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Schizophrenic Number
A schizophrenic number (also known as mock rational number) is an irrational number that displays certain characteristics of rational numbers. Definition ''The Universal Book of Mathematics'' defines "schizophrenic number" as: The sequence of numbers generated by the recurrence relation ''f''(''n'') = 10 ''f''(''n'' − 1) + ''n'' described above is: :0, 1, 12, 123, 1234, 12345, 123456, 1234567, 12345678, 123456789, 1234567900, ... . :''f''(49) = 1234567901234567901234567901234567901234567901229 The integer parts of their square roots, :1, 3, 11, 35, 111, 351, 1111, 3513, 11111, 35136, 111111, 351364, 1111111, ... , alternate between numbers with irregular digits and numbers with repeating digits, in a similar way to the alternations appearing within the fractional part of each square root. Characteristics The ''schizophrenic number'' shown above is the special case of a more general phenomenon that appears in the b-ary expansions of square roots of the solutions of the recurre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Irrational Number
In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integers. When the ratio of lengths of two line segments is an irrational number, the line segments are also described as being '' incommensurable'', meaning that they share no "measure" in common, that is, there is no length ("the measure"), no matter how short, that could be used to express the lengths of both of the two given segments as integer multiples of itself. Among irrational numbers are the ratio of a circle's circumference to its diameter, Euler's number ''e'', the golden ratio ''φ'', and the square root of two. In fact, all square roots of natural numbers, other than of perfect squares, are irrational. Like all real numbers, irrational numbers can be expressed in positional notation, notably as a decimal number. In the cas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wonders Of Numbers
Clifford Alan Pickover (born August 15, 1957) is an American author, editor, and columnist in the fields of science, mathematics, science fiction, innovation, and creativity. For many years, he was employed at the IBM Thomas J. Watson Research Center in Yorktown, New York where he was Editor-in-Chief of the '' IBM Journal of Research and Development''. He has been granted more than 700 U.S. patents, is an elected Fellow for the Committee for Skeptical Inquiry, and is author of more than 50 books, translated into more than a dozen languages. Pickover.com Biography Pickover was elected as a Fellow for the Committee for Skeptical Inquiry for his “significant contributions to the general public’s understanding of science, reason, and critical inquiry through their scholarship, writing, and work in the media.” Other Fellows have included Carl Sagan and Isaac Asimov. He has been awarded almost 700 United States patents, and his ''The Math Book'' was winner of the 2011 Ne ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Six Nines In Pi
A sequence of six consecutive nines occurs in the decimal representation of the number pi (), starting at the 762nd decimal place.. It has become famous because of the mathematical coincidence and because of the idea that one could memorize the digits of up to that point, which seems to suggest that is rational. The earliest known mention of this idea occurs in Douglas Hofstadter's 1985 book ''Metamagical Themas'', where Hofstadter states This sequence of six nines is sometimes called the "Feynman point", after physicist Richard Feynman, who allegedly stated this same idea in a lecture.. It is not clear when, or even if, Feynman made such a statement, however; it is not mentioned in published biographies or in his autobiographies, and is unknown to his biographer, James Gleick. Related statistics is conjectured to be, but not known to be, a normal number. For a normal number sampled uniformly at random, the probability of a specific sequence of six digits occurring this ear ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal Number
In mathematics, a real number is said to be simply normal in an integer base b if its infinite sequence of digits is distributed uniformly in the sense that each of the b digit values has the same natural density 1/b. A number is said to be normal in base b if, for every positive integer n, all possible strings n digits long have density b−''n''. Intuitively, a number being simply normal means that no digit occurs more frequently than any other. If a number is normal, no finite combination of digits of a given length occurs more frequently than any other combination of the same length. A normal number can be thought of as an infinite sequence of coin flips ( binary) or rolls of a die (base 6). Even though there ''will'' be sequences such as 10, 100, or more consecutive tails (binary) or fives (base 6) or even 10, 100, or more repetitions of a sequence such as tail-head (two consecutive coin flips) or 6-1 (two consecutive rolls of a die), there will also be equally ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Almost Integer
In recreational mathematics, an almost integer (or near-integer) is any number that is not an integer but is very close to one. Almost integers are considered interesting when they arise in some context in which they are unexpected. Almost integers relating to the golden ratio and Fibonacci numbers Well-known examples of almost integers are high powers of the golden ratio \phi=\frac\approx 1.618, for example: : \begin \phi^ & =\frac\approx 3571.00028 \\ pt\phi^ & =2889+1292\sqrt5 \approx 5777.999827 \\ pt\phi^ & =\frac\approx 9349.000107 \end The fact that these powers approach integers is non-coincidental, because the golden ratio is a Pisot–Vijayaraghavan number. The ratios of Fibonacci or Lucas numbers can also make almost integers, for instance: * \operatorname(360)/\operatorname(216) \approx 1242282009792667284144565908481.999999999999999999999999999999195 * \operatorname(361)/\operatorname(216) \approx 2010054515457065378082322433761.0000000000000000000000000000 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Root Of 2
The square root of 2 (approximately 1.4142) is a positive real number that, when multiplied by itself, equals the number 2. It may be written in mathematics as \sqrt or 2^, and is an algebraic number. Technically, it should be called the principal square root of 2, to distinguish it from the negative number with the same property. Geometrically, the square root of 2 is the length of a diagonal across a square with sides of one unit of length; this follows from the Pythagorean theorem. It was probably the first number known to be irrational. The fraction (≈ 1.4142857) is sometimes used as a good rational approximation with a reasonably small denominator. Sequence in the On-Line Encyclopedia of Integer Sequences consists of the digits in the decimal expansion of the square root of 2, here truncated to 65 decimal places: : History The Babylonian clay tablet YBC 7289 (c. 1800–1600 BC) gives an approximation of in four sexagesimal figures, , which is accurate to about six ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nth Root
In mathematics, a radicand, also known as an nth root, of a number ''x'' is a number ''r'' which, when raised to the power ''n'', yields ''x'': :r^n = x, where ''n'' is a positive integer, sometimes called the ''degree'' of the root. A root of degree 2 is called a ''square root'' and a root of degree 3, a ''cube root''. Roots of higher degree are referred by using ordinal numbers, as in ''fourth root'', ''twentieth root'', etc. The computation of an th root is a root extraction. For example, 3 is a square root of 9, since 3 = 9, and −3 is also a square root of 9, since (−3) = 9. Any non-zero number considered as a complex number has different complex th roots, including the real ones (at most two). The th root of 0 is zero for all positive integers , since . In particular, if is even and is a positive real number, one of its th roots is real and positive, one is negative, and the others (when ) are non-real complex numbers; if is even and is a negative real numbe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transcendental Numbers
In mathematics, a transcendental number is a number that is not algebraic—that is, not the root of a non-zero polynomial of finite degree with rational coefficients. The best known transcendental numbers are and . Though only a few classes of transcendental numbers are known—partly because it can be extremely difficult to show that a given number is transcendental—transcendental numbers are not rare. Indeed, almost all real and complex numbers are transcendental, since the algebraic numbers comprise a countable set, while the set of real numbers and the set of complex numbers are both uncountable sets, and therefore larger than any countable set. All transcendental real numbers (also known as real transcendental numbers or transcendental irrational numbers) are irrational numbers, since all rational numbers are algebraic. The converse is not true: not all irrational numbers are transcendental. Hence, the set of real numbers consists of non-overlapping rational, algebraic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Usenet Newsgroup
A Usenet newsgroup is a repository usually within the Usenet system, for messages posted from users in different locations using the Internet. They are discussion groups and are not devoted to publishing news. Newsgroups are technically distinct from, but functionally similar to, discussion forums on the World Wide Web. Newsreader software is used to read the content of newsgroups. Before the adoption of the World Wide Web, Usenet newsgroups were among the most popular Internet services, and have retained their noncommercial nature in contrast to the increasingly ad-laden web. In recent years, this form of open discussion on the Internet has lost considerable ground to individually-operated browser-accessible forums and big media social networks such as Facebook and Twitter. Communication is facilitated by the Network News Transfer Protocol (NNTP) which allows connection to Usenet servers and data transfer over the internet. Similar to another early (yet still used) protocol S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clifford A
Clifford may refer to: People *Clifford (name), an English given name and surname, includes a list of people with that name * William Kingdon Clifford *Baron Clifford * Baron Clifford of Chudleigh *Baron de Clifford * Clifford baronets *Clifford family (bankers) *Jaryd Clifford *Justice Clifford (other) *Lord Clifford (other) Arts, entertainment, and media *''Clifford the Big Red Dog'', a series of children's books **Clifford (character), the central character of ''Clifford the Big Red Dog'' ** ''Clifford the Big Red Dog'' (2000 TV series), 2000 animated TV series **''Clifford's Puppy Days'', 2003 animated TV series **''Clifford's Really Big Movie'', 2004 animated movie ** ''Clifford the Big Red Dog'' (2019 TV series), 2019 animated TV series ** ''Clifford the Big Red Dog'' (film), 2021 live-action movie * ''Clifford'' (film), a 1994 film directed by Paul Flaherty *Clifford (Muppet) Mathematics *Clifford algebra, a type of associative algebra, named after William ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rational Number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rational numbers, also referred to as "the rationals", the field of rationals or the field of rational numbers is usually denoted by boldface , or blackboard bold \mathbb. A rational number is a real number. The real numbers that are rational are those whose decimal expansion either terminates after a finite number of digits (example: ), or eventually begins to repeat the same finite sequence of digits over and over (example: ). This statement is true not only in base 10, but also in every other integer base, such as the binary and hexadecimal ones (see ). A real number that is not rational is called irrational. Irrational numbers include , , , and . Since the set of rational numbers is countable, and the set of real numbers is uncountable ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Taylor Expansion
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor series are named after Brook Taylor, who introduced them in 1715. A Taylor series is also called a Maclaurin series, when 0 is the point where the derivatives are considered, after Colin Maclaurin, who made extensive use of this special case of Taylor series in the mid-18th century. The partial sum formed by the first terms of a Taylor series is a polynomial of degree that is called the th Taylor polynomial of the function. Taylor polynomials are approximations of a function, which become generally better as increases. Taylor's theorem gives quantitative estimates on the error introduced by the use of such approximations. If the Taylor series of a function is convergent, its sum is the limit of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
