30 (thirty) is the natural number following 29 and preceding 31.

In mathematics[link]

30 is the sum of the first four squares, which makes it a square pyramidal number.

It is a primorial and is the smallest Giuga number.

30 is the smallest sphenic number, and the smallest of the form Failed to parse (Missing texvc executable; please see math/README to configure.): 2\cdot 3\cdot r

where r is a prime greater than 3. 30 has an aliquot sum of 42; the second sphenic number and all sphenic numbers of this form have an aliquot sum 12 greater than themselves. The aliquot sequence of 30 is 16 members long, it comprises (30,42,54,66,78,90,144,259,45,33,15,9,4,3,1,0)

Thirty has but one number^{[citation needed]} for which it is the aliquot sum: the square number 841.

Adding up some subsets of its divisors (e.g., 5, 10 and 15) gives 30, hence 30 is a semiperfect number.

30 is the largest number such that all coprimes smaller than itself, except for 1, are prime.^[1]

A polygon with thirty sides is called a tricontagon.

The icosahedron and the dodecahedron are Platonic solids with 30 edges. The icosidodecahedron is an Archimedean solid with 30 vertices, and the Tutte–Coxeter graph is a symmetric graph with 30 vertices.

E₈ has Coxeter number 30.

30 is a Harshad number.

Since any group G such that |G| = pⁿm, where p does not divide m, has a subgroup of order pⁿ, and 30 is the only number less than 60 that is not either a prime or of the above form, it is the only candidate for the order of a simple group less than 60 that one needs other methods to reject.

Look up thirty in Wiktionary, the free dictionary.

In science[link]

• The atomic number of zinc is 30

Astronomy[link]

• Messier object M30, a magnitude 8.5 globular cluster in the constellation Capricornus
• The New General Catalogue object NGC 30, a double star in the constellation Pegasus
• The Saros number of the solar eclipse series which began on October 2, 2069 BC and ended on March 18, 572 BC. The duration of Saros series 30 was 1496.5 years, and it contained 84 solar eclipses. Further, the Saros number of the lunar eclipse series which began on June 19, 1803 BC and ended on August 18, 487 BC. The duration of Saros series 30 was 1316.2 years, and it contained 74 lunar eclipses.

In other fields[link]

Thirty is:

• Used (as –30–) to indicate the end of a newspaper (or broadcast) story, a copy editor's typographical notation.
• The number of days in the months April, June, September and November (and in unusual circumstances February—see February 30)
• The total number of major and minor keys in Western tonal music, including enharmonic equivalents
• The minimum age for United States senators
• In years of marriage, the pearl wedding anniversary
• The duration in years of the Thirty Years' War - 1618 to 1648.
• The code for international direct dial phone calls to Greece
• The house number of 30 St Mary Axe (The Gherkin)
• The designation of Interstate 30, a freeway that runs from Texas to Arkansas
• The designation of U.S. Route 30, a highway that runs from Oregon to New Jersey
• Various other routes have been numbered "30"; for example, New York State Route 30 which runs from the Pennsylvania border to the Canadian border
• The designation of E30, the European route from Cork to Samara
• The number of tracks on The Beatles' eponymous album, usually known as The White Album
• A stage in young adulthood
• Part of the name of:
  • 30 Odd Foot of Grunts, the band fronted by actor Russell Crowe
  • The movie title 13 Going on 30, starring Jennifer Garner
  • The title of the Food Network show 30 Minute Meals
  • 30 Days of Night, a comic book miniseries and film.
• Judas Iscariot betrayed Jesus for 30 pieces of silver. Matthew 26:15.
• The number of the French department Gard
• 30 was the route number of the bus blown up by terrorists in Tavistock Square during the 7 July 2005 London bombings^[2]

Historical years[link]

30 A.D., 30 B.C., 1930, 2030, etc.

History and literature[link]

• At age 30 (according to most biblical scholars) Jesus of Nazareth was baptized by John the Baptist, at the beginning of his public ministry of teaching and healing.
• One of the rallying-cries of the 1960s student/youth protest movement was the slogan, 'Don't trust anyone over thirty'.
• In 'The Myth of Sisyphus' the French existentialist Albert Camus comments that the age of thirty is a crucial period in the life of a man, for at that age he gains a new awareness of the meaning of time.
• In Franz Kafka's novel 'The Trial' Joseph wakes up on the morning of his thirtieth birthday to find himself under arrest for an unspecified crime. After making many futile attempts to find the nature of the crime or the name of his accuser, Joseph dies on the eve of his thirty-first birthday.
• The number of uprights that formed the Sarsen Circle at Stonehenge, also the supposed number of holes forming the arrays of Y and Z Holes at Stonehenge.
• Western Christianity's most prolific 20th century essayist, F. W. Boreham in 'Life at Thirty' ('Cliffs of Opal') mentions that in addition to Jesus commencing ministry at 30 (Luke 3:23), Joseph was 30 when he stood before Pharaoh, King of Egypt (Genesis 41:46), King David was 30 when he began to reign (2 Samuel 5:4), and the Levites were numbered from the age of 30 and upward (1 Chronicles 23:3). Also in that essay Boreham writes 'It was said of [the English poet] Keats, that "he ensphered himself in thirty perfect years and died, not young".'

Sports[link]

• In tennis, the number 30 represents the second point gained in a game.
• As of 2012, three of the four traditional major leagues in English-speaking North America (Major League Baseball, the NBA, and the NHL) have 30 teams each.
• The California Angels (now Los Angeles Angels of Anaheim) baseball team retired the number in honor of its most notable wearer, Nolan Ryan, on June 16, 1992. (Ryan is the only major league player to have his number retired by three different teams, though the other two teams retired number 34 for him rather than 30).^[3] The San Francisco Giants extended the same honor to Orlando Cepeda. While on the Chicago White Sox, Nick Swisher wore the number 30 and was known as "The Dirty 30".
• 30 is one of the more common numbers worn by hockey goaltenders (along with 31).
• The number of New Jersey Devils goalie, Martin Brodeur
• Lewis Hamilton became the 30th Formula One World Champion on November 2, 2008.

Music[link]

• The number of variations in Bach's Goldberg Variations
• 30, album from 2001 by Harry Connick, Jr.
• 30 Seconds to Mars
• 30 Minutes, a song by t.A.T.u

References[link]

1. ^ Michael Slone, Every positive integer greater than 30 has at least one composite totative from PlanetMath. Accessed 24 April 2007
2. ^ Report of the 7 July Review Committee Accessed 18 Dec 2011
3. ^ Texas Rangers Retired Number History on mlb.com Retrieved May 18, 2006. Note however that Jackie Robinson's number 42 was retired by Major League Baseball for all teams.

A number is a mathematical object used to count and measure. In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers.

Mathematical operations are certain procedures that take one or more numbers as input and produce a number as output. Unary operations take a single input number and produce a single output number. For example, the successor operation adds one to an integer, thus the successor of 4 is 5. Binary operations take two input numbers and produce a single output number. Examples of binary operations include addition, subtraction, multiplication, division, and exponentiation. The study of numerical operations is called arithmetic.

A notational symbol that represents a number is called a numeral. In addition to their use in counting and measuring, numerals are often used for labels (telephone numbers), for ordering (serial numbers), and for codes (e.g., ISBNs).

In common use, the word number can mean the abstract object, the symbol, or the word for the number.

Classification of numbers[link]

Different types of numbers are used in many cases. Numbers can be classified into sets, called number systems. (For different methods of expressing numbers with symbols, such as the Roman numerals, see numeral systems.)

Natural numbers[link]

The most familiar numbers are the natural numbers or counting numbers: one, two, three, and so on. Traditionally, the sequence of natural numbers started with 1 (0 was not even considered a number for the Ancient Greeks.) However, in the 19th century, set theorists and other mathematicians started including 0 (cardinality of the empty set, i.e. 0 elements, where 0 is thus the smallest cardinal number) in the set of natural numbers.^{[citation needed]} Today, different mathematicians use the term to describe both sets, including zero or not. The mathematical symbol for the set of all natural numbers is N, also written Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{N} .

In the base ten numeral system, in almost universal use today for mathematical operations, the symbols for natural numbers are written using ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. In this base ten system, the rightmost digit of a natural number has a place value of one, and every other digit has a place value ten times that of the place value of the digit to its right.

In set theory, which is capable of acting as an axiomatic foundation for modern mathematics,^[1] natural numbers can be represented by classes of equivalent sets. For instance, the number 3 can be represented as the class of all sets that have exactly three elements. Alternatively, in Peano Arithmetic, the number 3 is represented as sss0, where s is the "successor" function (i.e., 3 is the third successor of 0). Many different representations are possible; all that is needed to formally represent 3 is to inscribe a certain symbol or pattern of symbols three times.

Integers[link]

The negative of a positive integer is defined as a number that produces zero when it is added to the corresponding positive integer. Negative numbers are usually written with a negative sign (a minus sign). As an example, the negative of 7 is written −7, and 7 + (−7) = 0. When the set of negative numbers is combined with the set of natural numbers (which includes zero), the result is defined as the set of integer numbers, also called integers, Z also written Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{Z} . Here the letter Z comes from German Zahl, meaning "number".

The set of integers forms a ring with operations addition and multiplication.^[2]

Rational numbers[link]

A rational number is a number that can be expressed as a fraction with an integer numerator and a non-zero natural number denominator. Fractions are written as two numbers, the numerator and the denominator, with a dividing bar between them. In the fraction written ^m⁄_n or

Failed to parse (Missing texvc executable; please see math/README to configure.): m \over n \,

m represents equal parts, where n equal parts of that size make up one whole. Two different fractions may correspond to the same rational number; for example ¹⁄₂ and ²⁄₄ are equal, that is:

Failed to parse (Missing texvc executable; please see math/README to configure.): {1 \over 2} = {2 \over 4}.\,

If the absolute value of m is greater than n, then the absolute value of the fraction is greater than 1. Fractions can be greater than, less than, or equal to 1 and can also be positive, negative, or zero. The set of all rational numbers includes the integers, since every integer can be written as a fraction with denominator 1. For example −7 can be written ⁻⁷⁄₁. The symbol for the rational numbers is Q (for quotient), also written Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{Q} .

Real numbers[link]

The real numbers include all of the measuring numbers. Real numbers are usually written using decimal numerals, in which a decimal point is placed to the right of the digit with place value one. Each digit to the right of the decimal point has a place value one-tenth of the place value of the digit to its left. Thus

Failed to parse (Missing texvc executable; please see math/README to configure.): 123.456\,

represents 1 hundred, 2 tens, 3 ones, 4 tenths, 5 hundredths, and 6 thousandths. In saying the number, the decimal is read "point", thus: "one two three point four five six ". In the US and UK and a number of other countries, the decimal point is represented by a period, whereas in continental Europe and certain other countries the decimal point is represented by a comma. Zero is often written as 0.0 when it must be treated as a real number rather than an integer. In the US and UK a number between −1 and 1 is always written with a leading zero to emphasize the decimal. Negative real numbers are written with a preceding minus sign:

Failed to parse (Missing texvc executable; please see math/README to configure.): -123.456.\,

Every rational number is also a real number. It is not the case, however, that every real number is rational. If a real number cannot be written as a fraction of two integers, it is called irrational. A decimal that can be written as a fraction either ends (terminates) or forever repeats, because it is the answer to a problem in division. Thus the real number 0.5 can be written as ¹⁄₂ and the real number 0.333... (forever repeating threes, otherwise written 0.3) can be written as ¹⁄₃. On the other hand, the real number π (pi), the ratio of the circumference of any circle to its diameter, is

Failed to parse (Missing texvc executable; please see math/README to configure.): \pi = 3.14159265358979\dots.\,

Since the decimal neither ends nor forever repeats, it cannot be written as a fraction, and is an example of an irrational number. Other irrational numbers include

Failed to parse (Missing texvc executable; please see math/README to configure.): \sqrt{2} = 1.41421356237 \dots\,

(the square root of 2, that is, the positive number whose square is 2).

Thus 1.0 and 0.999... are two different decimal numerals representing the natural number 1. There are infinitely many other ways of representing the number 1, for example ²⁄₂, ³⁄₃, 1.00, 1.000, and so on.

Every real number is either rational or irrational. Every real number corresponds to a point on the number line. The real numbers also have an important but highly technical property called the least upper bound property. The symbol for the real numbers is R, also written as Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{R} .

When a real number represents a measurement, there is always a margin of error. This is often indicated by rounding or truncating a decimal, so that digits that suggest a greater accuracy than the measurement itself are removed. The remaining digits are called significant digits. For example, measurements with a ruler can seldom be made without a margin of error of at least 0.001 meters. If the sides of a rectangle are measured as 1.23 meters and 4.56 meters, then multiplication gives an area for the rectangle of 5.6088 square meters. Since only the first two digits after the decimal place are significant, this is usually rounded to 5.61.

In abstract algebra, it can be shown that any complete ordered field is isomorphic to the real numbers. The real numbers are not, however, an algebraically closed field.

Complex numbers[link]

Moving to a greater level of abstraction, the real numbers can be extended to the complex numbers. This set of numbers arose, historically, from trying to find closed formulas for the roots of cubic and quartic polynomials. This led to expressions involving the square roots of negative numbers, and eventually to the definition of a new number: the square root of negative one, denoted by i, a symbol assigned by Leonhard Euler, and called the imaginary unit. The complex numbers consist of all numbers of the form

Failed to parse (Missing texvc executable; please see math/README to configure.): \,a + b i

where a and b are real numbers. In the expression a + bi, the real number a is called the real part and b is called the imaginary part. If the real part of a complex number is zero, then the number is called an imaginary number or is referred to as purely imaginary; if the imaginary part is zero, then the number is a real number. Thus the real numbers are a subset of the complex numbers. If the real and imaginary parts of a complex number are both integers, then the number is called a Gaussian integer. The symbol for the complex numbers is C or Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{C} .

In abstract algebra, the complex numbers are an example of an algebraically closed field, meaning that every polynomial with complex coefficients can be factored into linear factors. Like the real number system, the complex number system is a field and is complete, but unlike the real numbers it is not ordered. That is, there is no meaning in saying that i is greater than 1, nor is there any meaning in saying that i is less than 1. In technical terms, the complex numbers lack the trichotomy property.

Complex numbers correspond to points on the complex plane, sometimes called the Argand plane.

Each of the number systems mentioned above is a proper subset of the next number system. Symbolically, Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C} .

Computable numbers[link]

Moving to problems of computation, the computable numbers are determined in the set of the real numbers. The computable numbers, also known as the recursive numbers or the computable reals, are the real numbers that can be computed to within any desired precision by a finite, terminating algorithm. Equivalent definitions can be given using μ-recursive functions, Turing machines or λ-calculus as the formal representation of algorithms. The computable numbers form a real closed field and can be used in the place of real numbers for many, but not all, mathematical purposes.

Other types[link]

Algebraic numbers are those that can be expressed as the solution to a polynomial equation with integer coefficients. The complement of the algebraic numbers are the transcendental numbers.

Hyperreal numbers are used in non-standard analysis. The hyperreals, or nonstandard reals (usually denoted as *R), denote an ordered field that is a proper extension of the ordered field of real numbers R and satisfies the transfer principle. This principle allows true first order statements about R to be reinterpreted as true first order statements about *R.

Superreal and surreal numbers extend the real numbers by adding infinitesimally small numbers and infinitely large numbers, but still form fields.

The p-adic numbers may have infinitely long expansions to the left of the decimal point, in the same way that real numbers may have infinitely long expansions to the right. The number system that results depends on what base is used for the digits: any base is possible, but a prime number base provides the best mathematical properties.

For dealing with infinite collections, the natural numbers have been generalized to the ordinal numbers and to the cardinal numbers. The former gives the ordering of the collection, while the latter gives its size. For the finite set, the ordinal and cardinal numbers are equivalent, but they differ in the infinite case.

A relation number is defined as the class of relations consisting of all those relations that are similar to one member of the class.^[3]

Sets of numbers that are not subsets of the complex numbers are sometimes called hypercomplex numbers. They include the quaternions H, invented by Sir William Rowan Hamilton, in which multiplication is not commutative, and the octonions, in which multiplication is not associative. Elements of function fields of non-zero characteristic behave in some ways like numbers and are often regarded as numbers by number theorists.

Specific uses[link]

There are also other sets of numbers with specialized uses. Some are subsets of the complex numbers. For example, algebraic numbers are the roots of polynomials with rational coefficients. Complex numbers that are not algebraic are called transcendental numbers.

An even number is an integer that is "evenly divisible" by 2, i.e., divisible by 2 without remainder; an odd number is an integer that is not evenly divisible by 2. (The old-fashioned term "evenly divisible" is now almost always shortened to "divisible".) A formal definition of an odd number is that it is an integer of the form n = 2k + 1, where k is an integer. An even number has the form n = 2k where k is an integer.

A perfect number is a positive integer that is the sum of its proper positive divisors—the sum of the positive divisors not including the number itself. Equivalently, a perfect number is a number that is half the sum of all of its positive divisors, or σ(n) = 2 n. The first perfect number is 6, because 1, 2, and 3 are its proper positive divisors and 1 + 2 + 3 = 6. The next perfect number is 28 = 1 + 2 + 4 + 7 + 14. The next perfect numbers are 496 and 8128 (sequence A000396 in OEIS). These first four perfect numbers were the only ones known to early Greek mathematics.

A figurate number is a number that can be represented as a regular and discrete geometric pattern (e.g. dots). If the pattern is polytopic, the figurate is labeled a polytopic number, and may be a polygonal number or a polyhedral number. Polytopic numbers for r = 2, 3, and 4 are:

• P₂(n) = ¹⁄₂ n(n + 1) (triangular numbers)
• P₃(n) = ¹⁄₆ n(n + 1)(n + 2) (tetrahedral numbers)
• P₄(n) = ¹⁄₂₄ n(n + 1)(n + 2)(n + 3) (pentatopic numbers)

Numerals[link]

Numbers should be distinguished from numerals, the symbols used to represent numbers. Boyer showed that Egyptians created the first ciphered numeral system.^{[citation needed]} Greeks followed by mapping their counting numbers onto Ionian and Doric alphabets. The number five can be represented by both the base ten numeral '5', by the Roman numeral 'Ⅴ' and ciphered letters. Notations used to represent numbers are discussed in the article numeral systems. An important development in the history of numerals was the development of a positional system, like modern decimals, which can represent very large numbers. The Roman numerals require extra symbols for larger numbers.

History[link]

First use of numbers[link]

Bones and other artifacts have been discovered with marks cut into them that many believe are tally marks.^[4] These tally marks may have been used for counting elapsed time, such as numbers of days, lunar cycles or keeping records of quantities, such as of animals.

A tallying system has no concept of place value (as in modern decimal notation), which limits its representation of large numbers. Nonetheless tallying systems are considered the first kind of abstract numeral system.

The first known system with place value was the Mesopotamian base 60 system (ca. 3400 BC) and the earliest known base 10 system dates to 3100 BC in Egypt.^[5]

[edit] Zero

The use of zero as a number should be distinguished from its use as a placeholder numeral in place-value systems. Many ancient texts used zero. Babylonian and Egyptian texts used it. Egyptians used the word nfr to denote zero balance in double entry accounting entries. Indian texts used a Sanskrit word Shunye to refer to the concept of void. In mathematics texts this word often refers to the number zero.^[6]

Records show that the Ancient Greeks seemed unsure about the status of zero as a number: they asked themselves "how can 'nothing' be something?" leading to interesting philosophical and, by the Medieval period, religious arguments about the nature and existence of zero and the vacuum. The paradoxes of Zeno of Elea depend in large part on the uncertain interpretation of zero. (The ancient Greeks even questioned whether 1 was a number.)

The late Olmec people of south-central Mexico began to use a true zero (a shell glyph) in the New World possibly by the 4th century BC but certainly by 40 BC, which became an integral part of Maya numerals and the Maya calendar. Mayan arithmetic used base 4 and base 5 written as base 20. Sanchez in 1961 reported a base 4, base 5 'finger' abacus.

By 130 AD, Ptolemy, influenced by Hipparchus and the Babylonians, was using a symbol for zero (a small circle with a long overbar) within a sexagesimal numeral system otherwise using alphabetic Greek numerals. Because it was used alone, not as just a placeholder, this Hellenistic zero was the first documented use of a true zero in the Old World. In later Byzantine manuscripts of his Syntaxis Mathematica (Almagest), the Hellenistic zero had morphed into the Greek letter omicron (otherwise meaning 70).

Another true zero was used in tables alongside Roman numerals by 525 (first known use by Dionysius Exiguus), but as a word, nulla meaning nothing, not as a symbol. When division produced zero as a remainder, nihil, also meaning nothing, was used. These medieval zeros were used by all future medieval computists (calculators of Easter). An isolated use of their initial, N, was used in a table of Roman numerals by Bede or a colleague about 725, a true zero symbol.

An early documented use of the zero by Brahmagupta (in the Brahmasphutasiddhanta) dates to 628. He treated zero as a number and discussed operations involving it, including division. By this time (the 7th century) the concept had clearly reached Cambodia as Khmer numerals, and documentation shows the idea later spreading to China and the Islamic world.

[edit] Negative numbers

The abstract concept of negative numbers was recognised as early as 100 BC – 50 BC. The Chinese Nine Chapters on the Mathematical Art (Chinese: Jiu-zhang Suanshu) contains methods for finding the areas of figures; red rods were used to denote positive coefficients, black for negative.^[7] This is the earliest known mention of negative numbers in the East; the first reference in a Western work was in the 3rd century in Greece. Diophantus referred to the equation equivalent to Failed to parse (Missing texvc executable; please see math/README to configure.): 4x + 20 = 0

(the solution is negative) in Arithmetica, saying that the equation gave an absurd result.

During the 600s, negative numbers were in use in India to represent debts. Diophantus’ previous reference was discussed more explicitly by Indian mathematician Brahmagupta, in Brahma-Sphuta-Siddhanta 628, who used negative numbers to produce the general form quadratic formula that remains in use today. However, in the 12th century in India, Bhaskara gives negative roots for quadratic equations but says the negative value "is in this case not to be taken, for it is inadequate; people do not approve of negative roots."

European mathematicians, for the most part, resisted the concept of negative numbers until the 17th century, although Fibonacci allowed negative solutions in financial problems where they could be interpreted as debts (chapter 13 of Liber Abaci, 1202) and later as losses (in Flos). At the same time, the Chinese were indicating negative numbers either by drawing a diagonal stroke through the right-most nonzero digit of the corresponding positive number's numeral.^[8] The first use of negative numbers in a European work was by Chuquet during the 15th century. He used them as exponents, but referred to them as “absurd numbers”.

As recently as the 18th century, it was common practice to ignore any negative results returned by equations on the assumption that they were meaningless, just as René Descartes did with negative solutions in a Cartesian coordinate system.

[edit] Rational numbers

It is likely that the concept of fractional numbers dates to prehistoric times. The Ancient Egyptians used their Egyptian fraction notation for rational numbers in mathematical texts such as the Rhind Mathematical Papyrus and the Kahun Papyrus. Classical Greek and Indian mathematicians made studies of the theory of rational numbers, as part of the general study of number theory. The best known of these is Euclid's Elements, dating to roughly 300 BC. Of the Indian texts, the most relevant is the Sthananga Sutra, which also covers number theory as part of a general study of mathematics.

The concept of decimal fractions is closely linked with decimal place-value notation; the two seem to have developed in tandem. For example, it is common for the Jain math sutras to include calculations of decimal-fraction approximations to pi or the square root of two. Similarly, Babylonian math texts had always used sexagesimal (base 60) fractions with great frequency.

[edit] Irrational numbers

The earliest known use of irrational numbers was in the Indian Sulba Sutras composed between 800–500 BC.^[9] The first existence proofs of irrational numbers is usually attributed to Pythagoras, more specifically to the Pythagorean Hippasus of Metapontum, who produced a (most likely geometrical) proof of the irrationality of the square root of 2. The story goes that Hippasus discovered irrational numbers when trying to represent the square root of 2 as a fraction. However Pythagoras believed in the absoluteness of numbers, and could not accept the existence of irrational numbers. He could not disprove their existence through logic, but he could not accept irrational numbers, so he sentenced Hippasus to death by drowning.

The sixteenth century brought final European acceptance of negative integral and fractional numbers. By the seventeenth century, mathematicians generally used decimal fractions with modern notation. It was not, however, until the nineteenth century that mathematicians separated irrationals into algebraic and transcendental parts, and once more undertook scientific study of irrationals. It had remained almost dormant since Euclid. 1872 brought publication of the theories of Karl Weierstrass (by his pupil Kossak), Heine (Crelle, 74), Georg Cantor (Annalen, 5), and Richard Dedekind. In 1869, Méray had taken the same point of departure as Heine, but the theory is generally referred to the year 1872. Weierstrass's method was completely set forth by Salvatore Pincherle (1880), and Dedekind's has received additional prominence through the author's later work (1888) and endorsement by Paul Tannery (1894). Weierstrass, Cantor, and Heine base their theories on infinite series, while Dedekind founds his on the idea of a cut (Schnitt) in the system of real numbers, separating all rational numbers into two groups having certain characteristic properties. The subject has received later contributions at the hands of Weierstrass, Kronecker (Crelle, 101), and Méray.

Continued fractions, closely related to irrational numbers (and due to Cataldi, 1613), received attention at the hands of Euler, and at the opening of the nineteenth century were brought into prominence through the writings of Joseph Louis Lagrange. Other noteworthy contributions have been made by Druckenmüller (1837), Kunze (1857), Lemke (1870), and Günther (1872). Ramus (1855) first connected the subject with determinants, resulting, with the subsequent contributions of Heine, Möbius, and Günther, in the theory of Kettenbruchdeterminanten. Dirichlet also added to the general theory, as have numerous contributors to the applications of the subject.

[edit] Transcendental numbers and reals

The first results concerning transcendental numbers were Lambert's 1761 proof that π cannot be rational, and also that eⁿ is irrational if n is rational (unless n = 0). (The constant e was first referred to in Napier's 1618 work on logarithms.) Legendre extended this proof to show that π is not the square root of a rational number. The search for roots of quintic and higher degree equations was an important development, the Abel–Ruffini theorem (Ruffini 1799, Abel 1824) showed that they could not be solved by radicals (formula involving only arithmetical operations and roots). Hence it was necessary to consider the wider set of algebraic numbers (all solutions to polynomial equations). Galois (1832) linked polynomial equations to group theory giving rise to the field of Galois theory.

The existence of transcendental numbers^[10] was first established by Liouville (1844, 1851). Hermite proved in 1873 that e is transcendental and Lindemann proved in 1882 that π is transcendental. Finally Cantor shows that the set of all real numbers is uncountably infinite but the set of all algebraic numbers is countably infinite, so there is an uncountably infinite number of transcendental numbers.

[edit] Infinity and infinitesimals

The earliest known conception of mathematical infinity appears in the Yajur Veda, an ancient Indian script, which at one point states, "If you remove a part from infinity or add a part to infinity, still what remains is infinity." Infinity was a popular topic of philosophical study among the Jain mathematicians c. 400 BC. They distinguished between five types of infinity: infinite in one and two directions, infinite in area, infinite everywhere, and infinite perpetually.

Aristotle defined the traditional Western notion of mathematical infinity. He distinguished between actual infinity and potential infinity—the general consensus being that only the latter had true value. Galileo's Two New Sciences discussed the idea of one-to-one correspondences between infinite sets. But the next major advance in the theory was made by Georg Cantor; in 1895 he published a book about his new set theory, introducing, among other things, transfinite numbers and formulating the continuum hypothesis. This was the first mathematical model that represented infinity by numbers and gave rules for operating with these infinite numbers.

In the 1960s, Abraham Robinson showed how infinitely large and infinitesimal numbers can be rigorously defined and used to develop the field of nonstandard analysis. The system of hyperreal numbers represents a rigorous method of treating the ideas about infinite and infinitesimal numbers that had been used casually by mathematicians, scientists, and engineers ever since the invention of infinitesimal calculus by Newton and Leibniz.

A modern geometrical version of infinity is given by projective geometry, which introduces "ideal points at infinity," one for each spatial direction. Each family of parallel lines in a given direction is postulated to converge to the corresponding ideal point. This is closely related to the idea of vanishing points in perspective drawing.

[edit] Complex numbers

The earliest fleeting reference to square roots of negative numbers occurred in the work of the mathematician and inventor Heron of Alexandria in the 1st century AD, when he considered the volume of an impossible frustum of a pyramid. They became more prominent when in the 16th century closed formulas for the roots of third and fourth degree polynomials were discovered by Italian mathematicians such as Niccolo Fontana Tartaglia and Gerolamo Cardano. It was soon realized that these formulas, even if one was only interested in real solutions, sometimes required the manipulation of square roots of negative numbers.

This was doubly unsettling since they did not even consider negative numbers to be on firm ground at the time. When René Descartes coined the term "imaginary" for these quantities in 1637, he intended it as derogatory. (See imaginary number for a discussion of the "reality" of complex numbers.) A further source of confusion was that the equation

Failed to parse (Missing texvc executable; please see math/README to configure.): \left ( \sqrt{-1}\right )^2 =\sqrt{-1}\sqrt{-1}=-1

seemed capriciously inconsistent with the algebraic identity

Failed to parse (Missing texvc executable; please see math/README to configure.): \sqrt{a}\sqrt{b}=\sqrt{ab},

which is valid for positive real numbers a and b, and was also used in complex number calculations with one of a, b positive and the other negative. The incorrect use of this identity, and the related identity

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{1}{\sqrt{a}}=\sqrt{\frac{1}{a}}

in the case when both a and b are negative even bedeviled Euler. This difficulty eventually led him to the convention of using the special symbol i in place of Failed to parse (Missing texvc executable; please see math/README to configure.): \sqrt{-1}

to guard against this mistake.

The 18th century saw the work of Abraham de Moivre and Leonhard Euler. de Moivre's formula (1730) states:

Failed to parse (Missing texvc executable; please see math/README to configure.): (\cos \theta + i\sin \theta)^{n} = \cos n \theta + i\sin n \theta \,

and to Euler (1748) Euler's formula of complex analysis:

Failed to parse (Missing texvc executable; please see math/README to configure.): \cos \theta + i\sin \theta = e ^{i\theta }. \,

The existence of complex numbers was not completely accepted until Caspar Wessel described the geometrical interpretation in 1799. Carl Friedrich Gauss rediscovered and popularized it several years later, and as a result the theory of complex numbers received a notable expansion. The idea of the graphic representation of complex numbers had appeared, however, as early as 1685, in Wallis's De Algebra tractatus.

Also in 1799, Gauss provided the first generally accepted proof of the fundamental theorem of algebra, showing that every polynomial over the complex numbers has a full set of solutions in that realm. The general acceptance of the theory of complex numbers is due to the labors of Augustin Louis Cauchy and Niels Henrik Abel, and especially the latter, who was the first to boldly use complex numbers with a success that is well known.

Gauss studied complex numbers of the form a + bi, where a and b are integral, or rational (and i is one of the two roots of x² + 1 = 0). His student, Gotthold Eisenstein, studied the type a + bω, where ω is a complex root of x³ − 1 = 0. Other such classes (called cyclotomic fields) of complex numbers derive from the roots of unity x^k − 1 = 0 for higher values of k. This generalization is largely due to Ernst Kummer, who also invented ideal numbers, which were expressed as geometrical entities by Felix Klein in 1893. The general theory of fields was created by Évariste Galois, who studied the fields generated by the roots of any polynomial equation F(x) = 0.

In 1850 Victor Alexandre Puiseux took the key step of distinguishing between poles and branch points, and introduced the concept of essential singular points. This eventually led to the concept of the extended complex plane.

[edit] Prime numbers

Prime numbers have been studied throughout recorded history. Euclid devoted one book of the Elements to the theory of primes; in it he proved the infinitude of the primes and the fundamental theorem of arithmetic, and presented the Euclidean algorithm for finding the greatest common divisor of two numbers.

In 240 BC, Eratosthenes used the Sieve of Eratosthenes to quickly isolate prime numbers. But most further development of the theory of primes in Europe dates to the Renaissance and later eras.

In 1796, Adrien-Marie Legendre conjectured the prime number theorem, describing the asymptotic distribution of primes. Other results concerning the distribution of the primes include Euler's proof that the sum of the reciprocals of the primes diverges, and the Goldbach conjecture, which claims that any sufficiently large even number is the sum of two primes. Yet another conjecture related to the distribution of prime numbers is the Riemann hypothesis, formulated by Bernhard Riemann in 1859. The prime number theorem was finally proved by Jacques Hadamard and Charles de la Vallée-Poussin in 1896. Goldbach and Riemann's conjectures remain unproven and unrefuted.

Word alternatives[link]

Some numbers traditionally have alternative words to express them, including the following:

• Nothing: 0
• Single: 1
• Pair, couple, brace: 2
• Trio: 3
• Half-Dozen: 6
• Decade: 10
• Dozen: 12
• Baker's dozen: 13
• Score: 20
• Half-century: 50
• Century: 100
• Gross: 144
• Ream: 480 (old measure) 500 (new measure)
• Millenium: 1000
• Great gross: 1728
• "n-figure", as in digit, generally for larger-number ranges, also written without a hyphen; often used in financial discussion. For example

• "five-figure": 10,000 to 99,999 (five digits); ten-thousands
• "six-figure": 100,000 to 999,999 (six digits); hundred-thousands
• "seven-figure": 1,000,000 to 9,999,999 (seven digits); millions

See also[link]

Wikimedia Commons has media related to: Numbers

Notes[link]

References[link]

• Tobias Dantzig, Number, the language of science; a critical survey written for the cultured non-mathematician, New York, The Macmillan company, 1930.
• Erich Friedman, What's special about this number?
• Steven Galovich, Introduction to Mathematical Structures, Harcourt Brace Javanovich, 23 January 1989, ISBN 0-15-543468-3.
• Paul Halmos, Naive Set Theory, Springer, 1974, ISBN 0-387-90092-6.
• Morris Kline, Mathematical Thought from Ancient to Modern Times, Oxford University Press, 1972.
• Alfred North Whitehead and Bertrand Russell, Principia Mathematica to *56, Cambridge University Press, 1910.
• George I. Sanchez, Arithmetic in Maya,Austin-Texas, 1961.

External links[link]

Look up number in Wiktionary, the free dictionary.

Wikiversity has learning materials about Primary mathematics:Numbers

• Nechaev, V.I. (2001), "Number", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4, http://www.encyclopediaofmath.org/index.php?title=Number 
• Mesopotamian and Germanic numbers
• BBC Radio 4, In Our Time: Negative Numbers
• '4000 Years of Numbers', lecture by Robin Wilson, 07/11/07, Gresham College (available for download as MP3 or MP4, and as a text file).
• http://planetmath.org/encyclopedia/MayanMath2.html
• "What's the World's Favorite Number?". 2011-06-22. http://www.npr.org/blogs/krulwich/2011/07/22/138493147/what-s-your-favorite-number-world-wide-survey-v1. Retrieved 2011-09-17. ; "Cuddling With 9, Smooching With 8, Winking At 7". 2011-08-11. http://www.npr.org/templates/transcript/transcript.php?storyId=139797360. Retrieved 2011-09-17. 

Calvin Coolidge
30th President of the United States
In office August 2, 1923 – March 4, 1929
Vice President	None (1923-1925) Charles G. Dawes (1925-1929)
Preceded by	Warren G. Harding
Succeeded by	Herbert Hoover
29th Vice President of the United States
In office March 4, 1921 – August 2, 1923
President	Warren G. Harding
Preceded by	Thomas R. Marshall
Succeeded by	Charles G. Dawes
48th Governor of Massachusetts
In office January 2, 1919 – January 6, 1921
Lieutenant	Channing Cox
Preceded by	Samuel W. McCall
Succeeded by	Channing H. Cox
46th Lieutenant Governor of Massachusetts
In office January 6, 1916 – January 2, 1919
Governor	Samuel W. McCall
Preceded by	Grafton D. Cushing
Succeeded by	Channing H. Cox
Personal details
Born	John Calvin Coolidge, Jr. (1872-07-04)July 4, 1872 Plymouth, Windsor County, Vermont, United States
Died	January 5, 1933(1933-01-05) (aged 60) Northampton, Hampshire County, Massachusetts, United States
Political party	Republican
Spouse(s)	Grace Goodhue
Children	John Calvin
Alma mater	Amherst College
Profession	Lawyer
Religion	Congregationalism
Signature

John Calvin Coolidge, Jr. (July 4, 1872 – January 5, 1933) was the 30th President of the United States (1923–1929). A Republican lawyer from Vermont, Coolidge worked his way up the ladder of Massachusetts state politics, eventually becoming governor of that state. His conduct during the Boston Police Strike of 1919 thrust him into the national spotlight and gave him a reputation as a man of decisive action. Soon after, he was elected as the 29th Vice President in 1920 and succeeded to the Presidency upon the sudden death of Warren G. Harding in 1923. Elected in his own right in 1924, he gained a reputation as a small-government conservative, and also as a man who said very little.

Coolidge restored public confidence in the White House after the scandals of his predecessor's administration, and left office with considerable popularity.^[1] As a Coolidge biographer put it, "He embodied the spirit and hopes of the middle class, could interpret their longings and express their opinions. That he did represent the genius of the average is the most convincing proof of his strength."^[2] Coolidge praised the achievement of widespread prosperity in 1928, saying: "The requirements of existence have passed beyond the standard of necessity into the region of luxury."^[3] Some later criticized Coolidge as part of a general criticism of laissez-faire government.^[4] His reputation underwent a renaissance during the Ronald Reagan Administration,^[5] but the ultimate assessment of his presidency is still divided between those who approve of his reduction of the size of government programs and those who believe the federal government should be more involved in regulating and controlling the economy.^[6]

Birth and family history[link]

Coolidge as an Amherst undergraduate

John Calvin Coolidge, Jr., was born in Plymouth Notch, Windsor County, Vermont, on July 4, 1872, the only U.S. President to be born on Independence Day. He was the elder of the two children of John Calvin Coolidge, Sr. (1845–1926) and Victoria Josephine Moor (1846–1885). Coolidge Senior engaged in many occupations, and ultimately enjoyed a statewide reputation as a prosperous farmer, storekeeper and public servant; he farmed, taught school, ran a local store, served in the Vermont House of Representatives and the Vermont Senate, and held various local offices including justice of the peace and tax collector.^[7] Coolidge's mother was the daughter of a Plymouth Notch farmer.^[8] Coolidge's chronically ill mother died, perhaps from tuberculosis, when he was twelve years old. His sister, Abigail Grace Coolidge (1875–1890), died at the age of fifteen, when Coolidge was eighteen. Coolidge's father remarried in 1891, to a schoolteacher, and lived to the age of eighty.

Coolidge's family had deep roots in New England. His earliest American ancestor, John Coolidge, emigrated from Cottenham, Cambridgeshire, England, around 1630 and settled in Watertown, Massachusetts.^[9] Another ancestor, Edmund Rice, arrived at Watertown in 1638.^[10] Coolidge's great-great-grandfather, also named John Coolidge, was an American military officer in the Revolutionary War and one of the first selectmen of the town of Plymouth Notch.^[11] Most of Coolidge's ancestors were farmers. Other well-known Coolidges, architect Charles Allerton Coolidge, General Charles Austin Coolidge, and diplomat Archibald Cary Coolidge among them, were descended from branches of the family that had remained in Massachusetts.^[9] Coolidge's grandmother Sarah Almeda Brewer had two famous first cousins: Arthur Brown, a United States Senator, and Olympia Brown, a women's suffragist. It is through Sarah Brewer that Coolidge believed that he inherited American Indian blood, but this descent has never been established by modern genealogists.^[12]

Early career and marriage[link]

Western Massachusetts lawyer[link]

Coolidge attended Black River Academy and then Amherst College, where he joined the Phi Gamma Delta fraternity.^[13] At his father's urging, Coolidge moved to Northampton, Massachusetts, after graduating to take up the practice of law. Avoiding the costly alternative of attending a law school, Coolidge followed the more common practice of the time, apprenticing with a local law firm, Hammond & Field, and reading law with them. John C. Hammond and Henry P. Field, both Amherst graduates, introduced Coolidge to the law practice in the county seat of Hampshire County. In 1897, Coolidge was admitted to the bar, becoming a country lawyer. With his savings and a small inheritance from his grandfather, Coolidge was able to open his own law office in Northampton in 1898. He practiced transactional law, believing that he served his clients best by staying out of court. As his reputation as a hard-working and diligent attorney grew, local banks and other businesses began to retain his services.^[14]

Marriage and family[link]

In 1905, Coolidge met and married a fellow Vermonter, Grace Anna Goodhue, who was working as a teacher at the Clarke School for the Deaf. While Grace was watering flowers outside the school one day in 1903, she happened to look up at the open window of Robert N. Weir's boardinghouse and caught a glimpse of Calvin Coolidge shaving in front of a mirror with nothing on but long underwear and a hat.^[15] After a more formal introduction sometime later, the two were quickly attracted to each other.^[15] They were married on October 4, 1905, in the parlor of her parents' home in Burlington, Vermont.

They were opposites in personality: she was talkative and fun-loving, while he was quiet and serious.^[16] Not long after their marriage, Coolidge handed her a bag with fifty-two pairs of socks in it, all of them full of holes. Grace's reply was "Did you marry me to darn your socks?" Without cracking a smile and with his usual seriousness, Calvin answered, "No, but I find it mighty handy."^[17] They had two sons: John, born in 1906, and Calvin, Jr., born in 1908.^[18] The marriage was, by most accounts, a happy one.^[19] As Coolidge wrote in his Autobiography, "We thought we were made for each other. For almost a quarter of a century she has borne with my infirmities, and I have rejoiced in her graces."^[20]

Local political office[link]

City offices[link]

The Republican Party was dominant in New England in Coolidge's time, and he followed Hammond's and Field's example by becoming active in local politics.^[21] Coolidge campaigned locally for Republican presidential candidate William McKinley in 1896, and the next year he was selected to be a member of the Republican City Committee.^[22] In 1898, he won election to the City Council of Northampton, placing second in a ward where the top three candidates were elected.^[22] The position offered no salary, but gave Coolidge experience in the political world.^[23] In 1899, he declined renomination, running instead for City Solicitor, a position elected by the City Council. He was elected for a one-year term in 1900, and reelected in 1901.^[24] This position gave Coolidge more experience as a lawyer, and paid a salary of $600.^[24] In 1902, the city council selected a Democrat for city solicitor, and Coolidge returned to an exclusively private practice.^[25] Soon thereafter, however, the clerk of courts for the county died, and Coolidge was chosen to replace him. The position paid well, but barred him from practicing law, so he only remained at the job for one year.^[25] The next year, 1904, Coolidge met with his only defeat before the voters, losing an election to the Northampton school board. When told that some of his neighbors voted against him because he had no children in the schools he would govern, Coolidge replied "Might give me time!"^[25]

State legislator and mayor[link]

Calvin and Grace Coolidge, about 1918.

In 1906 the local Republican committee nominated Coolidge for election to the state House of Representatives. He won a close victory over the incumbent Democrat, and reported to Boston for the 1907 session of the Massachusetts General Court.^[26] In his freshman term, Coolidge served on minor committees and, although he usually voted with the party, was known as a Progressive Republican, voting in favor of such measures as women's suffrage and the direct election of Senators.^[27] Throughout his time in Boston, Coolidge found himself allied primarily with the western Winthrop Murray Crane faction of the state Republican Party, as against the Henry Cabot Lodge-dominated eastern faction.^[28] In 1907, he was elected to a second term. In the 1908 session, Coolidge was more outspoken, but was still not one of the leaders in the legislature.^[29]

Instead of vying for another term in the State House, Coolidge returned home to his growing family and ran for mayor of Northampton when the incumbent Democrat retired. He was well liked in the town, and defeated his challenger by a vote of 1,597 to 1,409.^[30] During his first term (1910 to 1911), he increased teachers' salaries and retired some of the city's debt while still managing to effect a slight tax decrease.^[31] He was renominated in 1911, and defeated the same opponent by a slightly larger margin.^[32]

In 1911, the State Senator for the Hampshire County area retired and encouraged Coolidge to run for his seat for the 1912 session. He defeated his Democratic opponent by a large margin.^[33] At the start of that term, Coolidge was selected to be chairman of a committee to arbitrate the "Bread and Roses" strike by the workers of the American Woolen Company in Lawrence, Massachusetts.^[34] After two tense months, the company agreed to the workers' demands in a settlement the committee proposed.^[35] The other major issue for Republicans that year was the party split between the progressive wing, which favored Theodore Roosevelt, and the conservative wing, which favored William Howard Taft. Although he favored some progressive measures, Coolidge refused to leave the Republican party.^[36] When the new Progressive Party declined to run a candidate in his state senate district, Coolidge won reelection against his Democratic opponent by an increased margin.^[36]

The 1913 session was less eventful, and Coolidge's time was mostly spent on the railroad committee, of which he was the chairman.^[37] Coolidge intended to retire after the 1913 session, as two terms were the norm, but when the President of the State Senate, Levi H. Greenwood, considered running for Lieutenant Governor, Coolidge decided to run again for the Senate in the hopes of being elected as its presiding officer.^[38] Although Greenwood later decided to run for reelection to the Senate, he was defeated and Coolidge was elected, with Crane's help, as the President of a closely divided Senate.^[39] After his election in January 1914, Coolidge delivered a speech entitled Have Faith in Massachusetts, which summarized his philosophy of government. It was later published in a book, and frequently quoted.^[40]

Coolidge's speech was well received, and he attracted some admirers on its account.^[42] Towards the end of the term, many of them were proposing his name for nomination to lieutenant governor. After winning reelection to the Senate by an increased margin in the 1914 elections, Coolidge was reelected unanimously to be President of the Senate.^[43] As the 1915 session ended, Coolidge's supporters, led by fellow Amherst alumnus Frank Stearns, encouraged him again to run for lieutenant governor. This time, he accepted their advice.^[44]

Lieutenant Governor[link]

Coolidge entered the primary election for lieutenant governor and was nominated to run alongside gubernatorial candidate Samuel W. McCall. Coolidge was the leading vote-getter in the Republican primary, and balanced the Republican ticket by adding a western presence to McCall's eastern base of support.^[45] McCall and Coolidge won the 1915 election, with Coolidge defeating his opponent by more than 50,000 votes.^[46]

Coolidge's duties as lieutenant governor were few; in Massachusetts, the lieutenant governor does not preside over the state Senate, although Coolidge did become an ex officio member of the governor's cabinet.^[47] As a full-time elected official, Coolidge no longer practiced law after 1916, though his family continued to live in Northampton.^[48] McCall and Coolidge were both reelected in 1916 and again in 1917 (At the time, both offices were held for one-year terms). When McCall decided that he would not stand for a fourth term, Coolidge announced his intention to run for governor.^[49]

Governor of Massachusetts[link]

1918 election[link]

Coolidge was unopposed for the Republican nomination for Governor of Massachusetts in 1918. He and his running mate, Channing Cox, a Boston lawyer and Speaker of the Massachusetts House of Representatives, ran on the previous administration's record: fiscal conservatism, a vague opposition to Prohibition, support for women's suffrage, and support for American involvement in World War I.^[50] The issue of the war proved divisive, especially among Irish- and German-Americans.^[51] Coolidge was elected by a margin of 16,773 votes over his opponent, Richard H. Long, in the smallest margin of victory of any of his state-wide campaigns.^[52]

Boston police strike[link]

Governor Coolidge inspects militia

In 1919, in response to rumors that policemen of the Boston Police Department planned to form a union, Police Commissioner Edwin U. Curtis issued a statement saying that such a move would not be tolerated. In August of that year, the American Federation of Labor issued a charter to the Boston Police Union.^[53] Curtis said the union's leaders were insubordinate and planned to relieve them of duty, but said that he would suspend the sentence if the union was dissolved by September 4.^[54] The mayor of Boston, Andrew Peters, convinced Curtis to delay his action for a few days, but Curtis ultimately suspended the union leaders on September 8.^[55]

The following day, about three-quarters of the policemen in Boston went on strike.^[57] Coolidge had observed the situation throughout the conflict, but had not yet intervened. That night and the next, there was sporadic violence and rioting in the lawless city.^[58] Peters, concerned about sympathy strikes, had called up some units of the Massachusetts National Guard stationed in the Boston area and relieved Curtis of duty.^[59] Coolidge, furious that the mayor had called out state guard units, finally acted.^[60] He called up more units of the National Guard, restored Curtis to office, and took personal control of the police force.^[61] Curtis proclaimed that all of the strikers were fired from their jobs, and Coolidge called for a new police force to be recruited.^[62]

That night Coolidge received a telegram from AFL leader Samuel Gompers. "Whatever disorder has occurred", Gompers wrote, "is due to Curtis's order in which the right of the policemen has been denied …"^[63] Coolidge publicly answered Gompers's telegram with the response that would launch him into the national consciousness (quoted, above left).^[63] Newspapers across the nation picked up on Coolidge's statement and he became the newest hero to opponents of the strike. In the midst of the First Red Scare, many Americans were terrified of the spread of communist revolution, like those that had taken place in Russia, Hungary, and Germany. While Coolidge had lost some friends among organized labor, conservatives across the nation had seen a rising star. Although he usually acted with deliberation, the Boston police strike gave him a national reputation as a man who would take decisive action.

1919 election[link]

Coolidge and Cox were renominated for their respective offices in 1919. By this time Coolidge's supporters (especially Stearns) had publicized his actions in the Police Strike around the state and the nation and some of Coolidge's speeches were published in book form.^[40] He faced the same opponent as in 1918, Richard Long, but this time Coolidge defeated him by 125,101 votes, more than seven times his margin of victory from a year earlier.^[64] His actions in the police strike, combined with the massive electoral victory, led to suggestions that Coolidge run for President in 1920.^[65]

Legislation and vetoes as governor[link]

Governor Coolidge, laying the cornerstone at Suffolk Law School in Boston in August 1920.

By the time Coolidge was inaugurated on January 2, 1919, the First World War had ended, and Coolidge pushed the legislature to give a $100 bonus to Massachusetts veterans. He also signed a bill reducing the work week for women and children from fifty-four hours to forty-eight, saying, "We must humanize the industry, or the system will break down."^[66] He signed into law a budget that kept the tax rates the same, while trimming four million dollars from expenditures, thus allowing the state to retire some of its debt.^[67]

Coolidge also wielded the veto pen as governor. His most publicized veto was of a bill that would have increased legislators' pay by 50%.^[68] Although Coolidge was personally opposed to Prohibition, he vetoed a bill in May 1920 that would have allowed the sale of beer or wine of 2.75% alcohol or less, in Massachusetts in violation of the Eighteenth Amendment to the United States Constitution. "Opinions and instructions do not outmatch the Constitution," he said in his veto message, "Against it, they are void."^[69]

Vice Presidency[link]

1920 election[link]

Coolidge before his nomination as Vice President

At the 1920 Republican National Convention most of the delegates were selected by state party conventions, not primaries. As such, the field was divided among many local favorites.^[70] Coolidge was one such candidate, and while he placed as high as sixth in the voting, the powerful party bosses never considered him a serious candidate. After ten ballots, the delegates settled on Senator Warren G. Harding of Ohio as their nominee for President.^[71] When the time came to select a Vice Presidential nominee, the party bosses had also made a decision on who they would nominate: Senator Irvine Lenroot of Wisconsin.^[72] A delegate from Oregon, Wallace McCamant, having read Have Faith in Massachusetts, proposed Coolidge for Vice President instead.^[72] The suggestion caught on quickly, and Coolidge found himself unexpectedly nominated.^[73]

The Democrats nominated another Ohioan, James M. Cox, for President and the Assistant Secretary of the Navy, Franklin D. Roosevelt, for Vice President. The question of the United States joining the League of Nations was a major issue in the campaign, as was the unfinished legacy of Progressivism.^[74] Harding ran a "front-porch" campaign from his home in Marion, Ohio, but Coolidge took to the campaign trail in the Upper South, New York, and New England.^[75] On November 2, 1920, Harding and Coolidge were victorious in a landslide, winning every state outside the South.^[76] They also won in Tennessee, the first time a Republican ticket had won a Southern state since Reconstruction.^[76]

"Silent Cal"[link]

President Harding and Vice President Coolidge and their wives.

The Vice-Presidency did not carry many official duties, but Coolidge was invited by President Harding to attend cabinet meetings, making him the first Vice President to do so.^[77] He gave speeches around the country, but none were especially noteworthy.^[78]

As Vice-President, Coolidge and his vivacious wife Grace were invited to quite a few parties, where the legend of "Silent Cal" was born. It is from this time that most of the jokes and anecdotes involving Coolidge originate. Although Coolidge was known to be a skilled and effective public speaker, in private he was a man of few words and was therefore commonly referred to as "Silent Cal." A possibly apocryphal story has it that Dorothy Parker, seated next to him at a dinner, said to him, "Mr. Coolidge, I've made a bet against a fellow who said it was impossible to get more than two words out of you." His famous reply: "You lose."^[79] It was also Parker who, upon learning that Coolidge had died, reportedly remarked, "How can they tell?"^[80] Coolidge often seemed uncomfortable among fashionable Washington society; when asked why he continued to attend so many of their dinner parties, he replied, "Got to eat somewhere."^[81] Alice Roosevelt Longworth, a leading Republican wit, underscored Coolidge's silence and his dour personality: "When he wished he were elsewhere, he pursed his lips, folded his arms, and said nothing. He looked then precisely as though he had been weaned on a pickle.^[82]

As President, Coolidge's reputation as a quiet man continued. "The words of a President have an enormous weight," he would later write, "and ought not to be used indiscriminately."^[83] Coolidge was aware of his stiff reputation; indeed, he cultivated it. "I think the American people want a solemn ass as a President," he once told Ethel Barrymore, "and I think I will go along with them."^[84] However, he did hold a then-record number of presidential press conferences, 520 during his presidency.^[85] Some historians would later suggest that Coolidge's image was created deliberately as a campaign tactic,^[86] while others believe his withdrawn and quiet behavior to be natural, deepening after the death of his son in 1924.^[87]

Presidency 1923–1929[link]

Succession to the Presidency[link]

On August 2, 1923, President Harding died while on a speaking tour in California.^[88] Vice-President Coolidge was in Vermont visiting his family home, which had neither electricity nor a telephone, when he received word by messenger of Harding's death.^[89] Coolidge dressed, said a prayer, and came downstairs to greet the reporters who had assembled.^[89] His father, a notary public, administered the oath of office in the family's parlor by the light of a kerosene lamp at 2:47 am on August 3, 1923; Coolidge then went back to bed. He returned to Washington the next day, and was re-sworn in by Justice Adolph A. Hoehling, Jr. of the Supreme Court of the District of Columbia, as there was some confusion over whether a state notary public had the authority to administer the presidential oath.^[90][91]

Finishing Harding's term[link]

Coolidge signing the Immigration Act and some appropriation bills. General John J. Pershing looks on.

The nation did not know what to make of its new President; Coolidge had not stood out in the Harding administration and many had expected him to be replaced on the ballot in 1924.^[92] He appointed C. Bascom Slemp, a Virginia Congressman and experienced federal politician^[93] to work jointly with Edward T. Clark, a Massachusetts Republican organizer whom he retained from his vice presidential staff, as Secretaries to the President (a position equivalent to the modern White House Chief of Staff).^[94] Although a few of Harding's cabinet appointees were scandal-tarred, Coolidge announced that he would not demand any of their resignations, believing that since the people had elected Harding, he should carry on Harding's presidency, at least until the next election.^[94]

He addressed Congress when it reconvened on December 6, 1923, giving a speech that echoed many of Harding's themes, including immigration restriction and the need for the government to arbitrate the coal strikes then ongoing in Pennsylvania.^[95] Coolidge's speech was the first Presidential speech to be broadcast to the nation over the radio.^[96] The Washington Naval Treaty was proclaimed just one month into Coolidge's term, and was generally well received in the country.^[94] In May 1924, the World War I veterans' World War Adjusted Compensation Act or "Bonus Bill" was passed over his veto.^[97] Coolidge signed the Immigration Act later that year, which was aimed at restricting southern and eastern European immigration, though he appended a signing statement expressing his unhappiness with the bill's specific exclusion of Japanese immigrants.^[98] Just before the Republican Convention began, Coolidge signed into law the Revenue Act of 1924, which decreased personal income tax rates while increasing the estate tax, and creating a gift tax to reinforce the transfer tax system.^[99]

1924 election[link]

John W. Davis

The Republican Convention was held from June 10–12, 1924 in Cleveland, Ohio; President Coolidge was nominated on the first ballot.^[100] The convention nominated Frank Lowden of Illinois for Vice President on the second ballot, but he declined by telegram.^[101] Former Brigadier General Charles G. Dawes, who would win the Nobel Peace Prize in 1925, was nominated on the third ballot; he accepted.^[101]

The Democrats held their convention from June 24 to July 9 in New York City. The convention soon deadlocked, and after 103 ballots, the delegates finally agreed on a compromise candidate, John W. Davis, with Charles W. Bryan nominated for Vice President. The Democrats' hopes were buoyed when Robert M. La Follette, Sr., a Republican Senator from Wisconsin, split from his party to form a new Progressive Party. Many believed that the split in the Republican party, like the one in 1912, would allow a Democrat to win the Presidency.^[102]

Shortly after the conventions Coolidge experienced a personal tragedy. Coolidge's younger son, Calvin, Jr., developed a blister from playing tennis on the White House courts. The blister became infected, and within days Calvin, Jr. developed sepsis and died. After that Coolidge became withdrawn. He later said that "when he died, the power and glory of the Presidency went with him."^[103] In spite of his sadness, Coolidge ran his conventional campaign; he never maligned his opponents (or even mentioned them by name) and delivered speeches on his theory of government, including several that were broadcast over radio.^[104] It was easily the most subdued campaign since 1896, partly because the President was grieving for his son, but partly because Coolidge's style was naturally non-confrontational.^[105] The other candidates campaigned in a more modern fashion, but despite the split in the Republican party, the results were very similar to those of 1920. Coolidge and Dawes won every state outside the South except for Wisconsin, La Follette's home state. Coolidge had a popular vote majority of 2.5 million over his opponents' combined total.^[106]

Industry and trade[link]

During Coolidge's presidency the United States experienced the period of rapid economic growth known as the "Roaring Twenties". He left the administration's industrial policy in the hands of his activist Secretary of Commerce, Herbert Hoover, who energetically used government auspices to promote business efficiency and develop airlines and radio.^[108] With the exception of favoring increased tariffs, Coolidge disdained regulation, and carried about this belief by appointing commissioners to the Federal Trade Commission and the Interstate Commerce Commission who did little to restrict the activities of businesses under their jurisdiction.^[109] The regulatory state under Coolidge was, as one biographer described it, "thin to the point of invisibility."^[110]

Coolidge's economic policy has often been misquoted as "generally speaking, the business of the American people is business" (full quotation at right). Some have criticized Coolidge as an adherent of the laissez-faire ideology, which they claim led to the Great Depression.^[111] On the other hand, historian Robert Sobel offers some context based on Coolidge's sense of federalism: "As Governor of Massachusetts, Coolidge supported wages and hours legislation, opposed child labor, imposed economic controls during World War I, favored safety measures in factories, and even worker representation on corporate boards. Did he support these measures while president? No, because in the 1920s, such matters were considered the responsibilities of state and local governments."^[112]

Taxation[link]

Coolidge's taxation policy was that of his Secretary of the Treasury, Andrew Mellon: taxes should be lower and fewer people should have to pay them.^[113] Congress agreed, and the taxes were reduced in Coolidge's term.^[113] In addition to these tax cuts, Coolidge proposed reductions in federal expenditures and retiring some of the federal debt.^[113] Coolidge's ideas were shared by the Republicans in Congress, and in 1924 Congress passed the Revenue Act of 1924, which reduced income tax rates and eliminated all income taxation for some two million people.^[113] They reduced taxes again by passing the Revenue Acts of 1926 and 1928, all the while continuing to keep spending down so as to reduce the overall federal debt.^[114] By 1927, only the richest 2% of taxpayers paid any federal income tax.^[114] Although federal spending remained flat during Coolidge's administration, allowing one-fourth of the federal debt to be retired, state and local governments saw considerable growth, surpassing the federal budget in 1927.^[115]

Farm subsidies[link]

Coolidge with his Vice President, Charles G. Dawes.

Perhaps the most contentious issue of Coolidge's presidency was that of relief for farmers. Some in Congress proposed a bill designed to fight falling agricultural prices by allowing the federal government to purchase crops to sell abroad at lowered prices.^[116] Agriculture Secretary Henry C. Wallace and other administration officials favored the bill when it was introduced in 1924, but rising prices convinced many in Congress that the bill was unnecessary, and it was defeated just before the elections that year.^[117] In 1926, with farm prices falling once more, Senator Charles L. McNary and Representative Gilbert N. Haugen—both Republicans—proposed the McNary-Haugen Farm Relief Bill. The bill proposed a federal farm board that would purchase surplus production in high-yield years and hold it (when feasible) for later sale, or sell it abroad.^[118] Coolidge opposed McNary-Haugen, declaring that agriculture must stand "on an independent business basis," and said that "government control cannot be divorced from political control."^[118] He favored instead Herbert Hoover's proposal to modernize agriculture to create profits, instead of manipulating prices. Secretary Mellon wrote a letter denouncing the McNary-Haugen measure as unsound and likely to cause inflation, and it was defeated.^[119]

After McNary-Haugen's defeat, Coolidge supported a less radical measure, the Curtis-Crisp Act, which would have created a federal board to lend money to farm co-operatives in times of surplus; the bill did not pass.^[119] In February 1927, Congress took up the McNary-Haugen bill again, this time narrowly passing it.^[120] Coolidge vetoed it.^[120] In his veto message, he expressed the belief that the bill would do nothing to help farmers, benefitting only exporters and expanding the federal bureaucracy.^[121] Congress did not override the veto, but passed the bill again in May 1928 by an increased majority; again, Coolidge vetoed it.^[120] "Farmers never have made much money," said Coolidge, the Vermont farmer's son, "I do not believe we can do much about it."^[122]

Flood control[link]

Coolidge has often been criticized for his actions during the Great Mississippi Flood of 1927, the worst natural disaster to hit the Gulf Coast until Hurricane Katrina in 2005.^[123] Although he did eventually name Secretary Hoover to a commission in charge of flood relief, Coolidge's lack of interest in federal flood control has been criticized.^[123] Coolidge did not believe that personally visiting the region after the floods would accomplish anything, but would be seen only as political grandstanding. He also did not want to incur the federal spending that flood control would require; he believed property owners should bear much of the cost.^[124] On the other hand, Congress wanted a bill that would place the federal government completely in charge of flood mitigation.^[125] When Congress passed a compromise measure in 1928, Coolidge declined to take credit for it and signed the bill in private on May 15.^[126]

Civil rights[link]

Coolidge spoke out in favor of the civil rights of African Americans and Catholics.^[127] He appointed no known members of the Ku Klux Klan to office; indeed the Klan lost most of its influence during his term.^[128]

In 1924, Coolidge responded to a letter that claimed the United States was a "white man's country":

....I was amazed to receive such a letter. During the war 500,000 colored men and boys were called up under the draft, not one of whom sought to evade it. [As president, I am] one who feels a responsibility for living up to the traditions and maintaining the principles of the Republican Party. Our Constitution guarantees equal rights to all our citizens, without discrimination on account of race or color. I have taken my oath to support that Constitution....^[129]

On June 2, 1924, Coolidge signed the Indian Citizenship Act, which granted full U.S. citizenship to all American Indians, while permitting them to retain tribal land and cultural rights. However, the act was unclear on whether the federal government or the tribal leaders retained tribal sovereignty.^[130] Coolidge repeatedly called for anti-lynching laws to be enacted, but most Congressional attempts to pass this legislation were filibustered by Southern Democrats. Coolidge appointed some African Americans to federal office. He retained Harding's choice of Walter L. Cohen of New Orleans, Louisiana, as the comptroller of customs and offered Cohen the post of minister to Liberia, which the businessman declined.^[131]

Foreign policy[link]

Official White House portrait of Calvin Coolidge

Although not an isolationist, Coolidge was reluctant to enter into foreign alliances.^[132] Coolidge saw the landslide Republican victory of 1920 as a rejection of the Wilsonian idea that the United States should join the League of Nations.^[133] While not completely opposed to the idea, Coolidge believed the League, as then constituted, did not serve American interests, and he did not advocate membership in it.^[133] He spoke in favor of the United States joining the Permanent Court of International Justice, provided that the nation would not be bound by advisory decisions.^[134] The Senate eventually approved joining the Court (with reservations) in 1926.^[135] The League of Nations accepted the reservations, but suggested some modifications of their own.^[136] The Senate failed to act; the United States never joined the World Court.^[136]

Coolidge's best-known initiative was the Kellogg–Briand Pact of 1928, named for Coolidge's Secretary of State, Frank B. Kellogg, and French foreign minister Aristide Briand. The treaty, ratified in 1929, committed signatories including the U.S., the United Kingdom, France, Germany, Italy, and Japan to "renounce war, as an instrument of national policy in their relations with one another."^[137] The treaty did not achieve its intended result – the outlawry of war – but did provide the founding principle for international law after World War II.^[138]

Coolidge continued the previous administration's policy not to recognize the Soviet Union.^[139] He also continued the United States' support for the elected government of Mexico against the rebels there, lifting the arms embargo on that country.^[140] He sent his close friend Dwight Morrow to Mexico as the American ambassador.^[141] Coolidge represented the U.S. at the Pan American Conference in Havana, Cuba, making him the only sitting U.S. President to visit the country. The United States' occupation of Nicaragua and Haiti continued under his administration, but Coolidge withdrew American troops from the Dominican Republic in 1924.^[142]

1928 election[link]

Collection of video clips of the president

In the summer of 1927, Coolidge vacationed in the Black Hills of South Dakota, where he engaged in horseback riding and fly fishing and attended rodeos. He made Custer State Park his "summer White House". News coverage of Coolidge's time in the Black Hills soon increased tourism in the general region and promoted the popularity of Wind Cave National Park.^[143] While on vacation, Coolidge surprisingly issued his terse statement that he would not seek a second full term as President in 1928: "I do not choose to run for President in 1928."^[144] After allowing the reporters to take that in, Coolidge elaborated. "If I take another term, I will be in the White House till 1933 … Ten years in Washington is longer than any other man has had it—too long!"^[145] In his memoirs, Coolidge explained his decision not to run: "The Presidential office takes a heavy toll of those who occupy it and those who are dear to them. While we should not refuse to spend and be spent in the service of our country, it is hazardous to attempt what we feel is beyond our strength to accomplish."^[146] After leaving office, he and Grace returned to Northampton, where he wrote his memoirs. The Republicans retained the White House in 1928 in the person of Coolidge's Secretary of Commerce, Herbert Hoover.

Coolidge had been reluctant to choose Hoover as his successor; on one occasion he remarked that "for six years that man has given me unsolicited advice—all of it bad."^[147] Even so, Coolidge had no desire to split the party by publicly opposing the popular Commerce Secretary's nomination.^[148] The delegates did consider nominating Vice President Charles Dawes to be Hoover's running mate. But Coolidge (who hated Dawes) remarked that this would be "a personal affront" to him, and the convention selected Senator Charles Curtis instead.^[149]

Cabinet[link]

Coolidge's cabinet in 1924, outside the White House
Front row, left to right: Harry Stewart New, John W. Weeks, Charles Evans Hughes, Coolidge, Andrew Mellon, Harlan F. Stone, Curtis D. Wilbur
Back row, left to right, James J. Davis, Henry C. Wallace, Herbert Hoover, Hubert Work

Judicial appointments[link]

Coolidge appointed Harlan Fiske Stone Attorney General and then Supreme Court Justice

Supreme Court[link]

Coolidge appointed one Justice to the Supreme Court of the United States, Harlan Fiske Stone in 1925. Stone was Coolidge's fellow Amherst alumnus, a Wall Street lawyer and conservative Republican. Stone was serving as dean of Columbia Law School when Coolidge appointed him to be Attorney General in 1924 to restore the reputation tarnished by Harding's Attorney General, Harry M. Daugherty. Coolidge named Stone to the Supreme Court in 1925.^[150] Stone was later appointed Chief Justice by President Franklin D. Roosevelt.

Other courts[link]

Along with his Supreme Court appointment, Coolidge successfully nominated 17 judges to the United States Courts of Appeals, and 61 judges to the United States district courts. He appointed judges to various specialty courts as well, including Genevieve R. Cline, who became the first woman named to the Federal judiciary when Coolidge placed her on the United States Customs Court in 1928.^[151] Coolidge also signed the Judiciary Act of 1925 into law, allowing the Supreme Court more discretion over its workload.

Retirement and death[link]

Coolidge addressing a crowd at Arlington National Cemetery's Roman style Memorial Amphitheater in 1924.

After his presidency, Coolidge retired to the modest rented house on residential Massasoit Street in Northampton before moving to a more spacious mansion, "The Beeches." He kept a Hacker runabout boat on the Connecticut River and was often observed on the water by local boating enthusiasts. During this period he also served as chairman of the non-partisan Railroad Commission, as honorary president of the American Foundation for the Blind, as a director of New York Life Insurance Company, as president of the American Antiquarian Society, and as a trustee of Amherst College.^[152] Coolidge received an honorary Doctor of Laws from Bates College in Lewiston, Maine.

Coolidge published his autobiography in 1929 and wrote a syndicated newspaper column, "Calvin Coolidge Says," from 1930 to 1931.^[153] Faced with looming defeat in 1932, some Republicans spoke of rejecting Herbert Hoover as their party's nominee, and instead drafting Coolidge to run, but the former President made it clear that he was not interested in running again, and that he would publicly repudiate any effort to draft him, should it come about.^[154] Hoover was renominated, and Coolidge made several radio addresses in support of him.^[155]

He died suddenly of a heart attack at "The Beeches," at 12:45 pm, January 5, 1933.^[156] Shortly before his death, Coolidge confided to an old friend: "I feel I no longer fit in with these times."^[157]