Mathematical fallacy

In mathematics, certain kinds of mistaken proof are often exhibited, and sometimes collected, as illustrations of a concept of mathematical fallacy. There is a distinction between a simple mistake and a mathematical fallacy in a proof: a mistake in a proof leads to an invalid proof just in the same way, but in the best-known examples of mathematical fallacies, there is some concealment in the presentation of the proof. For example, the reason validity fails may be a division by zero that is hidden by algebraic notation. There is a striking quality of the mathematical fallacy: as typically presented, it leads not only to an absurd result, but does so in a crafty or clever way. Therefore, these fallacies, for pedagogic reasons, usually take the form of spurious proofs of obvious contradictions. Although the proofs are flawed, the errors, usually by design, are comparatively subtle, or designed to show that certain steps are conditional, and should not be applied in the cases that are the exceptions to the rules.

Donald Trump

Donald John Trump, Sr. (born June 14, 1946) is an American businessperson and media personality. He is the chairman and president of The Trump Organization and the founder of Trump Entertainment Resorts. Trump's career, branding efforts, personal life, wealth, and outspoken manner have made him famous throughout the country. Since 2015, he is also a candidate for the Republican nomination for President of the United States in the 2016 election.

Trump is a native of New York City and a son of Fred Trump, who inspired him to enter real estate development. After two years at Fordham University and while studying at Wharton School of the University of Pennsylvania, Trump worked for his father's firm, Elizabeth Trump & Son. Upon graduating in 1968 he joined the company, and in 1971 was given control, renaming the company "The Trump Organization". Since then he has built hotels, golf courses, and other properties, many of which bear his name. He is a major figure in the American business scene and has received prominent media exposure. The NBC reality show The Apprentice bolstered his fame, and his three marriages were extensively reported in tabloids.

Kanye West

Kanye Omari West (/ˈkɑːnjeɪ/; born June 8, 1977) is an American recording artist, record producer, fashion designer and entrepreneur. He is the founder of the record label GOOD Music and creative content company DONDA. West is one of the most acclaimed musicians of the 21st century. He has attracted both praise and criticism for his work and his controversial, outspoken public persona.

Raised in Chicago, West first became known as a producer for Roc-A-Fella Records in the early 2000s, producing hit singles for musical artists such as Jay-Z and Alicia Keys, before pursing a solo career as a rapper. In 2004, he released his debut album The College Dropout, to widespread commercial and critical success, and founded record label and management company GOOD Music. He went on to explore a variety of different musical styles on subsequent albums that included the baroque-inspired Late Registration (2005), the electronic-tinged Graduation (2007), and the starkly polarizing 808s & Heartbreak (2008). In 2010, he released his critically acclaimed fifth album My Beautiful Dark Twisted Fantasy. He collaborated with Jay-Z on Watch the Throne (2011), and released his sixth album, Yeezus, to further critical praise in 2013. Following a series of recording delays and work on non-musical projects, West's seventh album, The Life of Pablo, was released in 2016.

Paul McCartney

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Sir James Paul McCartney, MBE (born 18 June 1942) is an English singer-songwriter, multi-instrumentalist, and composer. With John Lennon, George Harrison, and Ringo Starr, he gained worldwide fame as the bassist of the rock band the Beatles, one of the most popular and influential groups in the history of pop music; his songwriting partnership with Lennon is one of the most celebrated of the 20th century. After the band's break-up, he pursued a solo career and formed Wings with his first wife, Linda, and Denny Laine.

McCartney has been recognised as one of the most successful composers and performers of all time, with 60 gold discs and sales of over 100 million albums and 100 million singles of his work with the Beatles and as a solo artist. More than 2,200 artists have covered his Beatles song "Yesterday", more than any other copyrighted song in history. Wings' 1977 release "Mull of Kintyre" is one of the all-time best-selling singles in the UK. A two-time inductee into the Rock and Roll Hall of Fame (as a member of the Beatles in 1988, and as a solo artist in 1999), and a 21-time Grammy Award winner (having won both individually and with the Beatles), McCartney has written, or co-written 32 songs that have reached number one on the Billboard Hot 100, and as of 2014 he has sold more than 15.5 million RIAA-certified units in the United States. McCartney, Lennon, Harrison and Starr received MBEs in 1965, and in 1997, McCartney was knighted for his services to music.