Counting is the action of finding the number of elements of a finite set of objects. The traditional way of counting consists of continually increasing a (mental or spoken) counter by a unit for every element of the set, in some order, while marking (or displacing) those elements to avoid visiting the same element more than once, until no unmarked elements are left; if the counter was set to one after the first object, the value after visiting the final object gives the desired number of elements. The related term enumeration refers to uniquely identifying the elements of a finite (combinatorial) set or infinite set by assigning a number to each element.

Counting using tally marks at Hanakapiai Beach

Counting sometimes involves numbers other than one; for example, when counting money, counting out change, when "counting by twos" (2, 4, 6, 8, 10, 12, ...) or when "counting by fives" (5, 10, 15, 20, 25, ...).

There is archeological evidence suggesting that humans have been counting for at least 50,000 years.^[1] Counting was primarily used by ancient cultures to keep track of economic data such as debts and capital (i.e., accountancy). The development of counting led to the development of mathematical notation, numeral systems and writing.

Forms of counting[link]

Counting can occur in a variety of forms.

Counting can be verbal; that is, speaking every number out loud (or mentally) to keep track of progress. This is often used to count objects that are present already, instead of counting a variety of things over time.

Counting can also be in the form of tally marks, making a mark for each number and then counting all of the marks when done tallying. This is useful when counting objects over time, such as the number of times something occurs during the course of a day. Tallying is base 1 counting; normal counting is done in base 10. Computers use base 2 counting (0's and 1's).

Counting can also be in the form of finger counting, especially when counting small numbers. This is often used by children to facilitate counting and simple mathematical operations. Finger-counting uses unary notation (one finger = one unit), and is thus limited to counting 10 (unless you start in with your toes). Other hand-gesture systems are also in use, for example the Chinese system by which one can count 10 using only gestures of one hand. By using finger binary (base 2 counting), it is possible to keep a finger count up to 1023 = 2¹⁰ − 1.

Various devices can also be used to facilitate counting, such as hand tally counters and abacuses.

Inclusive counting[link]

Inclusive counting is usually encountered when counting days in a calendar. Normally when counting "8" days from Sunday, Monday will be day 1, Tuesday day 2, and the following Monday will be the eighth day. When counting "inclusively," the Sunday (the start day) will be day 1 and therefore the following Sunday will be the eighth day. For example, the French phrase for "fortnight" is en quinze (in 15 [days]), and similar words are present in Greek (δεκαπενθήμερο, dekapenthímero), Spanish (quincena) and Portuguese (quinzena) - whereas "a fortnight" derives from "a fourteen-night", as the archaic "a sennight" does from "a seven-night". This practice appears in other calendars as well; in the Roman calendar the nones (meaning "nine") is 8 days before the ides; and in the Christian calendar Quinquagesima (meaning 50) is 49 days before Easter Sunday.

The Jewish people also counted^[when?] days inclusively. For instance, Jesus announced he would die and resurrect "on the third day," i.e. two days later. Scholars^[who?] most commonly place his crucifixion on a Friday afternoon and his resurrection on Sunday before sunrise, spanning three different days but a period of around 36–40 hours.^{[citation needed]}

Musical terminology also uses inclusive counting of intervals between notes of the standard scale: going up one note is a second interval, going up two notes is a third interval, etc., and going up seven notes is an octave.

Education and development[link]

Learning to count is an important educational/developmental milestone in most cultures of the world. Learning to count is a child's very first step into mathematics, and constitutes the most fundamental idea of that discipline. However, some cultures in Amazonia and the Australian Outback do not count,^[2][3] and their languages do not have number words.

Many children at just 2 years of age have some skill in reciting the count list (i.e., saying "one, two, three, ..."). They can also answer questions of ordinality for small numbers, e.g., "What comes after three?". They can even be skilled at pointing to each object in a set and reciting the words one after another. This leads many parents and educators to the conclusion that the child knows how to use counting to determine the size of a set.^[4] Research suggests that it takes about a year after learning these skills for a child to understand what they mean and why the procedures are performed.^[5][6] In the mean time, children learn how to name cardinalities that they can subitize.

Children with Williams syndrome often display serious delays in learning to count.^{[citation needed]}

Counting in mathematics[link]

In mathematics, the essence of counting a set and finding a result n, is that it establishes a one to one correspondence (or bijection) of the set with the set of numbers {1, 2, ..., n}. A fundamental fact, which can be proved by mathematical induction, is that no bijection can exist between {1, 2, ..., n} and {1, 2, ..., m} unless n = m; this fact (together with the fact that two bijections can be composed to give another bijection) ensures that counting the same set in different ways can never result in different numbers (unless an error is made). This is the fundamental mathematical theorem that gives counting its purpose; however you count a (finite) set, the answer is the same. In a broader context, the theorem is an example of a theorem in the mathematical field of (finite) combinatorics—hence (finite) combinatorics is sometimes referred to as "the mathematics of counting."

Many sets that arise in mathematics do not allow a bijection to be established with {1, 2, ..., n} for any natural number n; these are called infinite sets, while those sets for which such a bijection does exist (for some n) are called finite sets. Infinite sets cannot be counted in the usual sense; for one thing, the mathematical theorems which underlie this usual sense for finite sets are false for infinite sets. Furthermore, different definitions of the concepts in terms of which these theorems are stated, while equivalent for finite sets, are inequivalent in the context of infinite sets.

The notion of counting may be extended to them in the sense of establishing (the existence of) a bijection with some well understood set. For instance, if a set can be brought into bijection with the set of all natural numbers, then it is called "countably infinite." This kind of counting differs in a fundamental way from counting of finite sets, in that adding new elements to a set does not necessarily increase its size, because the possibility of a bijection with the original set is not excluded. For instance, the set of all integers (including negative numbers) can be brought into bijection with the set of natural numbers, and even seemingly much larger sets like that of all finite sequences of rational numbers are still (only) countably infinite. Nevertheless there are sets, such as the set of real numbers, that can be shown to be "too large" to admit a bijection with the natural numbers, and these sets are called "uncountable." Sets for which there exists a bijection between them are said to have the same cardinality, and in the most general sense counting a set can be taken to mean determining its cardinality. Beyond the cardinalities given by each of the natural numbers, there is an infinite hierarchy of infinite cardinalities, although only very few such cardinalities occur in ordinary mathematics (that is, outside set theory that explicitly studies possible cardinalities).

Counting, mostly of finite sets, has various applications in mathematics. One important principle is that if two sets X and Y have the same finite number of elements, and a function f: X → Y is known to be injective, then it is also surjective, and vice versa. A related fact is known as the pigeonhole principle, which states that if two sets X and Y have finite numbers of elements n and m with n > m, then any map f: X → Y is not injective (so there exist two distinct elements of X that f sends to the same element of Y); this follows from the former principle, since if f were injective, then so would its restriction to a strict subset S of X with m elements, which restriction would then be surjective, contradicting the fact that for x in X outside S, f(x) cannot be in the image of the restriction. Similar counting arguments can prove the existence of certain objects without explicitly providing an example. In the case of infinite sets this can even apply in situations where it is impossible to give an example; for instance there must exists real numbers that are not computable numbers, because the latter set is only countably infinite, but by definition a non-computable number cannot be precisely specified.

The domain of enumerative combinatorics deals with computing the number of elements of finite sets, without actually counting them; the latter usually being impossible because infinite families of finite sets are considered at once, such as the set of permutations of {1, 2, ..., n} for any natural number n.

See also: counting games

See also[link]

References[link]

External links[link]

• History of Counting-PlainMath.Net

19 Kids and Counting (formerly 17 Kids and Counting and 18 Kids and Counting) is an American reality television show on TLC. The show is about the Duggar family, which consists of parents Jim Bob and Michelle Duggar and their 19 children—nine girls and ten boys (including two sets of fraternal twins), who all have names beginning with the letter "J". The series began on September 29, 2008.^[1] Season nine ended on March 27, 2012.^[2]

Series overview[link]

Background[link]

The family lives in Tontitown, Arkansas, and originally appeared in several TLC and Discovery Health one-hour specials, most of which focused on four of Michelle's last five deliveries.^[3][4][5] Jim Bob has one older sister, Deanna, who (along with her daughter Amy) occasionally appears on the show. Michelle has six siblings. Jim Bob and Michelle met in the early 1980s when Jim Bob, along with another church member, were sent for a follow-up visit after Michelle had experienced a religious conversion.^[6] Jim Bob and Michelle were married on July 21, 1984.^[7]

The Duggars elected to delay having children and practiced birth control. It was three years before Josh, their eldest child, was born. They then resumed using birth control; despite precautions, they conceived again but Michelle had a miscarriage. On December 9, 2011, Michelle revealed that they had named the miscarried child Caleb, even though they did not know whether the baby was a boy or a girl.^[8] Believing the miscarriage was due to the contraceptive, the Duggars quit birth control and embraced the Quiverfull movement.^[9] As they explained, they "decided to allow God to determine the number of children" they would have. Shortly thereafter, Michelle became pregnant again, this time with her first set of twins, Jana and John-David. Thereafter, Michelle has given birth approximately every year and a half.

The Duggars are conservative Christians, and due to their fundamentalist biblical beliefs, they practice the following: only watching wholesome family television on DVD, and various historical events; restricted Internet use; wearing modest clothing—which includes modest-wear swimsuits, and for the girls, skirts that go below the knees; male family members keeping their hair cut short, and the females keeping their hair long. Instead of dating, they practice chaperoned courtship, which includes the couple abstaining from physical contact. In the event of engagement, the suitor seeks permission from his girlfriend's father before asking her to be his wife. The children are home-schooled using a mix of materials, including those of Switched On Schoolhouse, Institute in Basic Life Principles (IBLP), and Accelerated Christian Education (ACE).^[10] For college studies, several of the older kids use the CollegePlus! program.^[11]

The Duggars raise their children using a buddy system, wherein an older sibling is assigned to a younger sibling and assists in their primary care. According to Michelle, "they help them with their little phonics lessons and games during the day and help them practice their music lessons. They will play with them or help them pick out the color of their outfit that day and just all of those types of things."^[12] In 2004, Michelle Duggar won the "Young Mother of the Year Award" in Arkansas, which is sponsored by American Mothers Incorporated.^[13]

Jim Bob served in the Arkansas House of Representatives from 1999 to 2002. Currently, he is a real estate agent and investor. The Duggars' income is derived from the rental proceeds of the commercial properties they own.^[14][15] They live debt-free,^[16][17] which Jim Bob has said is "the fruit of Jim Sammons' Financial Freedom Seminar" he attended years ago (Sammons' Seminar is endorsed by IBLP). The construction of their 7,000 square foot (650 square meter) house was started by the family in 2000 when they bought the land and ordered the frame. Discovery Networks completed it, by finding local Arkansas construction workers to donate their skills and time. The home was completed on January 20, 2006. The painting, decorating, furnishings, appliances, and other finishing touches—such as a stocked pantry—were provided by Discovery Networks and corporate sponsors as part of the one-hour television special^[15] entitled 16 Children and Moving In.

During the series[link]

Jim Bob and Michelle Duggar

On December 10, 2009, Michelle gave birth via emergency C-section to Josie Brooklyn Duggar, three months prematurely. Michelle was rushed to the hospital for gallstones and it was there that doctors discovered she had pre-eclampsia and performed an emergency delivery. Josie weighed 1 lb, 6oz at birth. A week after the birth, Michelle reported that Josie was "doing as well as can be expected for a baby at her age. We are taking each hour as it comes." On April 6, 2010, after nearly four months in the hospital, Michelle and Jim Bob were able to take Josie to their rental in Little Rock, while they awaited the okay to take her home to Tontitown.^[18][19] However, on April 8, 2010, Josie Duggar was readmitted to the hospital after her vital signs dropped. Fortifiers that had been added to breast milk to help her gain weight appeared to be the cause.^[20]

The family was able to take Josie home to Tontitown on June 23, 2010. Michelle Duggar reported that Josie is doing very well, and weighs about 9 pounds. "She has a double chin now," Michelle said, "It is so precious." Josie reportedly weighed in at a healthy 15 lbs. 12 ounces just over 5 months later in December 2010.^[21]

The family had discussed the possibility of having more children in an May 2010 interview with Radar Online.^[22] Michelle was quoted as saying:

"We would love more! I'm 43, almost 44 this September. I know that my mommy years are probably numbered, and I don't know how many more children God will see fit to give me. "It is something we've been praying about because we do love children. Each child really is a gift and that doesn't mean just our children. We asked the Lord to give us a love of children the way He loves children. That is something that we've prayed about, and we'll just see what the Lord has in store for our family in the future."

On November 8, 2011, Michelle Duggar and family appeared on NBC's Today Show to announce that she was expecting their 20th child.^[23] On December 8, 2011, Michelle and Jim Bob Duggar revealed that she had miscarried the baby.^[24] Six days later, they held a memorial service for their unborn girl whom they named Jubilee Shalom.^[25][26]

Duggar family[link]

^[27]

Parents[link]

James Robert "Jim Bob" Duggar [born (1965-07-18) July 18, 1965 (age 46)]

Michelle Annette Duggar (née Ruark) [born (1966-09-13) September 13, 1966 (age 45)]

Children[link]

Josh's family[link]

Guests appearing[link]

Family members[link]

Non-family[link]

The most often recurring guests (other than relatives) are the Bates family from Tennessee. Because they are so similar to the Duggars (having 19 kids of their own), they have become close friends.

Other notable guests have been:

• Meredith Vieira of Today
• Kirk Cameron – season 2, episode 10
• Dolly Parton – two episodes in season 3
• Charles Stanley (minister) – season 6, episode 3
• Dan Harris of Good Morning America – season 7, episode 12
• Bethany Hamilton (professional surfer) – season 8, episode 6

Duggar appearances on other shows[link]

^[30]

• The Early Show 2006–2008 (CBS)
• Fox & Friends 2007 (Fox News Channel)
• The Today Show – 10 appearances between 2007 and 2011 (NBC)
• American Morning 2008 & 2009 (CNN)
• Larry King Live February 19, 2009 & April 13, 2009 (CNN)
• Huckabee August 1, 2009 (Fox News Channel)
• Say Yes to the Dress October 2, 2009 (TLC)

Availability[link]

DVDs[link]

Online releases[link]

Episodes of the series are available for download from the iTunes Store, Amazon Instant Video, and Microsoft's Zune Marketplace.

References[link]

External links[link]

• Official site
• Duggar Family's website
• Influence-Music.com (Show's composers Scott Pearson & David Imhof)
• Official family facebook page
• Josh & Anna Duggar's website
• 19 Kids and Counting at the Internet Movie Database
• 19 Kids and Counting at TV.com

Mark D
Birth name	Mark Randall
Nationality	British
Field	Painting
Movement	Stuckism

Mark D, born Mark Randall,^[1] is a British artist and punk musician (guitarist and songwriter). He is also associated with the Stuckist group of artists.

Life and work[link]

Mark D was born and spent his childhood in Peterborough. He now lives in Nottingham. From university onwards, Mark D (D standing for "degenerate") played in various bands including the Fat Tulips, Confetti (when he was known as David), the Pleasure Heads (when he was known as Mark Randyhead), Oscar, Servalan and Sundress, and appeared on dozens of releases. He published and edited fanzines, including the underground C86 fanzine Two Pint Take Home. He is a co-owner of Heaven Records.^[2]

The Fat Tulips were formed in 1987 and have been described as "incredibly skilled in the art of buzzing electric guitars, ferocious tempos, pristine pop melodies and lyrics that weren’t nearly as sweet as they sounded at first listen." ^[3]

Stella Vine: Go Fuck Yourself by Mark D

A few years after the Fat Tulips broke up in the mid-1990s, he started a dealership in art and antiques with his wife, Tully, focusing on work from the 1960s and 1970s, including John Piper, Stig Lindberg, Lucienne Day and John Clappison.

An interest in the music of Billy Childish led onto Childish's paintings, which he acquired, and subsequently also the work of other artists in the Stuckist group. He contacted two artists who had been exhibited by the Stuckists, Stella Vine and Gina Bold, both of whom refused to sell him work, because of his connection with Childish. Vine responded with an email, which said, "Go fuck yourself".^[1][2]

The Hypocrisy of Myners, a painting by Mark D, satirising the Tate gallery (Paul Myners was the Tate Chairman at the time).

Frustrated at not being able to acquire that work for his collection and inspired by an Andy Warhol quote, "If you want to sell paintings you have to make paintings", Mark D began in May 2005 to make his own paintings, for which he had no formal training or qualifications ("my only qualification is a love of art"). He was encouraged and given advice by Stuckist co-founder, Charles Thomson, without whom Mark D said, "I could easily have given up on doing my own work."^[4]

His paintings make visual and verbal comments on well-known figures in the art world, starting with Billy Childish, Tracey Emin, Stella Vine and Damien Hirst. He describes Emin as "without doubt a very talented artist whose work I greatly admire."^[2] In response to the controversy over the Tate's purchase of its trustee Chris Ofili's work, The Upper Room, he painted a picture of the Tate Chairman, Paul Myners—who had defended the purchase—called The Hypocrisy of Myners. A reproduction of this work was featured on a placard outside the Turner Prize in December 2005 during a Stuckist demonstration against the purchase.

Mark D's painting in the window of the Stuckist show, "Painting is the Medium of Yesterday", 2005.

Mark D first exhibited in September 2005 with the Stuckists at La Viande gallery, Shoreditch, in a show ''Painting Is the Medium of Yesterday"—Paul Myners CBE, Chairman of Tate Gallery, Chairman of Marks and Spencer, Chairman of Aspen Insurance, Chairman of Guardian Media, Director of Bank of England, Director of Bank of New York. A Show of Paintings by the Stuckists, as Refused by the Tate Gallery. Guaranteed 100% Free of Elephant Dung.^[2]

Mark D. Victoria Beckham: America Doesn't Love Me.

His solo show, Mark D and the Stuckists v Tracey Emin and Damien Hirst, was at the Art Organisation gallery in March 2007. It included a display of material about Stuckist demonstrations.^[5] As well as his own paintings he showed paintings by other Stuckist artists from his collection,^[2] which he jointly owns with his wife, Tully, and states is "the world's largest collection of Stuckist paintings".^[6]

In February 2008, he staged a show of his paintings, which satirise Stella Vine's images of Princess Diana and Kate Moss, replacing the former with Victoria Beckham and showing Moss eating slugs.^[1] The exhibition, held jointly with Abby Jackson at La Viande gallery in Shoreditch, London, was called Disney Heroines Committing Suicide.^[7]

He names Alfred Wallis, John Bellany, Bryan Pearce, and Stella Vine as influences, and sums up his artistic and musical philosophy with a derivation from punk:

Some outstanding musicians have proved that all you need are 3 chords and a cheap guitar to come up with a classic. I find it much more interesting to listen to some low budget production record full of life and energy with lots of good ideas, (e.g. early Clash and Velvet Underground, The Vaselines, The White Stripes and, of course, Billy Childish with the Buff Medways etc.) than some over-produced bland pomp like Athlete, Coldplay or U2.^[2]

Charles Thomson described his work as "brilliant punk paintings".^[8]

Gallery[link]

• 

  Mark D. Kate Moss: Kate Loves Slugs.

• 

  Heather Mills: Beauty and the Beatles Money

• 

  Paris Hilton: Panties Please Paris

• 

  Tracey Emin:I Could Be Carrie

• 

  Billy Childish: Truth Lies and Audiotape

• 

  Gina Bold: I Blame Charles

• 

  Damien Hirst: Money for Old Rope

• 

  Shed Boat Shed Shed

See also[link]

Mark D's painting on a placard during a Stuckists demonstration at Tate Britain, December 6, 2005.

• Stuckism
• The Upper Room (paintings)
• Fat Tulips
• Confetti (band)

References[link]

External links[link]

Wikimedia Commons has media related to: Mark D

• Mark D paintings on the Stuckism International site
• The Fat Tulips official site
• The Fat Tulips discography