Every $n$th root of unity is a primitive $n$th root of unity for exactly one positive divisor $n$ of $d$. This implies that

 * $$z^n - 1 = \prod_{d \,|\, n} \Phi_d(z).$$

This formula represents the factorization of the polynomial $z_{1}, z_{2}, z_{3}, …, z_{φ(n)}$ into irreducible factors:

 * $$\begin{align}

z^1 -1 &= z-1 \\ z^2 -1 &= (z-1)(z+1) \\ z^3 -1 &= (z-1) (z^2 + z + 1) \\ z^4 -1 &= (z-1)(z+1) (z^2+1) \\ z^5 -1 &= (z-1) (z^4 + z^3 +z^2 + z + 1) \\ z^6 -1 &= (z-1)(z+1) (z^2 + z + 1) (z^2 - z + 1)\\ z^7 -1 &= (z-1) (z^6+ z^5 + z^4 + z^3 + z^2 + z + 1) \\ z^8 -1 &= (z-1)(z+1) (z^2+1) (z^4+1) \\ \end{align}$$

Applying Möbius inversion to the formula gives

 * $$\Phi_n(z) = \prod_{d \,|\, n}\left(z^\frac{n}{d} - 1\right)^{\mu(d)} = \prod_{d \,|\, n}\left(z^d - 1\right)^{\mu\left(\frac{n}{d}\right)},$$

where $φ(n)$ is the Möbius function. So the first few cyclotomic polynomials are

If $d$ is a prime number, then all the $n$th roots of unity except 1 are primitive $p$th roots. Therefore, $$\Phi_p(z) = \frac{z^p - 1}{z - 1} = \sum_{k = 0}^{p - 1} z^k.$$ Substituting any positive integer ≥ 2 for $p$, this sum becomes a base $p$ repunit. Thus a necessary (but not sufficient) condition for a repunit to be prime is that its length be prime.

Note that, contrary to first appearances, not all coefficients of all cyclotomic polynomials are 0, 1, or −1. The first exception is $Φ_{n}(z)$. It is not a surprise it takes this long to get an example, because the behavior of the coefficients depends not so much on $z$ as on how many odd prime factors appear in $z$. More precisely, it can be shown that if $n$ has 1 or 2 odd prime factors (for example, $z^{n} − 1$) then the $n$th cyclotomic polynomial only has coefficients 0, 1 or −1. Thus the first conceivable $n$ for which there could be a coefficient besides 0, 1, or −1 is a product of the three smallest odd primes, and that is $μ$. This by itself doesn't prove the 105th polynomial has another coefficient, but does show it is the first one which even has a chance of working (and then a computation of the coefficients shows it does). A theorem of Schur says that there are cyclotomic polynomials with coefficients arbitrarily large in absolute value. In particular, if $$n = p_1 p_2 \cdots p_t,$$ where $$p_1 < p_2 < \cdots < p_t$$ are odd primes, $$p_1 +p_2>p_t,$$ and t is odd, then $Φ_{1}(z) = z − 1$ occurs as a coefficient in the $n$th cyclotomic polynomial.

Many restrictions are known about the values that cyclotomic polynomials can assume at integer values. For example, if $n$ is prime, then $Φ_{2}(z) = (z^{2} − 1)⋅(z − 1)^{−1} = z + 1$ if and only if $Φ_{3}(z) = (z^{3} − 1)⋅(z − 1)^{−1} = z^{2} + z + 1$.

Cyclotomic polynomials are solvable in radicals, as roots of unity are themselves radicals. Moreover, there exist more informative radical expressions for $n$th roots of unity with the additional property that every value of the expression obtained by choosing values of the radicals (for example, signs of square roots) is a primitive $p$th root of unity. This was already shown by Gauss in 1797. Efficient algorithms exist for calculating such expressions.

Cyclic groups
The $n$th roots of unity form under multiplication a cyclic group of order $n$, and in fact these groups comprise all of the finite subgroups of the multiplicative group of the complex number field. A generator for this cyclic group is a primitive $n$th root of unity.

The $n$th roots of unity form an irreducible representation of any cyclic group of order $n$. The orthogonality relationship also follows from group-theoretic principles as described in Character group.

The roots of unity appear as entries of the eigenvectors of any circulant matrix; that is, matrices that are invariant under cyclic shifts, a fact that also follows from group representation theory as a variant of Bloch's theorem. In particular, if a circulant Hermitian matrix is considered (for example, a discretized one-dimensional Laplacian with periodic boundaries ), the orthogonality property immediately follows from the usual orthogonality of eigenvectors of Hermitian matrices.

Cyclotomic fields
By adjoining a primitive $n$th root of unity to $$\Q,$$ one obtains the $n$th cyclotomic field $$\Q(\exp(2\pi i/n)).$$This field contains all $n$th roots of unity and is the splitting field of the $n$th cyclotomic polynomial over $$\Q.$$ The field extension $$\Q(\exp(2\pi i /n))/\Q$$ has degree φ(n) and its Galois group is naturally isomorphic to the multiplicative group of units of the ring $$\Z/n\Z.$$

As the Galois group of $$\Q(\exp(2\pi i /n))/\Q$$ is abelian, this is an abelian extension. Every subfield of a cyclotomic field is an abelian extension of the rationals. It follows that every nth root of unity may be expressed in term of k-roots, with various k not exceeding φ(n). In these cases Galois theory can be written out explicitly in terms of Gaussian periods: this theory from the Disquisitiones Arithmeticae of Gauss was published many years before Galois.

Conversely, every abelian extension of the rationals is such a subfield of a cyclotomic field – this is the content of a theorem of Kronecker, usually called the Kronecker–Weber theorem on the grounds that Weber completed the proof.

Relation to quadratic integers
For $Φ_{4}(z) = (z^{4} − 1)⋅(z^{2} − 1)^{−1} = z^{2} + 1$, both roots of unity and  are integers.

For three values of $n$, the roots of unity are quadratic integers:
 * For $Φ_{5}(z) = (z^{5} − 1)⋅(z − 1)^{−1} = z^{4} + z^{3} + z^{2} + z + 1$ they are Eisenstein integers ($Φ_{6}(z) = (z^{6} − 1)⋅(z^{3} − 1)^{−1}⋅(z^{2} − 1)^{−1}⋅(z − 1) = z^{2} − z + 1$).
 * For $Φ_{7}(z) = (z^{7} − 1)⋅(z − 1)^{−1} = z^{6} + z^{5} + z^{4} + z^{3} + z^{2} +z + 1$ they are Gaussian integers ($Φ_{8}(z) = (z^{8} − 1)⋅(z^{4} − 1)^{−1} = z^{4} + 1$): see Imaginary unit.

For four other values of $n$, the primitive roots of unity are not quadratic integers, but the sum of any root of unity with its complex conjugate (also an $n$th root of unity) is a quadratic integer.

For $Φ_$, none of the non-real roots of unity (which satisfy a quartic equation) is a quadratic integer, but the sum $n = 150$ of each root with its complex conjugate (also a 5th root of unity) is an element of the ring $3 ⋅ 5 ⋅ 7 = 105$ ($1 − t$). For two pairs of non-real 5th roots of unity these sums are inverse golden ratio and minus golden ratio.

For $d ∣ Φ_{p}(d)$, for any root of unity $d ≡ 1 (mod p)$ equals to either 0, ±2, or ±√2 ($n = 1, 2$).

For $n = 3, 6$, for any root of unity, $D = −3$ equals to either 0, ±1, ±2 or ±√3 ($n = 4$).

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

title: Root of unity


text:
Martin Luther King Jr. Day

Martin Luther King Jr. Day (officially Birthday of Martin Luther King Jr., and often referred to shorthand as MLK Day) is a federal holiday in the United States observed on the third Monday of January each year. King was chief spokesperson for nonviolent activism in the Civil Rights Movement, which protested racial discrimination in federal and state law and civil society. The movement led to several groundbreaking legislative reforms in the United States.

Born in 1929, Martin Luther King Jr.'s actual birthday is January 15 (which in 1929 fell on a Tuesday). The earliest Monday for this holiday is January 15 and the latest is January 21. The Monday observance is similar for those federal holidays which fall under the Uniform Monday Holiday Act.

The campaign for a federal holiday in King's honor began soon after his assassination in 1968. President Ronald Reagan signed the holiday into law in 1983, and it was first observed three years later on January 20, 1986. At first, some states resisted observing the holiday as such, giving it alternative names or combining it with other holidays. Official observance in each state's law as well as federal law occurred in 2000.

Proposals
The idea of Martin Luther King Jr. Day as a holiday was promoted by labor unions in contract negotiations. After King's death, Representative John Conyers (a Democrat from Michigan) and Senator Edward Brooke (a Republican from Massachusetts) introduced a bill in Congress to make King's birthday a national holiday. The bill first came to a vote in the U.S. House of Representatives in 1979. However, it fell five votes short of the number needed for passage. Two of the main arguments mentioned by opponents were that a paid holiday for federal employees would be too expensive and that a holiday to honor a private citizen would be contrary to longstanding tradition (King had never held public office). Only two other figures have national holidays in the U.S. honoring them: George Washington and Christopher Columbus.

Soon after, the King Center turned to support from the corporate community and the general public. The success of this strategy was cemented when musician Stevie Wonder released the single "Happy Birthday" to popularize the campaign in 1980 and hosted the Rally for Peace Press Conference in 1981. Six million signatures were collected for a petition to Congress to pass the law, termed by a 2006 article in The Nation as "the largest petition in favor of an issue in U.S. history".

Senators Jesse Helms and John Porter East (both North Carolina Republicans) led the opposition to the holiday and questioned whether King was important enough to receive such an honor. Helms criticized King's opposition to the Vietnam War and accused him of espousing "action-oriented Marxism". Helms led a filibuster against the bill and on October 3, 1983, submitted a 300-page document to the Senate alleging that King had associations with communists. Democratic New York Senator Daniel Patrick Moynihan declared Helms' document a "packet of filth", threw it on the Senate floor, and stomped on it.

Federal passage
President Ronald Reagan originally opposed the holiday, citing cost concerns. When asked to comment on Helms' accusations that King was a communist, the president said "We'll know in thirty-five years, won't we", referring to the eventual release of FBI surveillance tapes that had previously been sealed. But on November 2, 1983, Reagan signed a bill into law, proposed by Representative Katie Hall of Indiana, to create a federal holiday honoring King. The final vote in the House of Representatives on August 2, 1983, was 338–90 (242–4 in the House Democratic Caucus and 89–77 in the House Republican Conference) with 5 members voting present or abstaining, while the final vote in the Senate on October 19, 1983, was 78–22 (41–4 in the Senate Democratic Caucus and 37–18 in the Senate Republican Conference), both veto-proof margins. The holiday was observed for the first time on January 20, 1986. It is observed on the third Monday of January.

The bill also established the "Martin Luther King, Jr. Federal Holiday Commission" to oversee observance of the holiday, and Coretta Scott King, King's wife, was made a member of this commission for life by President George H. W. Bush in May 1989.

State-level passage
Although the federal holiday honoring King was signed into law in 1983 and took effect three years later, not every U.S. state chose to observe the January holiday at the state level until 1991, when the New Hampshire legislature created "Civil Rights Day" and abolished its April "Fast Day". In 1999, New Hampshire became the last state to name a holiday after King, which they first celebrated in January 2000 – the first nationwide celebration of the day with this name.

In 1986, Arizona Governor Bruce Babbitt, a Democrat, created a paid state MLK holiday in Arizona by executive order just before he left office, but in 1987, his Republican successor Evan Mecham, citing an attorney general's opinion that Babbitt's order was illegal, reversed Babbitt's decision days after taking office. Later that year, Mecham proclaimed the third Sunday in January to be "Martin Luther King Jr./Civil Rights Day" in Arizona, albeit as an unpaid holiday. This proposal was rejected by the state Senate the following year. In 1990, Arizona voters were given the opportunity to vote on giving state employees a paid MLK holiday. That same year, the National Football League threatened to move Super Bowl XXVII, which was planned for Arizona in 1993, if the MLK holiday was voted down. In the November 1990 election, the voters were offered two King Day options: Proposition 301, which replaced Columbus Day on the list of paid state holidays, and Proposition 302, which merged Lincoln's and Washington's birthdays into one paid holiday to make room for MLK Day. Both measures failed to pass, with only 49% of voters approving Prop 302, the more popular of the two options; although some who voted "no" on 302 voted "yes" on Prop 301. Consequently, the state lost the chance to host Super Bowl XXVII, which was subsequently held at the Rose Bowl in Pasadena, California. In a 1992 referendum, the voters, this time given only one option for a paid King Day, approved state-level recognition of the holiday.

On May 2, 2000, South Carolina governor Jim Hodges signed a bill to make King's birthday an official state holiday. South Carolina was the last state to recognize the day as a paid holiday for all state employees. Before the bill, employees could choose between celebrating Martin Luther King Jr. Day or one of three Confederate holidays.

Alternative names
While all states now observe the holiday, some did not name the day after King. For example, in New Hampshire, the holiday was known as "Civil Rights Day" until 1999, when the State Legislature voted to change the name of the holiday to Martin Luther King Day.

Several additional states have chosen to combine commemorations of King's birthday with other observances:
 * In Alabama: "Robert E. Lee/Martin Luther King Birthday".
 * In Arizona: "Martin Luther King Jr./Civil Rights Day".
 * In Arkansas: it was known as "Dr. Martin Luther King, Jr.'s Birthday and Robert E. Lee's Birthday" from 1985 to 2017. Legislation in March 2017 changed the name of the state holiday to "Dr. Martin Luther King, Jr.'s Birthday" and moved the commemoration of Lee to October.
 * In Idaho: "Martin Luther King Jr.–Idaho Human Rights Day".
 * In Mississippi: "Martin Luther King's and Robert E. Lee's Birthdays".
 * In New Hampshire: "Martin Luther King Jr. Civil Rights Day".
 * In Virginia: it was known as Lee–Jackson–King Day, combining King's birthday with the established Lee–Jackson Day. In 2000, Lee–Jackson Day was moved to the Friday before Martin Luther King Jr. Day, establishing Martin Luther King Jr. Day as a holiday in its own right. Lee-Jackson Day was eliminated in 2020.
 * In Wyoming: it is known as "Martin Luther King Jr./Wyoming Equality Day". Liz Byrd, the first black woman in the Wyoming legislature, introduced a bill in 1991 for Wyoming to recognize MLK day as a paid state holiday; however, she compromised on the name because her peers would not pass it otherwise.

Workplace leave

Overall, as of 2019, 45% of employers gave employees the day off. The reasons for not providing the day off have varied, ranging from the recent addition of the holiday to its occurrence just two weeks after the week between Christmas and New Year's Day, when many businesses are closed for part or all of it. The New York Stock Exchange and NASDAQ both close for trading, and banks are generally closed. Additionally, many schools and places of higher education are closed for classes; others remain open but may hold seminars or celebrations of King's message. The observance of MLK Day has led to some colleges and universities extending their Christmas break to include the day as part of the break. Some employers use MLK Day as a floating or movable holiday.

MLK Day of Service
The national "Martin Luther King, Jr., National Day of Service" was started by former Pennsylvania U.S. Senator Harris Wofford and Atlanta Congressman John Lewis, who co-authored the King Holiday and Service Act. The federal legislation challenges Americans to transform the King Holiday into a day of citizen action volunteer service in honor of King. The federal legislation was signed into law by President Bill Clinton on August 23, 1994. Since 1996, Wofford's former state office director, Todd Bernstein, has been directing the annual Greater Philadelphia King Day of Service, the largest event in the nation honoring King.

Since 1994, the day of service has been coordinated nationally by AmeriCorps, a federal agency, which provides grants to organizations that coordinate service activities on MLK Day.

The only other official national day of service in the U.S., as designated by the government, is September 11 National Day of Service (9/11 Day).

Speeches
Cesar Chavez campaigned with him to call attention to the economic needs of farmworkers in the United States. Chavez used his speech on this day in 1990 to again call attention to the similarity between his campaign regarding pesticide issues and King's campaigns. He later was honored with the creation of Cesar Chavez Day in imitation of this holiday.

Canada
The City of Toronto government in Ontario officially recognizes Martin Luther King Jr. Day, although not as a paid holiday: all government services and businesses remain open. The Ottawa municipal government in Ontario officially began observing this national holiday on January 26, 2005.

Israel
In 1984, during a visit by the U.S. Sixth Fleet, Navy chaplain Rabbi Arnold Resnicoff conducted the first Israeli presidential ceremony in commemoration of Martin Luther King Jr. Day, held in the President's Residence, Jerusalem. Aura Herzog, wife of Israel's then-President Chaim Herzog, noted that she was especially proud to host this special event, because Israel had a national forest in honor of King, and that Israel and King shared the idea of "dreams." Resnicoff continued this theme in his remarks during the ceremony, quoting the verse from Genesis, spoken by the brothers of Joseph when they saw their brother approach, "Behold the dreamer comes; let us slay him and throw him into the pit, and see what becomes of his dreams." Resnicoff noted that, from time immemorial, there have been those who thought they could kill the dream by slaying the dreamer, but – as the example of King's life shows – such people are always wrong.

Japan
Martin Luther King Jr. Day is observed in the Japanese city of Hiroshima. In January 2005, Mayor Tadatoshi Akiba held a special banquet at the mayor's office as an act of unifying his city's call for peace with King's message of human rights.

Netherlands
Every year since 1987, the Dr. Martin Luther King Tribute and Dinner has been held in Wassenaar, The Netherlands. The Tribute includes young people and veterans of the Civil Rights Movement as well as music. It always ends with everyone holding hands in a circle and singing "We Shall Overcome". The Tribute is held on the last Sunday in January.

Dates
 1986–2103

Observed on the third Monday in January.

General holidays

 * List of holidays by country
 * List of holidays commemorating individuals
 * List of multinational festivals and holidays
 * Public holidays in the United States

Volunteer day events

 * Global Youth Service Day
 * Good Deeds Day
 * International Volunteer Day
 * International Year of Volunteers
 * Mitzvah Day
 * National Philanthropy Day (U.S. and Canada)
 * Random Acts of Kindness Day
 * Sewa Day
 * Education and Sharing Day
 * Make A Difference Day
 * World Kindness Day

title: Martin Luther King Jr. Day


text:
Gabriela Mistral

Lucila Godoy Alcayaga (7 April 1889 – 10 January 1957), known by her pseudonym Gabriela Mistral, was a Chilean poet-diplomat, educator, and Catholic. She was a member of the Secular Franciscan Order or Third Franciscan order. She was the first Latin American author to receive a Nobel Prize in Literature in 1945, "for her lyric poetry which, inspired by powerful emotions, has made her name a symbol of the idealistic aspirations of the entire Latin American world". Some central themes in her poems are nature, betrayal, love, a mother's love, sorrow and recovery, travel, and Latin American identity as formed from a mixture of Native American and European influences. Her image is featured on the 5,000 Chilean peso banknote.

Early life
Mistral was born in Vicuña, Chile, but grew up in Montegrande, an Andean village where she attended a primary school taught by her older sister, Emelina Molina. Despite the financial problems caused by Emelina later on, Mistral held great respect for her. Her father, Juan Gerónimo Godoy Villanueva, was also a schoolteacher but left the family when she was three years old and died alone and estranged in 1911. Poverty was a constant presence in her early life. At the age of fifteen, she supported herself and her mother, Petronila Alcayaga, a seamstress, by working as a teacher's aide in Compañía Baja, a seaside town near La Serena, Chile.

In 1904, Mistral published some early poems, including Ensoñaciones ("Dreams"), Carta Íntima ("Intimate Letter"), and Junto al Mar ("By the Sea"), in the local newspapers El Coquimbo: Diario Radical and La Voz de Elqui, using different pseudonyms and variations of her name.

In 1906, Mistral met Romelio Ureta, a railway worker and her first love, who tragically took his own life in 1909. Shortly after, her second love married someone else. These heartbreaks were reflected in her early poetry and gained recognition with her first published literary work in 1914, Sonetos de la muerte ("Sonnets on Death"). To protect her job as a teacher, she used a pen name, fearing the consequences of revealing her true identity. Mistral won first prize in the national literary contest Juegos Florales held in Santiago, the capital of Chile. Exploring themes of death and life more broadly than previous Latin American poets, she expanded her poetic horizons. While Mistral had passionate friendships with both men and women, which influenced her writing, she kept her emotional life private.

Since June 1908, Mistral had been using the pen name Gabriela Mistral for most of her writing. After winning the Juegos Florales, she rarely used her given name, Lucila Godoy, for her publications. She constructed her pseudonym from the names of two of her favorite poets, Gabriele D'Annunzio and Frédéric Mistral, or, according to another account, as a combination of the Archangel Gabriel and the mistral wind of Provence.

In 1922, Mistral published her debut book, Desolación ("Desolation"), with assistance from Federico de Onis, the Director of the Hispanic Institute of New York. The collection of poems explored themes such as motherhood, religion, nature, morality, and love for children. Her personal sorrows were reflected in the poems, solidifying her international reputation. Departing from the modernist trends in Latin America, Mistral's work was hailed by critics as straightforward yet simplistic. Two years later, in 1924, she released her second book, Ternura ("Tenderness").

Career as an educator
During her adolescence, the scarcity of trained teachers, especially in rural areas, allowed anyone willing to work to find employment as a teacher. However, the young woman faced challenges in accessing good schools due to her lack of political and social connections. In 1907, she was rejected from the Normal School without explanation, which she later attributed to the school's chaplain, Father Ignacio Munizaga, who was aware of her publications advocating for educational reform and increased access to schools for all social classes.

Although her formal education ended in 1900, she secured teaching positions with the help of her older sister, Emelina, who had likewise begun as a teacher's aide and was responsible for much of the poet's early education. Through her publications in local and national newspapers and magazines, as well as her willingness to relocate, she advanced from one teaching position to another. Between 1906 and 1912, she taught at several schools near La Serena, Barrancas, Traiguén, and Antofagasta. In 1912, she began working at a liceo (high school) in Los Andes, where she remained for six years, frequently visiting Santiago. In 1918, Pedro Aguirre Cerda, the Minister of Education and future President of Chile, appointed her as the director of the Sara Braun Lyceum in Punta Arenas. She subsequently moved to Temuco in 1920 and then to Santiago in 1921, defeating a candidate associated with the Radical Party to become the director of Santiago's Liceo #6, the country's newest and most prestigious girls' school.

The controversy surrounding Gabriela Mistral's nomination for the coveted position in Santiago influenced her decision to accept an invitation to work in Mexico in 1922, under the guidance of Mexico's Minister of Education, José Vasconcelos. There, she contributed to the nation's plan to reform libraries and schools and establish a national education system. During this time, she gained international recognition through her journalism, public speaking, and the publication of her work Desolación in New York. She later published Lecturas para Mujeres (Readings for Women), a collection of prose and verse celebrating girls' education, featuring works by Latin American and European writers.

After spending nearly two years in Mexico, Mistral traveled to Washington D.C., where she addressed the Pan American Union, and then continued her journey to New York and Europe. In Madrid, she published Ternura (Tenderness), a collection of lullabies and rondas intended for children, parents, and fellow poets. She returned to Chile in early 1925, formally retiring from the country's education system and receiving a pension. Just in time, as the legislature had recently granted the demands of the teachers' union, led by Mistral's rival Amanda Labarca Hubertson, stipulating that only university-trained teachers could be appointed in schools. Despite her limited formal education, Mistral received the academic title of Spanish Professor from the University of Chile in 1923, which highlighted her remarkable self-education and her intellectual abilities, nurtured by the vibrant culture of newspapers, magazines, and books in provincial Chile.

Pablo Neruda, Chile's second Nobel Prize laureate in literature, met Mistral when she relocated to his hometown, Temuco. She introduced him to her poetry and recommended readings, leading to a lifelong friendship between the two poets.

International work and recognition
Mistral's international stature made it unlikely for her to remain in Chile. In mid-1925, she was invited to represent Latin America in the newly formed Institute for Intellectual Cooperation of the League of Nations. In early 1926, she relocated to France, effectively becoming an exile for the rest of her life. Initially, she made a living through journalism and giving lectures in the United States and Latin America, including Puerto Rico, the Caribbean, Brazil, Uruguay, and Argentina.

Between 1926 and 1932, Mistral primarily resided in France and Italy. During this period, she worked for the League for Intellectual Cooperation of the League of Nations, attending conferences throughout Europe and the Americas. She held a visiting professorship at Barnard College of Columbia University in 1930–1931, briefly worked at Middlebury College and Vassar College in 1931, and received a warm reception at the University of Puerto Rico at Rio Piedras, where she gave conferences and wrote in 1931, 1932, and 1933.

Like many Latin American artists and intellectuals, Mistral served as a consul from 1932 until her death, working in various locations including Naples, Madrid, Lisbon, Nice, Petrópolis, Los Angeles, Santa Barbara, Veracruz, Rapallo, and New York City. While serving as consul in Madrid, she had occasional professional interactions with fellow Chilean consul and Nobel Prize recipient Pablo Neruda. Mistral was among the early writers to recognize the importance and originality of Neruda's work, which she had known since he was a teenager and she was a school director in his hometown of Temuco.

Mistral published hundreds of articles in magazines and newspapers throughout the Spanish-speaking world. She had notable confidants such as Eduardo Santos, President of Colombia, all the elected Presidents of Chile from 1922 to her death in 1957, Eduardo Frei Montalva (who would be elected president in 1964), and Eleanor Roosevelt.

Her second major volume of poetry, Tala, was published in 1938 in Buenos Aires with the assistance of her longtime friend and correspondent Victoria Ocampo. The proceeds from the sale were dedicated to children orphaned by the Spanish Civil War. This volume contains poems that celebrate the customs and folklore of Latin America and Mediterranean Europe, reflecting Mistral's identification as "una mestiza de vasco," acknowledging her European Basque-Indigenous Amerindian background.

On 14 August 1943, Mistral's 17-year-old nephew, Juan Miguel Godoy, whom she considered as a son and called Yin Yin, tragically took his own life. The grief from this loss, along with her responses to the tensions of World War II and the Cold War in Europe and the Americas, are reflected in her last volume of poetry published during her lifetime, Lagar, which appeared in a truncated form in 1954. Her partner Doris Dana edited and published a final volume of poetry, Poema de Chile, posthumously in 1967. Poema de Chile depicts the poet's return to Chile after death, accompanied by an Indian boy from the Atacama desert and an Andean deer, the huemul. This collection of poetry foreshadows the interest in objective description and re-vision of the epic tradition that would emerge among poets of the Americas, all of whom Mistral carefully read.On 15 November 1945, Mistral became the first Latin American and the fifth woman to receive the Nobel Prize in Literature. King Gustav of Sweden presented her with the award in person on 10 December 1945. In 1947, she received an honorary doctorate from Mills College in Oakland, California. In 1951, she was awarded the National Literature Prize in Chile.

Poor health limited Mistral's travel in her final years. She resided in the town of Roslyn, New York, and then transferred to Hempstead, New York, where she died from pancreatic cancer on 10 January 1957 at the age of 67. Her remains were returned to Chile nine days later, and the Chilean government declared three days of national mourning, with hundreds of thousands of mourners paying their respects.

Some of Mistral's best-known poems include Piececitos de Niño, Balada, Todas Íbamos a ser Reinas, La Oración de la Maestra, El Ángel Guardián, Decálogo del Artista, and La Flor del Aire. She also wrote and published approximately 800 essays in magazines and newspapers. Mistral was renowned as a correspondent and highly regarded orator, both in person and through radio broadcasts.

Mistral may be most widely quoted in English for Su Nombre es Hoy ("His Name is Today"): "We are guilty of many errors and many faults, but our worst crime is abandoning the children, neglecting the fountain of life. Many of the things we need can wait. The child cannot. Right now is the time his bones are being formed, his blood is being made, and his senses are being developed. To him we cannot answer ‘Tomorrow,’ his name is today."

Characteristics of her work
Mistral's work incorporates gray tones and conveys recurring feelings of sadness and bitterness, reflecting her difficult childhood marked by deprivation and a lack of affection at home. Despite this, her writings also reveal her deep affection for children, which she developed during her early years as a teacher in a rural school. Catholicism, a significant influence in Mistral's life, is also evident in her literature; however, she maintains a neutral stance toward religion. Her writing skillfully combines religious themes with emotions of love and piety, solidifying her position as one of the most esteemed representatives of Latin American literature in the 20th century.

Death, posthumous tributes and legacy
During the 1970s and 1980s, the military dictatorship of General Augusto Pinochet appropriated Gabriela Mistral's image, portraying her as a symbol of "submission to authority" and "social order." Author Licia Fiol-Matta challenged the traditional views of Mistral as a saint-like celibate and suffering heterosexual woman, suggesting that she was a lesbian instead. In 2007, after the death of Mistral's alleged last romantic partner, Doris Dana, her archive was discovered, containing letters exchanged between Mistral and various occasional female lovers. The publication of these letters in the book Niña errante (2007), edited by Pedro Pablo Zegers, supported the notion of a long-lasting romantic relationship between Mistral and Dana during Mistral's final years. The letters were later translated into English by Velma García and published by the University of New Mexico Press in 2018. Despite these claims, Doris Dana, who was 31 years younger than Mistral, explicitly denied in her final interview that their relationship was ever romantic or erotic, describing it as that of a stepmother and stepdaughter. Dana also denied being a lesbian and expressed skepticism regarding Mistral's sexual orientation.

Mistral suffered from diabetes and heart problems, and she ultimately died of pancreatic cancer at the age of 67 on 10 January 1957, in Hempstead Hospital in New York City, with Doris Dana by her side.

On 7 April 2015, Google commemorated Gabriela Mistral's 126th birthday, honoring the Chilean poet and educator with a special doodle.

Themes
Gabriela Mistral has greatly influenced Latin American poetry. In a powerful speech by Swedish writer Hjalmar Gullberg, a member of the Swedish Academy, he provided insights into the perspective and emotions of Gabriela Mistral. Gullberg discussed how the language of troubadours, once unintelligible to Frédéric Mistral's own mother, became the language of poetry. This language continued to thrive with the birth of Gabriela Mistral, whose voice shook the world and opened the eyes and ears of those willing to listen.

Gullberg noted that after experiencing the suicide of her first love, Gabriela Mistral emerged as a poet whose words spread across South America and beyond. While little is known about her first love, his death influenced Mistral's poems, which often explored themes of death, despair, and possibly a resentment towards God. Her collection of poems titled Desolación, inspired by the loss of her first love and later the death of a beloved nephew, impacted many others. The fifteenth poem in Desolación expressed sorrow for the loss of a child and resonated with those who experienced the pain of losing loved ones.

However, Gabriela Mistral's books do not solely focus on themes of death, desolation, and loss. She also explored themes of love and motherhood, not only in relation to her beloved railroad employee and nephew but also in her interactions with the children she taught. Her collection of songs and rounds, titled Ternura, reflects her love for the children in her school. Published in Madrid in 1924, these heartfelt words were embraced by four thousand Mexican children who sang them as a tribute to Mistral. Her dedication to her children earned her the title of the Poet of Motherhood.

Having lived through two world wars and other violent conflicts, Mistral's experiences paved the way for her third major collection, Tala (meaning "ravage" according to Gullberg). Tala encompasses a blend of sacred hymns, simple songs for children, and poems that touch on subjects like water, corn, salt, and wine. Gullberg pays homage to Mistral, acknowledging her as the great singer of sorrow and motherhood in Latin America. Mistral's collections of poems and songs beautifully express her care for children and the sorrows she endured as a teacher and poet in Latin America. Every word in her work evokes themes of sorrow and motherhood.

Awards and honors

 * 1914: Juegos Florales, Sonetos de la Muerte
 * 1945: Nobel Prize in Literature
 * 1951: Chilean National Prize for Literature

The Venezuelan writer and diplomat who worked under the name Lucila Palacios took her nom de plume in honour of Mistral's original name.

Works

 * 1914: Sonetos de la muerte ("Sonnets of Death")
 * 1922: Desolación ("Despair"), including "Decalogo del artista", New York : Instituto de las Españas
 * 1923: Lecturas para Mujeres ("Readings for Women")
 * 1924: Ternura: canciones de niños, Madrid: Saturnino Calleja
 * 1934: Nubes Blancas y Breve Descripción de Chile (1934)
 * 1938: Tala ("Harvesting" ), Buenos Aires: Sur
 * 1941: Antología: Selección de Gabriela Mistral, Santiago, Chile: Zig Zag
 * 1952: Los sonetos de la muerte y otros poemas elegíacos, Santiago, Chile: Philobiblion
 * 1954: Lagar, Santiago, Chile
 * 1957: Recados: Contando a Chile, Santiago, Chile: Editorial del Pacífico Croquis mexicanos; Gabriela Mistral en México, México City: Costa-Amic
 * 1958: Poesías completas, Madrid : Aguilar
 * 1967: Poema de Chile ("Poem of Chile"), published posthumously
 * 1992: Lagar II, published posthumously, Santiago, Chile: Biblioteca Nacional

English
Several selections of Mistral's poetry have been published in English translation, including those by Doris Dana, Langston Hughes, and Ursula K. Le Guin. Two editions of her first book of poems, Desolación, have been translated into English and appear in bilingual volumes.
 * Selected Poems of Gabriela Mistral, trans. Langston Hughes (Bloomington: Indiana University Press, 1957)
 * Selected poems of Gabriela Mistral, trans. Doris Dana (Johns Hopkins Press, 1971), ISBN 978-0801811975
 * Selected Poems of Gabriela Mistral, trans. Ursula Le Guin (University of New Mexico Press, 2003), ISBN 978-0826328182
 * Gabriela Mistral: This Far Place, trans. John Gallas, Contemplative Poetry 8 (Oxford: SLG Press, 2023), ISBN 978-0728303409
 * Desolation: A Bilingual Edition of Desolación (1923), trans. Michael P. Predmore and Liliana Baltra (Pittsburgh: Latin American Literary Review Press, 2014), ISBN 9781891270246
 * Desolación (1922): Centennial Bilingual Edition, trans. Inés Bellina, Anne Freeland, and Alejandra Quintana Arocho, (New York: Sundial House, Columbia University Press, 2023), ISBN 9798987926437

Nepali
Some of Mistral's poems are translated into Nepali by Suman Pokhrel, and collected in an anthology titled Manpareka Kehi Kavita.

title: Gabriela Mistral


text:
1893 in music

Events in the year 1893 in music.

Specific locations

 * 1893 in Norwegian music

Events

 * February 9 – Première of Giuseppe Verdi's final opera Falstaff in La Scala in Milan with Victor Maurel in the title rôle.
 * Summer – Gustav Mahler's first summer composing at his Komponierhäuschen ("composition hut") at Steinbach am Attersee in the Salzkammergut region of Austria.
 * August 14–15 – America's oldest music organization, the Stoughton Musical Society performs at the World's Columbian Exposition.
 * October 16–28 – In Saint Petersburg (Russia), Pyotr Ilyich Tchaikovsky conducts the first performance of his Symphony No. 6 in B minor, Op. 74, Pathétique (Патетическая), nine days before his death (attributed to cholera). The second performance takes place 21 days later at a memorial concert conducted by Eduard Nápravník, incorporating minor revisions. Tchaikovsky wrote it between February and the end of August at Klin.
 * December 16 – Antonín Dvořák's Symphony No. 9 "From the New World" receives its première at Carnegie Hall in New York City.
 * December 29 – Claude Debussy's String Quartet is premièred in Paris.
 * American sisters Patty and Mildred J. Hill publish Song Stories for the Kindergarten including "Good Morning to All", which later becomes known as "Happy Birthday to You".
 * The first sousaphone is built by James Welsh Pepper at the request of bandmaster John Philip Sousa in the United States.

Published popular music

Selected compositions (words/music indicated by "w.m."):
 * "Can't Lose Me, Charlie" w.m. Harry S. Miller
 * "The Cat Came Back"    w.m. Harry S. Miller
 * "December And May"    w. Edward Marks m. William Lorraine
 * "Do Do My Huckleberry Do"    w. Harry Dillon m. John Dillon (the Dillon Brothers)
 * "The Fatal Wedding"    w. W. H. Windom m. Gussie L. Davis
 * "Good Morning to All"    w. Patty Smith Hill m. Mildred J. Hill
 * "I Long to See The Girl I Left Behind"    w.m. John T. Kelly
 * "The Liberty Bell" by John Philip Sousa
 * "Mamie, Come Kiss Your Honey"    w.m. May Irwin
 * "Marguerite" by Charles A. White
 * "Oh! Mr Porter"    w. Thomas Le Brunn m. George Le Brunn
 * "Private Tommy Atkins"    w. Henry Hamilton m. S. Potter
 * "Say 'Au Revoir', But Not 'Good-Bye'" by Harry Kennedy
 * "Sweet Marie"    w. Cy Warman m. Raymond Moore
 * "They All Take After Me"    w. T. W. Connor m. Harry Randall
 * "Two Little Girls in Blue"    w.m. Charles Graham
 * "The Volunteer Organist"    w. William G. Gray m. Henry Lamb
 * "When The Roll Is Called Up Yonder"    w.m. James M. Black
 * "Zacatecas" by Genaro Codina

Recorded popular music

 * "After the Ball (song)" – George J. Gaskin
 * "After the Fair (Parody)" – George H. Diamond
 * "Blind Tom" – Brilliant Quartette
 * "The Cat Came Back" – George H. Diamond
 * "Chinese Picnic" – Vess Ossman
 * "Cocoanut Dance" – Vess Ossman
 * "The Commodore Song" – Edward M. Favor
 * "Daisy Bell" – Dan W. Quinn
 * "Darkie Tickle" – Vess Ossman
 * "Down On The Farm" – Edward Clarance
 * "High School Cadets" – Vess Ossman
 * "The King's Song" – Edward M. Favor
 * "Love's Sweet Honor" – Vess Ossman
 * "Lovely Woman" – Al Reeves
 * "The Man That Broke the Bank at Monte Carlo” – George H. Diamond
 * "Marriage Bells" – Vess Ossman
 * "O Promise Me" – George J. Gaskin
 * "Parody On "After the Ball" – George H. Diamond
 * "Pat Brady and the World Fair at Chicago" – Dan Kelly
 * "The Washington Post (march)" – Vess Ossman
 * "When Summer Comes Again" – George H. Diamond
 * "Why Should I Keep From Whistling?" – John York AtLee & Fred Gaisberg

Classical music

 * Amy Beach – Gaelic Symphony
 * Johannes Brahms
 * Six Pieces for Piano, Op. 118
 * Four Pieces for Piano, Op. 119
 * Claude Debussy – String Quartet in G minor
 * Antonín Dvořák – Symphony no. 9 in E minor, "From the New World"
 * Edward German – Symphony in A minor, "Norwich"
 * Johan Halvorsen – Entry of the Boyars
 * Sergei Rachmaninoff
 * Fantaisie-Tableaux, for two pianos, Op. 5
 * Morceaux de salon for violin and piano, Op. 6
 * Jean Sibelius – Lemminkäinen Suite
 * William Stanley – Bay View Gavotte in A major
 * Josef Suk – Quintet for Piano and Strings in G minor
 * Pyotr Ilyich Tchaikovsky – Symphony no. 6 in B minor, "Pathétique"
 * Charles-Marie Widor - Symphony no. 3 for organ & orchestra, Op. 69

Opera

 * Granville Bantock – Caedmar
 * Julius Bechgaard – Frode premiered on May 11 in Copenhagen
 * Engelbert Humperdinck – Hänsel und Gretel
 * Isidore de Lara – Amy Robsart
 * Emile Pessard
 * Une nuit de Noël premiered at the Ambigu, Paris
 * Mam'zelle Carabin premiered on November 3 at the Théâtre des Bouffes Parisiens, Salle Choiseul, Paris
 * Giacomo Puccini – Manon Lescaut
 * Camille Saint-Saëns – Phryné
 * Giuseppe Verdi – Falstaff

Musical theater

 * A Gaiety Girl – London production opened at the Prince of Wales Theatre on October 14 and ran for 413 performances
 * Jane Annie – London production opened at the Savoy Theatre on May 13 and ran for 50 performances
 * Little Christopher Columbus – London production opened at the Lyric Theatre on October 10 and ran for 279 performances
 * Morocco Bound (music Frank Osmond Carr lyrics: Adrian Ross) – London production opened at the Shaftesbury Theatre on April 13 and transferred to the Trafalgar Square Theatre on January 8, 1894, for a total run of 295 performances.
 * A Trip To Chinatown – Broadway production
 * Utopia Limited – London production opened at the Savoy Theatre on October 7 and ran for 245 performances

Births

 * February 10 – Jimmy Durante, American comedian and singer (died 1980)
 * February 15 – Walter Donaldson, American songwriter (died 1947)
 * February 21 – Andrés Segovia, Spanish classical guitarist (died 1987)
 * March 8 – Mississippi John Hurt, American country blues singer and guitarist (died 1966)
 * April 2 – Sergei Protopopov, Russian composer and music theorist (died 1954)
 * April 16 – Federico Mompou, Spanish classical composer (died 1987)
 * April 18 – Georges Boulanger, Romanian violinist (died 1958)
 * June 10 – Hattie McDaniel, singer and actress (died 1952)
 * June 26 – Big Bill Broonzy, American blues singer, songwriter and guitarist (died 1958)
 * June 28 – Luciano Gallet, Brazilian composer, pianist and conductor (died 1931)
 * July 25 – Dorothy Dickson, American-born British singer and actress (died 1995)
 * July 28 – Rued Langgaard, Danish composer and organist (died 1952)
 * August 21 – Lili Boulanger, French composer (died 1918)
 * August 22 – Dorothy Parker, American writer, poet and lyricist (died 1953)
 * September 13 – Larry Shields, American Dixieland jazz clarinetist (died 1953)
 * September 24 – Blind Lemon Jefferson, blues musician (died 1929)
 * October 1 – Cliff Friend, American Tin Pan Alley songwriter (died 1974)
 * October 23 – Jean Absil, Belgian composer and organist (died 1974)
 * November 8 – Clarence Williams, American jazz pianist, composer, promoter, vocalist, theatrical producer and publisher (died 1965)
 * December 7 — Fay Bainter, American actress (d. 1968)
 * December 24 – Harry Warren, born Salvatore Antonio Guaragna, American film songwriter (died 1981)

Deaths

 * January 18 – Julius Eichberg, composer (b. 1824)
 * February 13 – George Lichtenstein, pianist and music teacher (b. 1827)
 * May 2 – Daniel Friedrich Eduard Wilsing, composer (b.1809)
 * May 25 – Johann Rufinatscha, composer and music teacher (b. 1812)
 * June 10 – Elek Erkel, Hungarian composer, son of Ferenc Erkel (b. 1843)
 * June 25 – Ferenc Erkel, Hungarian composer (b. 1810)
 * July 16 – Antonio Ghislanzoni, librettist (b. 1824)
 * August 7 – Alfredo Catalani, composer (b. 1854)
 * August 31 – Sir William Cusins, instrumentalist, conductor and composer; Master of the Queen's Music (b. 1833)
 * September 8 – Michel Lentz, lyricist of the national anthem of Luxembourg (b. 1820)
 * September 13 – Carl Ludvig Gerlach, opera singer and composer
 * October 16 – Carlo Pedrotti, conductor and composer (b. 1817)
 * October 18 – Charles Gounod, composer (b. 1818)
 * November 6 – Pyotr Ilyich Tchaikovsky, composer (b. 1840)
 * December 23 – Benedict Randhartinger, composer (b. 1802)
 * date unknown – Félix Battanchon, cellist (b. 1814)

title: 1893 in music


text:
Talk:Metropolitan statistical area

Changing the 2020 census numbers
The problem with never changing the 2020 census population on the chart is that some of the MSA definitions have changed since 2020, and comparing the 2023 estimates for the current definition of an MSA to the 2020 census report for a previous definition of that MSA may be grossly misleading about any changes in population. Donald Albury 21:08, 16 March 2024 (UTC)
 * When the new 2023 delineations were released, I manually went through each one of them and recalculated the MSAs given the new boundaries. Since MSAs are defined by county borders, I just used the source from the OMB which listed each county of the new boundaries and updated the 2020 Census figures to reflect those. The 2020 Census column in this chart is indeed the population figures given the new boundaries. Coulraphobic123 (talk) 01:12, 17 March 2024 (UTC)
 * Even worse, I see many pages with 2010 and even 2000 data. These should be removed unless there is no more recent data, right? Keystone18 (talk) 15:37, 25 April 2024 (UTC)
 * I would say the choices are: 1) update them; 2) place an Update or Update inline tag; 3) leave them alone because they are correct in their context (assuming they show the year the data is from). Stefen Towers among the rest!   Gab • Gruntwerk 15:05, 26 April 2024 (UTC)

The navbox
Minor question, but in U.S. city population tables, should the numbers include Puerto Rico? Kk.urban (talk) 22:45, 8 April 2024 (UTC)

 * As Puerto Rico is a part of the United States, that would seem to make sense. Stefen Towers among the rest!   Gab • Gruntwerk 23:45, 8 April 2024 (UTC)
 * The Census Bureau does separate statistical areas into two categories, the United States and Puerto Rico, and so do our lists. However, it makes sense to include Puerto Rico, because it's on our pages. Kk.urban (talk) 00:24, 9 April 2024 (UTC)

Question
Can the 2023 estimates be used on respective pages, or are the 2020 full census numbers what should be used? Keystone18 (talk) 15:35, 25 April 2024 (UTC)

 * You can add estimates in addition to the 2020 count, but the 2020 count is the primary number used. Criticalthinker (talk) 17:44, 26 April 2024 (UTC)

 * And preferably official estimate figures from the US Census Bureau, though if other estimates are used then the sources should be cited. Coulraphobic123 (talk) 02:43, 27 April 2024 (UTC)

 * Thank you! Keystone18 (talk) 21:20, 11 June 2024 (UTC)

Boise City is in Oklahoma
The link for Boise, ID says Boise City, ID. Boise city is in Oklahoma. The link correctly points to Boise, ID. The tag should probably be adjusted to the correct city so that there is no confusion. 64.89.105.195 (talk) 19:40, 1 May 2024 (UTC)

 * The name of the MSA, as defined by the Office of Management and Budget and reported by the Census Bureau, is "Boise City-Nampa, ID", and of the CSA, "Boise City-Mountain Home-Ontario, ID-OR". The entry in the list is correct. Donald Albury 20:56, 1 May 2024 (UTC)

New Map
The 2023 metropolitan and micropolitan statistical area map has now been published (https://www.census.gov/geographies/reference-maps/2023/geo/cbsa.html). Could you please update the map on the article? 141.156.171.41 (talk) 22:02, 1 July 2024 (UTC)

title: Talk:Metropolitan statistical area


text:
Vyatka

Vyatka may refer to:
 * Vyatka Land
 * Vyatka, former name of the city of Kirov, Kirov Oblast, Russia
 * Vyatka (river), a river in Russia
 * Vyatka Region, an informal name of Kirov Oblast of Russia
 * Vyatka (motor scooter), a Russian copy of Italy's Vespa Motor Scooter
 * Vyatka (horse), a sturdy breed of workhorse from the Vyatka Region

title: Vyatka