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10 (ten ⁱ/ˈtɛn/) is an even natural number following 9 and preceding 11. Ten is the base of the decimal numeral system, by far the most common system of denoting numbers in both spoken and written language. The reason for the choice of ten is assumed to be that humans have ten fingers (digits).

In common usage and derived terms[link]

• A collection of ten items (most often ten years) is called a decade.
• The ordinal adjective is denary.
• Increasing a quantity by one order of magnitude is most widely understood to mean multiplying the quantity by ten.
• To reduce something by one-tenth is to decimate. (In ancient Rome, the killing of one in ten soldiers in a cohort was the punishment for cowardice or mutiny; or, one-tenth of the able-bodied men in a village as a form of retribution, thus causing a labor shortage and threat of starvation in agrarian societies.)
• With ten being the base of the decimal system, a scale of 1 to 10 is often used to rank things, as a smaller version of a 1-to-100 scale (as is used in percentages and wine-tasting). Hence, something that scores perfectly is "a perfect ten". A person who is attractive and physically flawless is often said to be "a ten", from the idea of ranking that person's appearance and sex-appeal on a 1-to-10 scale.^{[citation needed]}

In mathematics[link]

Ten is a composite number, its proper divisors being 1, 2 and 5. Ten is the smallest noncototient, a number that cannot be expressed as the difference between any integer and the total number of coprimes below it.

Ten is the second discrete semiprime (2.5) and the second member of the (2.q) discrete semiprime family. Ten has an aliquot sum σ(n) of 8 and is accordingly the first discrete semiprime to be in deficit. All subsequent discrete semiprimes are in deficit. The aliquot sequence for 10 comprises five members (10,8,7,1,0) with this number being the second composite member of the 7-aliquot tree.

It is the aliquot sum of only one number the discrete semiprime 14.

Ten is a semi-meandric number.

Ten is the sum of the first three prime numbers, of the four first numbers (1 + 2 + 3 + 4), of the square of the two first odd numbers and also of the first four factorials (0! + 1! + 2! + 3!). Ten is the eighth Perrin number, preceded in the sequence by 5, 5, 7.

A polygon with ten sides is a decagon, and 10 is a decagonal number. But it is also a triangular number and a centered triangular number.

Ten is the number of n-Queens Problem solutions for n = 5.

Ten is the smallest number whose status as a possible friendly number is unknown.

In numeral systems[link]

Decimal system[link]

As is the case for any base in its system, ten is the first two-digit number in decimal and thus the lowest number where the position of a numeral affects its value. Any integer written in the decimal system can be multiplied by ten by adding a zero to the end (e.g. 855 * 10 = 8550).

Roman numerals[link]

The Roman numeral for ten is X (which looks like two V's [the Roman numeral for 5] put together); it is thought that the V for five is derived from an open hand (five digits displayed). Incidentally, the Chinese word numeral for ten, is a cross, too: 十.

Positional numeral systems other than decimal[link]

The digit '1' followed by '0' is how the value of p is written in base p. (E.g. 16 in hexadecimal is 10.)

Properties:

• Ten is a Harshad number in bases 2, 3, 5, 6, 9 and all others above.
• In bases 15, 18 and 30, ten is a 1-automorphic number (displayed as the numeral 'A')..

List of basic calculations[link]

In science[link]

The meaning "10" is part of the following terms:

• decapoda, an order of crustaceans with ten feet
• decane, a hydrocarbon with 10 carbon atoms

Also, the number 10 plays a role in the following:

• The atomic number of neon.
• The number of hydrogen atoms in butane, a hydrocarbon.
• The number of spacetime dimensions in some superstring theories.

Astronomy[link]

• The New General Catalogue object NGC 10, a magnitude 12.5 spiral galaxy in the constellation Sculptor.

• The Saros number of the solar eclipse series which began on -2467 February 28 and ended on -1169 April 18. The duration of Saros series 10 was 1298.1 years, and it contained 73 solar eclipses.

• The Saros number of the lunar eclipse series which began on -2454 June 17 and ended on -1138 August 15. The duration of Saros series 10 was 1316.2 years, and it contained 74 lunar eclipses.

• Messier object M10, a magnitude 6.4 globular cluster in the constellation Ophiuchus.

In religion and philosophy[link]

The tetractys

• References in the Bible, Judaism and Christianity:
  • The Ten Commandments of Exodus^[1] and Deuteronomy^[2] are considered a cornerstone of Judaism and Christianity.
  • People traditionally tithed one-tenth of their produce. The practice of tithing is still common in Christian churches today, though it is disputed in some circles as to whether or not it is required of Christians.
  • In Deuteronomy 26:12, the Torah commands Jews to give one-tenth of their produce to the poor (Maaser Ani). From this verse and from an earlier verse (Deut. 14:22) there derives a practice for Jews to give one-tenth of all earnings to the poor.^[3]
  • Ten Plagues were inflicted on Egypt in Exodus 7-12
  • Jews observe the annual Ten Days of Repentance beginning on Rosh Hashanah and ending on Yom Kippur.
  • In Jewish liturgy, Ten Martyrs are singled out as a group.
  • There are said to be Ten Lost Tribes of Israel (those other than Judah and Benjamin)
  • There are Ten Sephirot in the Kabbalistic Tree of Life
  • The Bible records 10 generations between Adam and Noah, and 10 generations between Noah and Abraham
  • In Judaism, ten men are the required quorum, called a minyan, for prayer services.

• In Pythagoreanism, the number 10 played an important role and was symbolized by the tetractys.

In money[link]

Most countries issue coins and bills with a denomination of 10 (See e.g. 10 dollar note‎). Of these, the dime, with the value of ten cents, or one tenth of a dollar, derives its name from the meaning "one tenth" - see Dime (United States coin)#Denomination history and etymology.

In music[link]

• The interval of a major tenth is an octave plus a major third.
• The interval of a minor tenth is an octave plus a minor third.
• The title of quite a few albums. See Ten (album).
• "Ten lords a-leaping" is the gift on the tenth day of Christmas in the carol "The Twelve Days of Christmas"

In sports and games[link]

The meaning "10" is part of the following terms:

• decathlon, a combined event in athletics consisting of ten track and field events.

Also, the number 10 plays a role in the following:

• In American football, the end zones are 10 yards deep.
• In association football, the number 10 is traditionally worn by the team's playmaker. This use has led to "Number 10" becoming a synonym for that particular player, even if he or she does not wear that number.
• In Australian rules football, considered the break even amount of games won in a regular 22 game season.
• In baseball, 10 is the minimum number of players on the field at any given time during play (including the batter).
• In basketball:
  • The top of the rim (goal) is 10 feet from the floor.
  • In standard full-court basketball, there are 10 players on the court (5 on each team).
• In blackjack, the Ten, Jack, Queen and King are all worth 10 points.
• In cricket, 10 is the number of wickets required to be taken by the bowling side for the batting side to be bowled out.
• In gridiron football, 10 is the number of yards the offense must advance to maintain possession in a single set of downs—four in American and three in Canadian.
• In most rugby league competitions, the number 10 is worn by one of the two starting props. (One exception to this rule is the European Super League, which uses static squad numbering.)
• In rugby union, the starting fly-half wears the 10 shirt.
• In ten-pin bowling, 10 pins are arranged in a triangular pattern and there are 10 frames per game.
• The highest score possible in Olympics gymnastics competitions.
• Driving a racing car at ten-tenths is driving as fast as possible, on the limit.

The jersey number 10 has been retired by several North American sports teams in honor of past playing greats or other key figures:

• In Major League Baseball:
  • The Chicago Cubs, for incoming Hall of Famer Ron Santo.
  • The Cincinnati Reds, for Hall of Fame manager Sparky Anderson.
  • The Kansas City Royals, for manager Dick Howser.
  • The Montreal Expos, first for Rusty Staub and later for Hall of Famer Andre Dawson. The team has since relocated and become the Washington Nationals; after initially announcing that it would reissue numbers retired in Montreal, the team reversed itself.
  • The New York Yankees, for Hall of Famer Phil Rizzuto, who went on to spend four decades as a Yankees radio and TV broadcaster.
  • The St. Louis Cardinals, for manager Tony La Russa.
  • The Minnesota Twins have announced they will retire the number for manager Tom Kelly on September 8, 2012.
• In the NBA:
  • The Boston Celtics, for Jo Jo White.
  • The Chicago Bulls, for Bob Love.
  • The Detroit Pistons, for Hall of Famer Dennis Rodman. Greg Monroe, who currently wears the number, is allowed to continue wearing it.
  • The Miami Heat, for Tim Hardaway.
  • The New York Knicks, for Hall of Famer Walt Frazier, who also served as a Knicks broadcaster after his playing career.
  • The Philadelphia 76ers, for Maurice Cheeks, who also served as the team's head coach.
  • The Seattle SuperSonics, for Nate McMillan, who also served as the team's head coach. The team has since relocated to become the Oklahoma City Thunder, but the Thunder have yet to issue any number retired by the franchise in Seattle.
  • The Washington Wizards, for Hall of Famer Earl Monroe, who played for the team in its past incarnation as the Baltimore Bullets.
• In the NFL:
  • The Atlanta Falcons, for Steve Bartkowski.
  • The Minnesota Vikings, for Hall of Famer Fran Tarkenton.
• In the NHL:
  • The Carolina Hurricanes, for Hall of Famer Ron Francis.
  • The Detroit Red Wings, for Hall of Famer Alex Delvecchio.
  • The Montreal Canadiens, for Hall of Famer Guy Lafleur.
  • The first NHL incarnation of the Winnipeg Jets, for Hall of Famer Dale Hawerchuk. The Jets have since reincarnated and have continued to honour the numbers retired by the initial franchise in Winnipeg.
  • The Toronto Maple Leafs have a policy of not retiring numbers unless the player honoured either died or suffered a career-ending incident while a member of the team. Other players whose numbers would otherwise be retired instead have their numbers enshrined by the team as "Honoured Numbers", which remain in circulation for future players. The number 10 is currently honoured for Hall of Famers Syl Apps and George Armstrong.
• The jersey number 10 has also been retired by the men's basketball program of the University of North Carolina for Lennie Rosenbluth.

In technology[link]

• Ten-codes are commonly used on emergency service radio systems.
• Ten refers to the "meter band" on the radio spectrum between 28 and 29.7 MHz, used by amateur radio.
• ASCII and Unicode code point for line feed
• In MIDI, Channel 10 is reserved for unpitched percussion instruments.
• In the Rich Text Format specification, all language codes for regional variants of the Spanish language are congruent to 10 mod 256.
• In Mac OS X, the F10 function key tiles all the windows of the current application and grays the windows of other applications.
• The IP addresses in the range 10.0.0.0/8 (meaning the interval between 10.0.0.0 and 10.255.255.255) is reserved for use by Private networks by RFC 1918.

In other fields[link]

10 playing cards of all four suits

• Blake Edwards' 1979 movie 10 (film).
• Series on HBO entitled "1st & Ten" which aired between December 1984 and January 1991.
• Series on ESPN and ESPN2 entitled 1st and 10 which launched on ESPN in October 2003 to 2008 and moved to ESPN2 from 2008 to present.
• Counting from one to ten before speaking is often done in order to cool one's temper.
• In astrology, Capricorn is the 10th astrological sign of the Zodiac.
• In Chinese astrology, the 10 Heavenly Stems, refer to a cyclic number system that is used also for time reckoning.
• An online show hosted by Microsoft.^[4]
• A 1977 short documentary film Powers of Ten depicts the relative scale of the Universe in factors of ten (orders of magnitude).
• CBS has a game show called Power of 10, where the player's prize goes up and down by either the previous or next power of ten.
• There are ten official inkblots in the Rorschach inkblot test.
• The traditional Snellen chart uses 10 different letters.

Ten is:

• Number of kingdoms in Five Dynasties and Ten Kingdoms Period
• House number of 10 Downing Street
• The number of Provinces in Canada
• The designation of United States Interstate 10, a freeway that runs from California to Florida.
• Number of dots in a tetractys.
• The number of the French department Aube
• The number of regions in Ghana.
• The state number of Virginia.

Historical years[link]

10 A.D., 10 B.C., 1910, 2010, etc.

See also[link]

Mathematics portal

References[link]

External links[link]

Look up ten in Wiktionary, the free dictionary.

Wikimedia Commons has media related to: 10 (number)

vep:10 (lugu)

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. 

The Right Honourable The Baroness Thatcher LG OM PC FRS
Prime Minister of the United Kingdom
In office 4 May 1979 – 28 November 1990
Monarch	Elizabeth II
Deputy	William Whitelaw Geoffrey Howe
Preceded by	James Callaghan
Succeeded by	John Major
Leader of the Opposition
In office 11 February 1975 – 4 May 1979
Monarch	Elizabeth II
Prime Minister	Harold Wilson James Callaghan
Preceded by	Edward Heath
Succeeded by	James Callaghan
Leader of the Conservative Party
In office 11 February 1975 – 28 November 1990
Preceded by	Edward Heath
Succeeded by	John Major
Secretary of State for Education and Science
In office 20 June 1970 – 4 March 1974
Prime Minister	Edward Heath
Preceded by	Edward Short
Succeeded by	Reginald Prentice
Member of Parliament for Finchley
In office 8 October 1959 – 9 April 1992
Preceded by	John Crowder
Succeeded by	Hartley Booth
Personal details
Born	Margaret Hilda Roberts (1925-10-13) 13 October 1925 (age 86) Grantham, Lincolnshire, United Kingdom
Political party	Conservative
Spouse(s)	Denis Thatcher (married 1951–2003, his death)
Children	Carol Thatcher Mark Thatcher
Residence	Chester Square
Alma mater	Somerville College, Oxford Inns of Court
Profession	Chemist Lawyer
Religion	Church of England (Since 1951)[1] Methodism (Before 1951)
Signature

Margaret Hilda Thatcher, Baroness Thatcher, LG, OM, PC, FRS, née Roberts (born 13 October 1925) is a British politician and the longest-serving (1979–1990) British prime minister of the 20th century, and the only woman ever to have held the post. A Soviet journalist nicknamed her the "Iron Lady", which later became associated with her uncompromising politics and leadership style. As prime minister, she implemented conservative policies that have come to be known as Thatcherism.

Originally a research chemist before becoming a barrister, Thatcher was elected Member of Parliament (MP) for Finchley in 1959. Edward Heath appointed her Secretary of State for Education and Science in his 1970 government. In 1975 Thatcher defeated Heath in the Conservative Party leadership election and became Leader of the Opposition, as well as the first woman to lead a major political party in the United Kingdom. She became prime minister after winning the 1979 general election.

After entering 10 Downing Street, Thatcher introduced a series of political and economic initiatives to reverse what she perceived as Britain's precipitous national decline.^{[nb 1]} Her political philosophy and economic policies emphasised deregulation (particularly of the financial sector), flexible labour markets, the privatisation of state-owned companies, and reducing the power and influence of trade unions. Thatcher's popularity during her first years in office waned amid recession and high unemployment, until economic recovery and the 1982 Falklands War brought a resurgence of support, resulting in her re-election in 1983. Thatcher was re-elected for a third term in 1987, but her Community Charge (popularly referred to as "poll tax") was widely unpopular and her views on the European Community were not shared by others in her Cabinet. She resigned as Prime Minister and party leader in November 1990, after Michael Heseltine launched a challenge to her leadership.

Thatcher holds a life peerage as Baroness Thatcher, of Kesteven in the County of Lincolnshire, which entitles her to sit in the House of Lords.

Early life and education[link]

Margaret Thatcher's birthplace, in Grantham

Commemorative plaque on the building in which Margaret Thatcher was born

Margaret Thatcher was born Margaret Roberts in Grantham, Lincolnshire, on 13 October 1925. Her father was Alfred Roberts, originally from Northamptonshire, and her mother was Beatrice Ethel (née Stephenson) from Lincolnshire.^[3] She spent her childhood in Grantham, where her father owned two grocery shops.^[4] She and her older sister Muriel were raised in the flat above the larger of the two, located near the railway line.^[4] Her father was active in local politics and the Christian church, serving as an alderman and a Methodist local preacher,^[5] and brought up his daughter as a strict Methodist.^[6] He came from a Liberal family but stood—as was then customary in local government—as an Independent. He was Mayor of Grantham in 1945–46 and lost his position as alderman in 1952 after the Labour Party won its first majority on Grantham Council in 1950.^[5]

Roberts attended Huntingtower Road Primary School and won a scholarship to Kesteven and Grantham Girls' School.^[7] Her school reports showed hard work and continual improvement; her extracurricular activities included the piano, field hockey, poetry recitals, swimming and walking.^[8][9] She was head girl in 1942–43.^[10] In her upper sixth year she applied for a scholarship to study chemistry at Somerville College, Oxford but was initially rejected, and was offered a place only after another candidate withdrew.^[11][12] She arrived at Oxford in 1943 and graduated in 1947 with Second Class Honours in the four-year Chemistry Bachelor of Science degree; in her final year she specialised in X-ray crystallography under the supervision of Dorothy Hodgkin.^[13][14]

Roberts became President of the Oxford University Conservative Association in 1946.^[15][16] She was influenced at university by political works such as Friedrich von Hayek's The Road to Serfdom (1944),^[17] which condemned economic intervention by government as a precursor to an authoritarian state.^[18]

After graduating, Roberts moved to Colchester in Essex to work as a research chemist for BX Plastics.^[19] She joined the local Conservative Association and attended the party conference at Llandudno in 1948, as a representative of the University Graduate Conservative Association.^[20] One of her Oxford friends was also a friend of the Chair of the Dartford Conservative Association in Kent, who were looking for candidates.^[20] Officials of the association were so impressed by her that they asked her to apply, even though she was not on the Conservative party's approved list: she was selected in January 1951 and added to the approved list post ante.^[21] At a dinner following her formal adoption as Conservative candidate for Dartford in February 1951 she met Denis Thatcher, a successful and wealthy divorced businessman, who drove her to her Essex train.^[20][21] In preparation for the election Roberts moved to Dartford, where she supported herself by working as a research chemist for J. Lyons and Co. in Hammersmith, part of a team developing emulsifiers for ice cream.^[20][22]

Early political career[link]

In the 1950 and 1951 general elections she was the Conservative candidate for the safe Labour seat of Dartford, where she attracted media attention as the youngest and the only female candidate.^[23][24] She lost both times to Norman Dodds, but reduced the Labour majority by 6,000, and then a further 1,000.^[23] (By an odd coincidence, Edward Heath was elected for the first time in the neighbouring constituency in 1950.) During the campaigns, she was supported by her parents and by Denis Thatcher, whom she married in December 1951.^[23][25] Denis funded his wife's studies for the bar;^[26] she qualified as a barrister in 1953 and specialised in taxation.^[27] That same year her twins, Carol and Mark, were born.^[28]

Member of Parliament (1959–1970)[link]

Thatcher was not a candidate in the 1955 general election as it came fairly soon after the birth of her children, although she was narrowly defeated as the candidate for the Orpington by-election, 1955.^[28] Afterwards, she began looking for a safe Conservative seat, and was selected as the candidate for Finchley in April 1958 (narrowly beating Ian Montagu Fraser). She was elected as MP for the seat after a hard campaign in the 1959 election.^[29] Her maiden speech was in support of her private member's bill (Public Bodies (Admission to Meetings) Act 1960), requiring local authorities to hold their council meetings in public. In 1961 she went against the Conservative Party's official position by voting for the restoration of birching.^[30] She regarded Finchley's Jewish residents as "her people" and became a founding member of the Anglo-Israel Friendship League of Finchley as well as a member of the Conservative Friends of Israel.^[31] But she nevertheless believed that Israel had to trade land for peace, and condemned Israel's bombing of Osirak as "a grave breach of international law".^[31]

In October 1961 Thatcher was promoted to the front bench as Parliamentary Undersecretary at the Ministry of Pensions and National Insurance in Harold Macmillan's administration.^[32] After the Conservatives lost the 1964 election she became spokeswoman on Housing and Land, in which position she advocated her party's policy of allowing tenants to buy their council houses.^[33] She moved to the Shadow Treasury team in 1966, and as Treasury spokeswoman opposed Labour's mandatory price and income controls, arguing that they would produce effects contrary to those intended and distort the economy.^[33]

At the Conservative Party Conference of 1966 she criticised the high-tax policies of the Labour Government as being steps "not only towards Socialism, but towards Communism".^[33] She argued that lower taxes served as an incentive to hard work.^[33] Thatcher was one of the few Conservative MPs to support Leo Abse's Bill to decriminalise male homosexuality^[34] and voted in favour of David Steel's bill to legalise abortion,^[1][35] as well as a ban on hare coursing.^[36][37] She supported the retention of capital punishment^[38] and voted against the relaxation of divorce laws.^[39][40]