Application For Honors English 9 Please Print Name: The: Fill & Download for Free

To Whom It May Concern,Is it true that Steven C. Harper and others in the history department of the Mormon church are transitioning the narrative about the Book of Mormon, so that it came through "revelation" and not from "translation"?Let’s First Identify the Person in Question“Steven C. Harper (M.A., Utah State University; Ph.D., Lehigh University) is an Associate Professor in the Department of Church History and Doctrine at Brigham Young University and a volume editor in the Joseph Smith Papers project. He previously taught at Brigham Young University-Hawaii, and among the honors he has received are the Juanita Brooks Award for Best Graduate Student Paper and the 1999 T. Edgar Lyon Award for the Best Article of the Year, both given by the Mormon History Association.Posted December 2010″ (Steven C. Harper - FairMormon)HISTORY AND THE BOOK OF MORMON(Problems with the Book of Mormon)(Problems with the Gold Plates of the Book of Mormon - Christian Research Institute)(https://www.apologeticspress.org/rr/reprints/Book-of-Mormon.pdf)The real issue is the Mormon Gospel and the Christian GospelPaul said it clearly,No Other Gospel6I am astonished that you are so quickly deserting the one who called you to live in the grace of Christ and are turning to a different gospel— 7which is really no gospel at all. Evidently, some people are throwing you into confusion and are trying to pervert the gospel of Christ. 8But even if we or an angel from heaven should preach a gospel other than the one we preached to you, let them be under God’s curse! 9As we have already said, so now I say again: If anybody is preaching to you a gospel other than what you accepted, let them be under God’s curse!10Am I now trying to win the approval of human beings, or of God? Or am I trying to please people? If I were still trying to please people, I would not be a servant of Christ. Galatians 1:6–10The Great Commission16Then the eleven disciples went to Galilee, to the mountain where Jesus had told them to go. 17When they saw him, they worshiped him; (they worshiped Jesus as God) but some doubted.18Then Jesus came to them and said, “All authority in heaven and on earth has been given to me. 19Therefore go and make disciples of all nations, baptizing them in the name of the Father and of the Son and of the Holy Spirit, (Trinity see singular name) 20and teaching them to obey everything I have commanded you. And surely I am with you always, to the very end of the age.”Who is Jesus? God in the FleshThe Apostle John wrote:The Word Became Flesh1In the beginning was the Word, and the Word was with God, and the Word was God. 2He was with God in the beginning. 3Through him all things were made; without him nothing was made that has been made. 4In him was life, and that life was the light of all mankind. 5The light shines in the darkness, and the darkness has not overcome a it.6There was a man sent from God whose name was John. 7He came as a witness to testify concerning that light, so that through him all might believe. 8He himself was not the light; he came only as a witness to the light.9The true light that gives light to everyone was coming into the world. 10He was in the world, and though the world was made through him, the world did not recognize him. 11He came to that which was his own, but his own did not receive him. 12Yet to all who did receive him, to those who believed in his name, he gave the right to become children of God— 13children born not of natural descent, nor of human decision or a husband’s will, but born of God.14The Word became flesh and made his dwelling among us. We have seen his glory, the glory of the one and only Son, who came from the Father, full of grace and truth. John 1:1–14Jon Gary Williams writes:“The Mormon church promotes the Book of Mormon as an inspired writing. However, after a fair and candid examination of the evidence, many thousands of honest Mormons have discovered that this claim simply is not true. What have they learned that caused them to reject the Book of Mormon?”People group issues Nephites ( dark skin the ancestors of American Indians) and Lamanites (destroyed). There is no historical support for this.Non-existence golden plates. The Bible has both the archaeological, textual, event and cultural context.Translation issueThe revelation in the Book of Mormon contradicts the Bible teaching.TranslationMisspelled words “journied” (for journeyed; 1 Nephi 4:38; 5:6; 7:6; 18:25; 2 Nephi 5:7; Omni 1:16) • “bellowses” (for bellows; 1 Nephi 17:11) • “feading” (for feeding; Enos 1:20) • “sayeth” (for saith; Mosiah 12:21) • “bablings” (for babblings; Alma 1:32) • “tempels” (for temples; Alma 16:13) • “yars” (for years; Alma 19:16) • “phrensied” (for frenzied; Alma 30:16) • “eigth” (for eighth; Alma 53:23) • “adhear” (for adhere; Alma 60:34) • “eatheth” (for eateth; 3 Nephi 20:8) • “rereward” (for rearward; 3 Nephi 20:42; 21:29)Joseph Smith himself said that any mistakes were corrected as they were detected. The misspelled words alone show any claim of inspiration for the Book of Mormon is fraudulent.Errors in grammar “Behold, for none of these I cannot hope” (2 Nephi 33:9). “And Mosiah, nor the people of Mosiah, could not understand them” (Omni 1:17). “And now behold the Lamanites could not retreat neither way” (Helaman 1:31). “Yea, if my days could have been in them days” (Helaman 7:8). “And it came to pass that there was certain men passing by” (Helaman 7:11). “That all might see the writing which he had wrote” (Alma 46:19). “That all might see the writing which he had wrote” (Alma 46:19). “I would cite your minds forward to the time when the Lord gave these commandments” (Alma 13:1). [still exists in recent editions] “They did not fight against God no more” (Alma 23:7). “I have wrote to them” (3 Nephi 26:8). • “I were about to write to them” (3 Nephi 26:11). • “...the gates of hell is...” (3 Nephi 18:13). “...the multitude had all eat” (3 Nephi 20:9). • “I Moroni have written the words which was commanded” (Ether 5:1). “The law had ought to be done away” (2 Nephi 25:27). “...which was wrote upon the plates...” (Alma 44:24). “Adam and Eve, which was our first parents...,” (1 Nephi 5:11). “...who was the most foremost among them” (Alma 32:5). “...that there might not be no more sorrow” (Alma 29:2). “And this he done” (Alma 2:10). In the first chapter alone of the Book of Mormon (1 Nephi 1, which has only 20 verses), there are no less than four such errors. Two examples are: “My father had read and saw,” and “the tender mercies of the Lord is....”Redundant Phrases and Words - The phrase, “and it came to pass,” is so common in the Book of Mormon (occurring over 1200 times) that it makes the reading monotonous. In 1 Nephi alone this phrase is used almost 200 times. In 1 Nephi 3:21-29 it is used to begin eight of the nine verses. In the sixty-nine verses of Ether 14 and 15, it is used thirty-nine times. Another phrase duplicated beyond reason is “the more part.” Is it reasonable to suggest that God, through inspiration, would employ such awkward repetition? • Numerous words in the Book of Mormon are overused. For example, the word “behold” is used repeatedly with no real need. Likewise the word “thereof ” is used to excess, as is the word “insomuch.” These and many similar words confirm that the writer was deficient in vocabulary. • Often the Book of Mormon is wordy, using far too many words to express a simple thought. A good illustration is 4 Nephi 6: “And thus did the thirty and eight years pass away, also the thirty and ninth, and forty and first, and the forty and second, yea even until forty and nine years had passed away, and also the fifty and second; yea, and even until fifty and nine years had passed away.” That is quite a mouthful just to say that 59 years had passed! Could anyone imagine the Bible reading like this? The writer definitely tended to ramble. • The use of colloquial terms is apparent in the Book of Mormon. The frequent use of “a” with various participles is noticeable in such phrases as: “a journeying,” “a preaching,” “a marching,” “a pointing,” “a preparing” “a coming.” Such terminology betrays the influence of the vernacular of the 1800s, and is not the language one would expect to find in writings of ancient times.Fabricated Word - “preparator” (for preparer; 1 Nephi 15:35) [still exists in later editions] • “arriven” (for arrived; Alma 20:30) • “numerority” (for a vast number; Alma 56:10) • “molten” (for melt; Ether 3:1) [still exists in recent editions] • “flatter” (for allure and instigate; Alma 52:19 and Helaman 1:7) • “enormity” (for enormous; Alma 52:5) “arrested” (for wrested; Alma 41:1) - 7 - • “consigned” (for convinced; Helaman 7:9) • “repair” (for recompense; Alma 27:8) [still exists in later editions] • “ezrom,” “senine,” and “limnah” (coins; Alma 11:6 and Alma 11:3) • “cimeter” (a sword; Enos 1:20) • “neas” (a plant; Mosiah 9:9) • “sheum” (a crop; Mosiah 9:9) • “deseret” (a bee; Ether 2:3) • “curelom” and “cummons” (unidentified animals; Ether 9:19)Fraudulent Language - 1 Nephi 18:25, dated at 589 B.C., speaks of “horses” and “asses.” But, these animals were unknown in the Western Hemisphere until the Spaniards introduced them about 450 years ago. Can anyone honestly believe that such a bungled mistake occurred as a result of divine revelation? • Ether 9:19 speaks of “elephants” being in America when the Jaredites arrived, which was supposed to have been around 2250 B.C. However, it is a well-known fact that elephants were not native to America. To suggest that they were is absurd, and proves the Book of Mormon to be erroneous. If someone were to argue that elephants might possibly have been brought to America in the Jaredites’ boats, such an argument could be disproved easily since elephants were not native to Bible lands either. • Surprising as it may seem, no less than six times the Book of Mormon employs the abbreviation “&c” (and so forth), a usage peculiar to the nineteenth century (subtitle of 2 Nephi; Jacob 1:11; Mosiah 8:8; 23:5; Alma 3:5). It can hardly be suggested that such a symbol is a “translation” from ancient writings. This kind of mistake is clear and compelling evidence of the recent origin of the book. • In Jacob 7:27, the French word adieu occurs. But how could a modern French word have found its way into those ancient plates? This is additional evidence of fraud, and presents grounds for rejecting the Book of Mormon. • In Jacob 3:11 and Mosiah 29:14, the word “faculties” appears. However, this is a term dating back no earlier than middle English. Strange, indeed, that it would be “translated” from a word on an ancient plate dating over 1,000 years earlier.2 Nephi 29:3 reads, “A Bible, A Bible, we have got a Bible...” This statement is made in reference to the Jewish Old Testament, which is dated at about 550 B.C. However, the word “Bible” is the English transliteration of the Greek term “biblos,” which came into use over 1,500 years later. In using the word “Bible,” the writer of the Book of Mormon made a serious blunder that shows the book to be of recent origin and, hence, fraudulent in its claims. • 3Nephi 15:21 is a word-for-word quote of John 10:16 (from the King James Version). However, this version is somewhat less than 400 years old. And, to make matters worse, the Book of Mormon even quotes the italicized word “and” that was supplied by the King James translators. Here, the writer of the Book of Mormon unwittingly demonstrates his work to be plagiarism. • The entire fourteenth chapter of Mosiah, made up of 12 verses, is a duplication of Isaiah 53:1-12. Interestingly, all eleven of the italicized words in the King James text are quoted, yet none is placed in italics, which indicates that the writer of the Book of Mormon apparently was unaware that the KJV translators used italics to highlight words that were not in the original manuscripts employed in the translation process. Thus, Mosiah 14 had to have been copied from the King James Bible. • Moroni 7:45, which is a quotation of 1 Corinthians 13:4-7 in the King James Version, is another example of fraud. In citing this verse, the writer included the italicized word “easily” (“...is not easily provoked”). However, the word “easily” is not in the original, but was placed there (incorrectly) by the King James translators. [It is omitted, correctly, from later versions.] That the writer included this word shows that Moroni 7:45 was copied from the KJV. • In 2 Nephi 31:13 and other places, reference is made to the “Holy Ghost.” But, the term “ghost” did not come into use until many hundreds of years after the Book of Mormon was supposed to have been inscribed on ancient plates. That the writer borrowed this from the King James Bible is indisputable. • The word “baptism” is found in 2 Nephi 31:13 and other places. But this cannot be an actual translation of a word found on ancient plates, because “baptism” is a transliteration of the Greek word baptisma, and was peculiar to the King James Version. This word is clearly a copy of an early English term, demonstrating again the fraudulent nature of the Book of Mormon. • The word “epistle” in 3 Nephi 3:5 is an obvious copy from the King James Version. Like baptism, the word “epistle” (epistolos) was left in its original Greek form, but given an English ending. This shows the writer of the Book of Mormon was not very careful in selecting his words. • The words “alpha” and “omega” appear in 3 Nephi 9:18. These, of course, are the English spellings of Greek words found in the Bible (Revelation 1:8; 21:6; 22:13). Since the Book of Mormon was not recorded in Greek, why were these words used? The simple fact is, they were copied from the King James Version. • 3 Nephi 20:23-26, dated at A.D. 34, refers to Moses’ prophecy about the Christ (Deuteronomy 18:15,18-19). However, the writer unwittingly used Peter’s New Testament paraphrase of this prophecy (Acts 2:22-26), which was not written until around A.D. 63. This was almost 30 years too soon, and thus proves the Book of Mormon is a hoax. • In the Book of Mormon there are numerous instances where the writer uses words that were not relevant to his time. Rather, these are words peculiar to the English spoken in the early 1600s (“prayest,” “durst,” “thou,” “thee,” “thy,” “thine,” “hast,” “doth,” “knoweth,” “hearest,” “cometh,” “thirsteth,” etc.). Did God really select these words for the Book of Mormon? This obviously shows the writer’s exposure to King James terminology. [NOTE: Scores of passages in the Book of Mormon, either in part or whole, exact or paraphrased, have been taken directly from the King James Version. Some researchers have estimated that as much as 4% can be traced to this English translation.]Changes in Context - In Helaman 1:16, there is an incomplete sentence that shows the writer of the Book of Mormon to be unfamiliar with proper sentence structure. In an effort to correct this embarrassing problem, two words were removed from the text, and a period was changed to a comma. The original reads, “...who was the son of Ammoron. Now Tobiah supposing that....” The corrected edition, however, reads, “who was the son of Ammoron, supposing that....” Obviously, this is not merely a typographical change, but a change to correct bad sentence structure. Would God have allowed such a mistake to be penned under His oversight? • Mosiah 21:28 contains a very serious mistake. In this passage, “king Benjamin” is spoken of as being alive. Apparently, however, the writer forgot that fifteen chapters earlier he recorded this man’s death. What he meant to write was “king Mosiah.” The second edition of the Book of Mormon made this change. How did such a conspicuous mistake slip into a translation being overseen by the power of God? In making this change, the Mormon church has admitted the error. • 1 Nephi 11:18 says of Mary, “Behold, the virgin which thou seest, is the mother of God.” This glaring error was changed in the second printing with the addition of three words. It now reads, “...the mother of the Son of God.” The question is, which version is to be believed—Smith’s original, or the later, corrected edition? • 1 Nephi 11:21 reads, “Behold the Lamb of God, yea, even the Eternal Father.” Such an idea is contrary to current Mormon theology, yet, this is the concept the original Book of Mormon presents. Thus, in order to remove the problem, later editions have changed the text by adding three words— “even the Son of the Eternal Father.” Again, one must ask: which is to be believed—Smith’s original, or the later, corrected edition? In Alma 5:48 is found the phrase, “...yea, the Son of the only begotten of the Father.” That the writer of the Book of Mormon states that Jesus had a son is another example of bungled writing. Later editions replaced the word “of” with a comma, making it read, “yea, the Son, the Only Begotten of the Father.” Can anyone imagine God making such an error?Absurdities - • 3 Nephi 28:4-8 speaks of three men who were told by Jesus that they would never die, but would remain on Earth until His second coming. However, the Bible is clear in explaining that God has “appointed” man to die (Hebrews 9:27). Such a tale is so imaginary many Mormons are reluctant to admit they believe it. • Ether 2:6-25 and Ether 6:6 relate the story of how the Jaredites came to America in eight boats called “barges.” They were said to be “the length of a tree,” whatever that may mean. These were strange barges, for they were “peaked” at the ends in the shape of footballs and were sealed “tight like unto a dish.” They were like submarines, for they were sometimes “buried in the depths of the sea” like “whales.” But the story is made even more absurd by the fact that God forgot to provide any ventilation for the boats. After having this pointed out to Him, He directed the people to put a hole in the top and bottom of each boat. This was commanded so they could alternate plugging and unplugging as they tumbled in the sea. But God also forgot to provide for light. After being reminded, He asked what to do. He was told to make sixteen illuminating stones, two for each boat. Such a story not only is foolish, but is an insult to God’s intelligence in that it portrays Him as completely inept. • Ether 15:30-31 relates one of the most peculiar and bizarre stories in the Book of Mormon. It tells of a man named Coriantumr killing a man named Shiz. “And it came to pass when he had smitten off the head of Shiz, which Shiz raised upon his hands and fell; and after that he had struggled for breath, he died.” This is amusing. Imagine, a decapitated man raising up and trying to breathe! Isn’t it a fact that decapitation brings immediate death? And one is made to wonder if this “struggle” to breathe was from the mouth or the neck? According to 2 Nephi 3:6-18 Joseph (of the Old Testament) allegedly prophesied that Joseph Smith would come from the “fruit of his loins.” In other words, he was to be a direct descendant of Joseph. Such an idea is so silly it hardly needs exposing. His own mother, in her biography of Joseph Smith, traced their family back five generations and 300 years to England (Joseph Smith, The Prophet, pp. 38-44). And, too, the Book of Mormon itself tells that the last of the Jewish descendants in America, the Lamanites, became Indians. Accordingly, Joseph Smith, to be a descendant of the Jews, had to be of Indian blood. [Because it is so unbelievable, Mormon writers claim this prophecy was “figurative.” However, in the inspired language of the Bible the phrase “the fruit of his loins” always refers to literal offspring.] 1 Nephi 5:14-16 tells that Lehi, a devout Jew, did not know from whom he descended. However, after reading some long lost plates, he “did discover the genealogy of his fathers” and “knew that he was a descendant of Joseph.” Now, since it is well known that the Jews were very careful in acknowledging and recording their ancestry, can it be believed that a Jew who “dwelt in Jerusalem all his days” (1 Nephi 1:4) would not be aware that he was descended from Joseph? Yet, this is what the Book of Mormon reports. It is evident that the writer of this book was not familiar with Jewish custom regarding ancestral records.Contradicts the Bible - The Bible relates that at the crucifixion there were three hours of darkness (Luke 23:44). However, the Book of Mormon states there was darkness “for the space of three days” (Helaman 14:20,27). Of course, this is a big difference. Which one is true? Can God be responsible for conflicting statements such as these? • The Book of Mormon relates that at the tower of Babel the Jaredites had their separate language (Esther 1:34-35). The Bible, however, plainly states that “the whole earth was of one language” (Genesis 11:1). Apparently, the writer of the Book of Mormon mistakenly thought there were many different languages and that God confounded them while sparing the language of the Jaredites. The fact is, there was only one language and God confounded the people by creating different languages. • 3 Nephi 11:8-10 claims that after Jesus ascended to heaven, He appeared in America to the Lamanites and Nephites in A.D. 34. However, this clearly contradicts the Bible. Of the ascension of Christ to heaven God said, “Sit thou on my right hand until I make thy foes thy footstools” (Acts 2:34-35). How long was He to be in heaven at God’s right hand? “For he must reign, until he hath put all enemies under his feet. The last enemy that shall be destroyed is death” (1 Corinthians 15:25-26). Furthermore, in referring to the ascension, the Bible speaks of Christ as He “whom the heavens must receive until the restitution of all things” (Acts 3:21). Christ did not come to America—because He has been in heaven since His ascension. • In the Bible, the name “Jesus” was announced first by an angel (to Mary; Luke 1:31). This was in 1 B.C. However, Alma 19:29, dated in the Book of Mormon at 90 B.C., has a woman speaking to the Lord and calling him “Jesus.” Which account is accurate? The Bible teaches that the church was established on the Day of Pentecost following the resurrection of Christ (Acts 2), which would have been approximately A.D. 33. However, Mosiah 18:17, dated at 145 B.C., has the church already in existence. This represents quite a discrepancy, to say the least. - 12 - Obviously, both cannot be correct. [NOTE: This same reference from Mosiah also mentions “baptism” as the means of being added to the church. However, according to the Bible the baptism taught by Christ did not begin until New Testament times (Matthew 28: 19).] The Bible clearly reveals that the disciples of Christ “were called Christians first in Antioch” (Acts 11:26). This was approximately A.D. 40. However, Alma 46:13,15, dated at 73 B.C., has people already wearing the name “Christian”—which represents a difference of over 100 years. Which account are people to believe? • The Book of Mormon teaches that “Melchizedek...did reign under his father” (Alma 13:18). Yet the Bible reveals that Melchizedek was a priest under no one. His priesthood typified the priesthood of Christ, and therefore was unique. In contrast to the Book of Mormon, the Bible states that Melchizedek was “without father, without mother, without descent”—emphasizing that he did not inherit his priesthood (Hebrews 7:3). The writer of the Book of Mormon did not know his Bible very well. • Ether 3:8-9 speaks of God having “flesh and blood.” Yet the Bible states clearly that God is a “spirit,” and thus does not possess a material body (John 4:24). In Doctrine and Covenants, another allegedly inspired writing of the Mormon church, Joseph Smith wrote: “The Father has a body of flesh and bones as tangible as man’s” (section 130:22). The Bible and the Book of Mormon both cannot be correct on this point. • Mosiah 2:3 states: “And they also took of the firstlings of their flocks, that they might offer sacrifice and burnt offerings, according to the law of Moses.” This represents a blatant contradiction with the Bible, because the firstlings of the flocks were to be reserved for the Lord and given to the priests. They were never used for sacrifice (see Exodus 13:2,12; 22:29-30; Numbers 3:13; 18:15-18; 2 Samuel 24:24). 3 Nephi 18:28-29 speaks of those who are “unworthy” to partake of the communion, and suggests that such people should be forbidden to partake. This is an obvious reference to Paul’s discussion in 1 Corinthians 11:27-29 (in fact, the Book of Mormon has a footnote to that effect). The application made here, however, shows that the writer of the Book of Mormon did not understand what Paul was saying, and as a result ended up with a serious misinterpretation. Paul was not discussing man’s personal worthiness, or lack thereof. Rather, he was discussing the manner in which the communion was being partaken. The context makes this clear. The word “unworthily” is an adverb of manner, and points to the way or manner in which a thing is done. The church at Corinth was abusing the communion in the manner they were observing it by turning it into a common meal. The American Standard Version evokes the correct idea by employing the phrase, “in an unworthy manner.” By misinterpreting the word “unworthy,” the writer unknowingly demonstrated that his work was manmade. (source: https://www.apologeticspress.org/rr/reprints/Book-of-Mormon.pdf)

Nein….. CloseTo all our readers in the U.S., We will get straight to the point: This Wednesday we ask you to protect Wikipedia's independence. Time is running out in 2017 to help us. We depend on donations averaging about $15. Only a tiny portion of our readers give. If everyone reading this gave $3, we could keep Wikipedia thriving for years to come. The price of your Wednesday coffee is all we need. When I made Wikipedia a non-profit, people warned me I’d regret it. Over a decade later, it’s the only top ten site run by a non-profit and a community of volunteers. Has it crossed my mind how much we could have made if it had ads? Sure. But it wouldn’t be the same. We wouldn’t trust it. Most people ignore my messages. But I hope you’ll think about how useful it is to have unlimited access to reliable, neutral information. Please help keep Wikipedia online and growing. Thank you. — Jimmy Wales, Wikipedia FounderHow often would you like to donate?Just OnceGive MonthlySelect an amount (USD)$3$5$10$20$30$50$100Select a payment methodSecure transactionMaybe laterNatural numberFrom Wikipedia, the free encyclopediaThis article is about “positive integers” and “non-negative integers”. For all the numbers …, −2, −1, 0, 1, 2, …, see Integer.Natural numbers can be used for counting (one apple, two apples, three apples, …)Real numbers (ℝ) include the rational (ℚ), which include the integers (ℤ), which include the natural numbers (ℕ)In mathematics, the natural numbers are those used for counting (as in "there are six coins on the table") and ordering (as in "this is the third largest city in the country"). In common language, words used for counting are "cardinal numbers" and words used for ordering are "ordinal numbers".Some definitions, including the standard ISO 80000-2,[1]begin the natural numbers with 0, corresponding to the non-negative integers 0, 1, 2, 3, …, whereas others start with 1, corresponding to the positive integers 1, 2, 3, ….[2][3][4][5]Texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, but in other writings, that term is used instead for the integers (including negative integers).[6]The natural numbers are the basis from which many other number sets may be built by extension: the integers, by including (if not yet in) the neutral element 0 and an additive inverse (−n) for each nonzero natural number n; the rational numbers, by including a multiplicative inverse (1/n) for each nonzero integer n (and also the product of these inverses by integers); the real numbers by including with the rationals the limits of (converging) Cauchy sequences of rationals; the complex numbers, by including with the real numbers the unresolved square root of minus one (and also the sums and products of thereof); and so on.[7][8]These chains of extensions make the natural numbers canonically embedded (identified) in the other number systems.Properties of the natural numbers, such as divisibility and the distribution of prime numbers, are studied in number theory. Problems concerning counting and ordering, such as partitioning and enumerations, are studied in combinatorics.In common language, for example in primary school, natural numbers may be called counting numbers[9]both to intuitively exclude the negative integers and zero, and also to contrast the discreteness of counting to the continuity of measurement, established by the real numbers.The natural numbers can, at times, appear as a convenient set of names (labels), that is, as what linguists call nominal numbers, foregoing many or all of the properties of being a number in a mathematical sense.Contents1 History 1.1 Ancient roots 1.2 Modern definitions2 Notation3 Properties 3.1 Addition 3.2 Multiplication 3.3 Relationship between addition and multiplication 3.4 Order 3.5 Division 3.6 Algebraic properties satisfied by the natural numbers4 Generalizations5 Formal definitions 5.1 Peano axioms 5.2 Constructions based on set theory 5.2.1 Von Neumann construction 5.2.2 Other constructions6 See also7 Notes8 References9 External linksHistoryAncient rootsThe Ishango bone (on exhibition at the Royal Belgian Institute of Natural Sciences)[10][11][12]is believed to have been used 20,000 years ago for natural number arithmetic.The most primitive method of representing a natural number is to put down a mark for each object. Later, a set of objects could be tested for equality, excess or shortage, by striking out a mark and removing an object from the set.The first major advance in abstraction was the use of numerals to represent numbers. This allowed systems to be developed for recording large numbers. The ancient Egyptians developed a powerful system of numerals with distinct hieroglyphs for 1, 10, and all the powers of 10 up to over 1 million. A stone carving from Karnak, dating from around 1500 BC and now at the Louvre in Paris, depicts 276 as 2 hundreds, 7 tens, and 6 ones; and similarly for the number 4,622. The Babylonians had a place-value system based essentially on the numerals for 1 and 10, using base sixty, so that the symbol for sixty was the same as the symbol for one, its value being determined from context.[13]A much later advance was the development of the idea that 0 can be considered as a number, with its own numeral. The use of a 0 digit in place-value notation (within other numbers) dates back as early as 700 BC by the Babylonians, but they omitted such a digit when it would have been the last symbol in the number.[14]The Olmec and Maya civilizations used 0 as a separate number as early as the 1st century BC, but this usage did not spread beyond Mesoamerica.[15][16]The use of a numeral 0 in modern times originated with the Indian mathematician Brahmagupta in 628. However, 0 had been used as a number in the medieval computus (the calculation of the date of Easter), beginning with Dionysius Exiguus in 525, without being denoted by a numeral (standard Roman numerals do not have a symbol for 0); instead nulla (or the genitive form nullae) from nullus, the Latin word for "none", was employed to denote a 0 value.[17]The first systematic study of numbers as abstractions is usually credited to the Greek philosophers Pythagoras and Archimedes. Some Greek mathematicians treated the number 1 differently than larger numbers, sometimes even not as a number at all.[18]Independent studies also occurred at around the same time in India, China, and Mesoamerica.[19]Modern definitionsThis section needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.(October 2014)(Learn how and when to remove this template message)In 19th century Europe, there was mathematical and philosophical discussion about the exact nature of the natural numbers. A school[which?]of Naturalism stated that the natural numbers were a direct consequence of the human psyche. Henri Poincaré was one of its advocates, as was Leopold Kronecker who summarized "God made the integers, all else is the work of man".[20]In opposition to the Naturalists, the constructivists saw a need to improve the logical rigor in the foundations of mathematics.[21]In the 1860s, Hermann Grassmann suggested a recursive definition for natural numbers thus stating they were not really natural but a consequence of definitions. Later, two classes of such formal definitions were constructed; later, they were shown to be equivalent in most practical applications.Set-theoretical definitions of natural numbers were initiated by Frege and he initially defined a natural number as the class of all sets that are in one-to-one correspondence with a particular set, but this definition turned out to lead to paradoxes including Russell's paradox. Therefore, this formalism was modified so that a natural number is defined as a particular set, and any set that can be put into one-to-one correspondence with that set is said to have that number of elements.[22]The second class of definitions was introduced by Giuseppe Peano and is now called Peano arithmetic. It is based on an axiomatization of the properties of ordinal numbers: each natural number has a successor and every non-zero natural number has a unique predecessor. Peano arithmetic is equiconsistent with several weak systems of set theory. One such system is ZFC with the axiom of infinity replaced by its negation. Theorems that can be proved in ZFC but cannot be proved using the Peano Axioms include Goodstein's theorem.[23]With all these definitions it is convenient to include 0 (corresponding to the empty set) as a natural number. Including 0 is now the common convention among set theorists[24]and logicians.[25]Other mathematicians also include 0[5]although many have kept the older tradition and take 1 to be the first natural number.[26]Computer scientists often start from zero when enumerating items like loop counters and string- or array- elements.[27][28]NotationThe double-struck capital N symbol, often used to denote the set of all natural numbers (see List of mathematical symbols).Mathematicians use N or ℕ (an N in blackboard bold) to refer to the set of all natural numbers. Older texts have also occasionally employed J as the symbol for this set.[29]This set is countably infinite: it is infinite but countable by definition. This is also expressed by saying that the cardinal number of the set is aleph-naught (ℵ0).[30]To be unambiguous about whether 0 is included or not, sometimes an index (or superscript) "0" is added in the former case, and a superscript "*" or subscript ">0" is added in the latter case:[1]ℕ0 = ℕ0 = {0, 1, 2, …}ℕ* = ℕ+ = ℕ1 = ℕ>0 = {1, 2, …}.Alternatively, natural numbers may be distinguished from positive integers with the index notation, but it must be understood by context that since both symbols are used, the natural numbers contain zero.[31]ℕ = {0, 1, 2, …}.ℤ+= {1, 2, …}.PropertiesAdditionOne can recursively define an addition operator on the natural numbers by setting a + 0 = a and a + S(b) = S(a + b) for all a, b. Here S should be read as "successor". This turns the natural numbers (ℕ, +) into a commutative monoid with identity element 0, the so-called free object with one generator. This monoid satisfies the cancellation property and can be embedded in a group (in the mathematical sense of the word group). The smallest group containing the natural numbers is the integers.If 1 is defined as S(0), then b + 1 = b + S(0) = S(b + 0) = S(b). That is, b + 1 is simply the successor of b.MultiplicationAnalogously, given that addition has been defined, a multiplication operator × can be defined via a × 0 = 0 and a × S(b) = (a × b) + a. This turns (ℕ*, ×) into a free commutative monoid with identity element 1; a generator set for this monoid is the set of prime numbers.Relationship between addition and multiplicationAddition and multiplication are compatible, which is expressed in the distribution law: a × (b + c) = (a × b) + (a × c). These properties of addition and multiplication make the natural numbers an instance of a commutative semiring. Semirings are an algebraic generalization of the natural numbers where multiplication is not necessarily commutative. The lack of additive inverses, which is equivalent to the fact that ℕ is not closed under subtraction (i.e., subtracting one natural from another does not always result in another natural), means that ℕ is not a ring; instead it is a semiring (also known as a rig).If the natural numbers are taken as "excluding 0", and "starting at 1", the definitions of + and × are as above, except that they begin with a + 1 = S(a) and a × 1 = a.OrderIn this section, juxtaposed variables such as ab indicate the product a × b, and the standard order of operations is assumed.A total order on the natural numbers is defined by letting a ≤ b if and only if there exists another natural number c where a + c = b. This order is compatible with the arithmetical operations in the following sense: if a, b and c are natural numbers and a ≤ b, then a + c ≤ b + c and ac ≤ bc.An important property of the natural numbers is that they are well-ordered: every non-empty set of natural numbers has a least element. The rank among well-ordered sets is expressed by an ordinal number; for the natural numbers, this is denoted as ω (omega).DivisionIn this section, juxtaposed variables such as ab indicate the product a × b, and the standard order of operations is assumed.While it is in general not possible to divide one natural number by another and get a natural number as result, the procedure of division with remainder is available as a substitute: for any two natural numbers a and b with b ≠ 0 there are natural numbers q and r such thata = bq + r and r < b.The number q is called the quotient and r is called the remainder of the division of a by b. The numbers q and r are uniquely determined by a and b. This Euclidean division is key to several other properties (divisibility), algorithms (such as the Euclidean algorithm), and ideas in number theory.Algebraic properties satisfied by the natural numbersThe addition (+) and multiplication (×) operations on natural numbers as defined above have several algebraic properties:Closure under addition and multiplication: for all natural numbers a and b, both a + b and a × b are natural numbers.Associativity: for all natural numbers a, b, and c, a + (b + c) = (a + b) + c and a × (b × c) = (a × b) × c.Commutativity: for all natural numbers a and b, a + b = b + a and a × b = b × a.Existence of identity elements: for every natural number a, a + 0 = a and a × 1 = a.Distributivity of multiplication over addition for all natural numbers a, b, and c, a × (b + c) = (a × b) + (a × c).No nonzero zero divisors: if a and b are natural numbers such that a × b = 0, then a = 0 or b = 0 (or both).GeneralizationsTwo important generalizations of natural numbers arise from the two uses of counting and ordering: cardinal numbers and ordinal numbers.A natural number can be used to express the size of a finite set; more precisely, a cardinal number is a measure for the size of a set, which is even suitable for infinite sets. This concept of "size" relies on maps between sets, such that two sets have the same size, exactly if there exists a bijection between them. The set of natural numbers itself, and any bijective image of it, is said to be countably infinite and to have cardinality aleph-null (ℵ0).Natural numbers are also used as linguistic ordinal numbers: "first", "second", "third", and so forth. This way they can be assigned to the elements of a totally ordered finite set, and also to the elements of any well-ordered countably infinite set. This assignment can be generalized to general well-orderings with a cardinality beyond countability, to yield the ordinal numbers. An ordinal number may also be used to describe the notion of "size" for a well-ordered set, in a sense different from cardinality: if there is an order isomorphism (more than a bijection!) between two well-ordered sets, they have the same ordinal number. The first ordinal number that is not a natural number is expressed as ω; this is also the ordinal number of the set of natural numbers itself.Many well-ordered sets with cardinal number ℵ0 have an ordinal number greater than ω (the latter is the lowest possible). The least ordinal of cardinality ℵ0 (i.e., the initial ordinal) is ω.For finite well-ordered sets, there is a one-to-one correspondence between ordinal and cardinal numbers; therefore they can both be expressed by the same natural number, the number of elements of the set. This number can also be used to describe the position of an element in a larger finite, or an infinite, sequence.A countable non-standard model of arithmetic satisfying the Peano Arithmetic (i.e., the first-order Peano axioms) was developed by Skolem in 1933. The hypernatural numbers are an uncountable model that can be constructed from the ordinary natural numbers via the ultrapower construction.Georges Reeb used to claim provocatively that The naïve integers don't fill up ℕ. Other generalizations are discussed in the article on numbers.Formal definitionsPeano axiomsMain article: Peano axiomsMany properties of the natural numbers can be derived from the Peano axioms.[32][33]Axiom One: 0 is a natural number.Axiom Two: Every natural number has a successor.Axiom Three: 0 is not the successor of any natural number.Axiom Four: If the successor of [math] x {\displaystyle x} [/math] equals the successor of [math] y {\displaystyle y} [/math], then [math] x {\displaystyle x} [/math] equals [math] y {\displaystyle y} [/math].Axiom Five (the axiom of induction): If a statement is true of 0, and if the truth of that statement for a number implies its truth for the successor of that number, then the statement is true for every natural number.These are not the original axioms published by Peano, but are named in his honor. Some forms of the Peano axioms have 1 in place of 0. In ordinary arithmetic, the successor of [math][/math][math] [/math][math][/math][math] [/math][math][/math][math] [/math][math][/math][math] x [/math][math][/math][math] [/math][math][/math][math] {\displaystyle x} [/math][math][/math][math] [/math][math][/math][math][/math] is [math][/math][math] [/math][math][/math][math] [/math][math][/math][math] [/math][math][/math][math] x [/math][math][/math][math]+ 1 [/math][math][/math][math] [/math][math][/math][math] {\displaystyle x+1} [/math][math][/math][math] [/math][math][/math][math][/math]. Replacing Axiom Five by an axiom schema one obtains a (weaker) first-order theory called Peano Arithmetic.Constructions based on set theoryMain article: Set-theoretic definition of natural numbersVon Neumann constructionIn the area of mathematics called set theory, a special case of the von Neumann ordinal construction[34]defines the natural numbers as follows:Set 0 = { }, the empty set,Define S(a) = a ∪ {a} for every set a. S(a) is the successor of a, and S is called the successor function.By the axiom of infinity, there exists a set which contains 0 and is closed under the successor function. Such sets are said to be 'inductive'. The intersection of all such inductive sets is defined to be the set of natural numbers. It can be checked that the set of natural numbers satisfies the Peano axioms.It follows that each natural number is equal to the set of all natural numbers less than it:0 = { },1 = 0 ∪ {0} = {0} = {{ }},2 = 1 ∪ {1} = {0, 1} = {{ }, {{ }}},3 = 2 ∪ {2} = {0, 1, 2} = {{ }, {{ }}, {{ }, {{ }}}},n = n−1 ∪ {n−1} = {0, 1, …, n−1} = {{ }, {{ }}, …, {{ }, {{ }}, …}}, etc.With this definition, a natural number n is a particular set with n elements, and n ≤ m if and only if n is a subset of m.Also, with this definition, different possible interpretations of notations like ℝn (n-tuples versus mappings of n into ℝ) coincide.Even if one does not accept the axiom of infinity and therefore cannot accept that the set of all natural numbers exists, it is still possible to define any one of these sets.Other constructionsAlthough the standard construction is useful, it is not the only possible construction. Zermelo's construction goes as follows:Set 0 = { }Define S(a) = {a},It then follows that0 = { },1 = {0} = {{ }},2 = {1} = {{{ }}},n = {n−1} = {{{…}}}, etc.Each natural number is then equal to the set containing just the natural number preceding it.See alsoMathematics portalIntegerSet-theoretic definition of natural numbersPeano axiomsCanonical representation of a positive integerCountable setNumber#Classification for other number systems (rational, real, complex etc.)Notes"Standard number sets and intervals". ISO 80000-2:2009. International Organization for Standardization. p. 6.Weisstein, Eric W. "Natural Number". MathWorld."natural number", Dictionary by Merriam-Webster: America's most-trusted online dictionary, Merriam-Webster, retrieved 4 October 2014Carothers (2000) says: "ℕ is the set of natural numbers (positive integers)" (p. 3)Mac Lane & Birkhoff (1999) include zero in the natural numbers: 'Intuitively, the set ℕ = {0, 1, 2, ...} of all natural numbers may be described as follows: ℕ contains an "initial" number 0; ...'. They follow that with their version of the Peano Postulates. (p. 15)Jack G. Ganssle & Michael Barr (2003). Embedded Systems Dictionary. p. 138 (integer), 247 (signed integer), & 276 (unsigned integer). ISBN 1578201209. integer 1. n. Any whole number.Mendelson (2008) says: "The whole fantastic hierarchy of number systems is built up by purely set-theoretic means from a few simple assumptions about natural numbers." (Preface, p. x)Bluman (2010): "Numbers make up the foundation of mathematics." (p. 1)Weisstein, Eric W. "Counting Number". MathWorld.Introduction, Royal Belgian Institute of Natural Sciences, Brussels, Belgium.Flash presentation, Royal Belgian Institute of Natural Sciences, Brussels, Belgium.The Ishango Bone, Democratic Republic of the Congo, on permanent display at the Royal Belgian Institute of Natural Sciences, Brussels, Belgium. UNESCO's Portal to the Heritage of AstronomyGeorges Ifrah, The Universal History of Numbers, Wiley, 2000, ISBN 0-471-37568-3"A history of Zero". MacTutor History of Mathematics. Retrieved 2013-01-23. … a tablet found at Kish … thought to date from around 700 BC, uses three hooks to denote an empty place in the positional notation. Other tablets dated from around the same time use a single hook for an empty placeMann, Charles C. (2005), 1491: New Revelations Of The Americas Before Columbus, Knopf, p. 19, ISBN 9781400040063.Evans, Brian (2014), "Chapter 10. Pre-Columbian Mathematics: The Olmec, Maya, and Inca Civilizations", The Development of Mathematics Throughout the Centuries: A Brief History in a Cultural Context, John Wiley & Sons, ISBN 9781118853979.Michael L. Gorodetsky (2003-08-25). "Cyclus Decemnovennalis Dionysii – Nineteen year cycle of Dionysius". Группа Квантовых и Прецизионных Измерений. Retrieved 2012-02-13.This convention is used, for example, in Euclid's Elements, see Book VII, definitions 1 and 2.Morris Kline, Mathematical Thought From Ancient to Modern Times, Oxford University Press, 1990 [1972], ISBN 0-19-506135-7The English translation is from Gray. In a footnote, Gray attributes the German quote to: "Weber 1891/92, 19, quoting from a lecture of Kronecker's of 1886."Gray, Jeremy (2008), Plato's Ghost: The Modernist Transformation of Mathematics, Princeton University Press, p. 153Weber, Heinrich L. 1891-2. Kronecker. Jahresbericht der Deutschen Mathematiker-Vereinigung 2:5-23. (The quote is on p. 19.)"Much of the mathematical work of the twentieth century has been devoted to examining the logical foundations and structure of the subject." (Eves 1990, p. 606)Eves 1990, Chapter 15L. Kirby; J. Paris, Accessible Independence Results for Peano Arithmetic, Bulletin of the London Mathematical Society 14 (4): 285. doi:10.1112/blms/14.4.285, 1982.Bagaria, Joan. "Set Theory". The Stanford Encyclopedia of Philosophy (Winter 2014 Edition).Goldrei, Derek (1998). "3". Classic set theory : a guided independent study (1. ed., 1. print ed.). Boca Raton, Fla. [u.a.]: Chapman & Hall/CRC. p. 33. ISBN 0-412-60610-0.This is common in texts about Real analysis. See, for example, Carothers (2000, p. 3) or Thomson, Bruckner & Bruckner (2000, p. 2).Brown, Jim (1978). "In Defense of Index Origin 0". ACM SIGAPL APL Quote Quad. 9 (2): 7. doi:10.1145/586050.586053. Retrieved 19 January 2015.Hui, Roger. "Is Index Origin 0 a Hindrance?". J Home. Retrieved 19 January 2015.Rudin, W. (1976). Principles of Mathematical Analysis (PDF). New York: McGraw-Hill. p. 25. ISBN 978-0-07-054235-8.Weisstein, Eric W. "Cardinal Number". MathWorld.Grimaldi, Ralph P. (2003). A review of discrete and combinatorial mathematics (5th ed.). Boston, MA: Addison-Wesley. p. 133. ISBN 978-0201726343.G.E. Mints (originator), "Peano axioms", Encyclopedia of Mathematics, Springer, in cooperation with the European Mathematical Society, retrieved 8 October 2014Hamilton (1988) calls them "Peano's Postulates" and begins with "1. 0 is a natural number." (p. 117f)Halmos (1960) uses the language of set theory instead of the language of arithmetic for his five axioms. He begins with "(I) 0 ∈ ω (where, of course, 0 = ∅" (ω is the set of all natural numbers). (p. 46)Morash (1991) gives "a two-part axiom" in which the natural numbers begin with 1. (Section 10.1: An Axiomatization for the System of Positive Integers)Von Neumann 1923ReferencesBluman, Allan (2010), Pre-Algebra DeMYSTiFieD (Second ed.), McGraw-Hill ProfessionalCarothers, N.L. (2000), Real analysis, Cambridge University Press, ISBN 0-521-49756-6Clapham, Christopher; Nicholson, James (2014), The Concise Oxford Dictionary of Mathematics (Fifth ed.), Oxford University PressDedekind, Richard (1963), Essays on the Theory of Numbers, Dover, ISBN 0-486-21010-3 Dedekind, Richard (2007), Essays on the Theory of Numbers, Kessinger Publishing, LLC, ISBN 0-548-08985-XEves, Howard (1990), An Introduction to the History of Mathematics (6th ed.), Thomson, ISBN 978-0-03-029558-4Halmos, Paul (1960), Naive Set Theory, Springer Science & Business MediaHamilton, A. G. (1988), Logic for Mathematicians (Revised ed.), Cambridge University PressJames, Robert C.; James, Glenn (1992), Mathematics Dictionary (Fifth ed.), Chapman & HallLandau, Edmund (1966), Foundations of Analysis (Third ed.), Chelsea Pub Co, ISBN 0-8218-2693-XMac Lane, Saunders; Birkhoff, Garrett (1999), Algebra (3rd ed.), American Mathematical SocietyMendelson, Elliott (2008) [1973], Number Systems and the Foundations of Analysis, Dover PublicationsMorash, Ronald P. (1991), Bridge to Abstract Mathematics: Mathematical Proof and Structures (Second ed.), Mcgraw-Hill CollegeMusser, Gary L.; Peterson, Blake E.; Burger, William F. (2013), Mathematics for Elementary Teachers: A Contemporary Approach (10th ed.), Wiley Global Education, ISBN 978-1118457443Szczepanski, Amy F.; Kositsky, Andrew P. (2008), The Complete Idiot's Guide to Pre-algebra, Penguin GroupThomson, Brian S.; Bruckner, Judith B.; Bruckner, Andrew M. (2008), Elementary Real Analysis (Second ed.), ClassicalRealAnalysis.info, ISBN 9781434843678Von Neumann, Johann (1923), "Zur Einführung der transfiniten Zahlen", Acta litterarum ac scientiarum Ragiae Universitatis Hungaricae Francisco-Josephinae, Sectio scientiarum mathematicarum, 1: 199–208 Von Neumann, John (January 2002) [1923], "On the introduction of transfinite numbers", in Jean van Heijenoort, From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931 (3rd ed.), Harvard University Press, pp. 346–354, ISBN 0-674-32449-8 - English translation of von Neumann 1923.External linksHazewinkel, Michiel, ed. (2001) [1994], "Natural number", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4Axioms and Construction of Natural NumbersEssays on the Theory of Numbers by Richard Dedekind at Project Gutenberg[show]vteNumber systems[show]vteClasses of natural numbersCategories:Cardinal numbersElementary mathematicsIntegersNumber theoryNumbersNavigation menuNot logged inTalkContributionsCreate accountLog inArticleTalkReadEditView historySearchMain pageContentsFeatured contentCurrent eventsRandom articleDonate to WikipediaWikipedia storeInteractionHelpAbout WikipediaCommunity portalRecent changesContact pageToolsWhat links hereRelated changesUpload fileSpecial pagesPermanent linkPage informationWikidata itemCite this pagePrint/exportCreate a bookDownload as PDFPrintable versionIn other projectsWikimedia CommonsLanguagesAfrikaansAlemannischአማርኛالعربيةAragonésঅসমীয়াAzərbaycancaتۆرکجهবাংলাBân-lâm-gúБашҡортсаБеларускаяБеларуская (тарашкевіца)‎Българскиབོད་ཡིགBosanskiBrezhonegCatalàЧӑвашлаČeštinaCymraegDanskDeutschEestiΕλληνικάEmiliàn e rumagnòlEspañolEsperantoEuskaraفارسیFøroysktFrançaisGaeilgeGalego贛語Хальмг한국어Հայերենहिन्दीHornjoserbsceHrvatskiBahasa IndonesiaInterlinguaИронIsiXhosaÍslenskaItalianoעבריתBasa Jawaಕನ್ನಡქართულიҚазақшаKiswahiliKurdîລາວLatinaLatviešuLëtzebuergeschLietuviųLa .lojban.LumbaartMagyarМакедонскиMalagasyമലയാളംमराठीمصرىBahasa MelayuMirandésМонголမြန်မာဘာသာNederlandsनेपालीनेपाल भाषा日本語NordfriiskNorskNorsk nynorskOʻzbekcha/ўзбекчаਪੰਜਾਬੀپنجابیPatoisភាសាខ្មែរPiemontèisPlattdüütschPolskiPortuguêsRomânăРусскийसंस्कृतम्SarduScotsShqipSicilianuසිංහලSimple EnglishSlovenčinaSlovenščinaŚlůnskiSoomaaligaکوردیСрпски / srpskiSrpskohrvatski / српскохрватскиSuomiSvenskaTagalogதமிழ்Татарча/tatarçaతెలుగుไทยТоҷикӣTürkçeTürkmençeУкраїнськаاردوTiếng ViệtVõro文言West-VlamsWinarayייִדישYorùbá粵語Žemaitėška中文Edit linksThis page was last edited on 28 November 2017, at 18:20.Text is available under the Creative Commons Attribution-ShareAlike License; additional terms may apply. By using this site, you agree to the Terms of Use and Privacy Policy. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization.Privacy policyAbout WikipediaDisclaimersContact WikipediaDevelopersCookie statementMobile viewEnable previews