Rader's NUMBERNUT.COM Ambiguously Defined Mathematical Terms at the High School Level Last revision: April 2, 2011 This page attempts to show some of the ambiguities in defining some of the mathematical terms that might be encountered at the high school level. A number of these issues may seem quite trivial, and some are the result of consulting out-of-date texts. ADJACENT ANGLES. AMPLITUDE OF A CURVE. ARCSECANT AND ARCCOSECANT, RANGE OF. Other textbooks give the range of the Arcsecant function to be [0, π/2) U [π, 3π/2). Some textbooks give the range of the Arccosecant function to be [-π/2, 0) U (0, π/2). Other textbooks give the range of the Arccosecant function to be (0, π/2] U (π, 3π/2]. BILLION, TRILLION, etc. CHARACTERISTIC AND MANTISSA. CONVERGENT and DIVERGENT SERIES. COUNTABLE has a mathematical definition that is very different from its meaning in other contexts. CRITICAL POINT. CURVATURE OF A FUNCTION. DEGENERATE CONIC SECTIONS. Various web pages say that a degenerate ellipse is a circle or a parabola. DOMAIN OF A FUNCTION. GEOMETRIC MEAN. IMAGINARY NUMBER.
Purplemath Earliest Uses of Various Mathematical Symbols These pages show the names of the individuals who first used various common mathematical symbols, and the dates the symbols first appeared. The most important written source is the definitive A History of Mathematical Notations by Florian Cajori. Symbols of operation, including +, -, X, division, exponents, radical symbol, dot and vector product Grouping symbols, including (), [], {}, vinculum Symbols of relation, including =, >, < Fractions, including decimals Symbols for various constants, such as π, i, e, 0 Symbols for variables Symbols to represent various functions, such as log, ln, γ, absolute value; also the f(x) notation Symbols used in geometry Symbols used in trigonometry; also symbols for hyperbolic functions Symbols used in calculus Symbols for matrices and vectors Set notation and logic Symbols used in number theory Symbols used in probability and statistics Written sources for these pages
How to Become a Pure Mathematician (or Statistician) Earliest Known Uses of Some of the Words of Mathematics A - B - C - D - E - F - G - H - I - J - K - L - M - N - O - P - Q - R - S - T - U - V - W - X - Y - Z - Sources RECENT CHANGES: On March 28 the cosecant entry was revised to show an earliest known use in English in 1658 by John Newton, and to show that Rheticus apparently did not use cosecans. This new information comes from Glen Van Brummelen. On March 28 an entry pointless was added, and the entry point was revised, both by John Aldrich. James Landau has found in a Google print search that Boole used conditional probability in 1854. The entry for that term will be updated soon. These pages attempt to show the first uses of various words used in mathematics. Mathematical Words: Origins and Sources by John Aldrich is an excellent article and companion to this web site. Please see also Earliest Uses of Various Mathematical Symbols, Images of Mathematicians on Postage Stamps, and Ambiguously Defined Mathematical Terms at the High School Level.
Modern X-ray technology reveals Archimedes' math theory under forged painting Stanford Report, August 9, 2006 Archimedes Palimpsest Project Bergmann, left, and Noel place fragments from the Archimedes Palimpsest in front of the X-ray beam. A scan of the palimpsest reveals vertical text hidden beneath the paint. Multispectral imaging revealed that Archimedes was well on his way to developing calculus, nearly 1,000 years before Isaac Newton. After more than 1,000 years in obscurity, the last unreadable pages of the works of ancient mathematician Archimedes are being deciphered, thanks to the X-ray vision at the Stanford Linear Accelerator Center (SLAC), which Stanford operates for the U.S. "It is amazing how much the Archimedes Palimpsest project has gained from Stanford University," said Will Noel, curator of manuscripts and rare books at the Walters Art Museum in Baltimore, where the document is housed. The manuscript uniquely records several of the works of the legendary 3rd century B.C.E. mathematician, who famously exclaimed "Eureka!" Archimedes rediscovered
Doron Zeilberger's 36th Opinion: Written: March 5, 1999 Rabbi Levi Ben Gerson, in his pre-algebra text (1321), Sefer Ma'asei Khosev, had about fifty theorems, complete with rigorous proofs. Nowadays, we no longer call them theorems, but rather (routine) algebraic identities. For example, proving (a+b)*c=a*c+b*c took him about half a page, while proving (a-b)*c+a*(b-c)=b*(a-c) took a page and a half, and proving a*(b*c*d)=d*(a*b*c) took him one page. The reason that it took him so long is that while he already had the algebraic concepts, he still was too hung-up on words, and while he used symbols, (denoted by dotted Hebrew letters), he did not quite utilize, systematically, the calculus of algebraic identities. Believe it not, our current proofs are just as long-winded and repetitive, since we use an informal language, a minor variation on our everyday speech. We are now on the brink of a much more significant revolution in mathematics, not of algebra, but of COMPUTER ALGEBRA. Doron Zeilberger's Homepage
Logic Puzzles - Solve Online or Print Your Own for Free! Number Theory with Python with Python Contents [Prev] [Python Intro] [Next] The Python programming language has basic commands which implement integer arithmetic. A particular strength of Python for number theory is its native support for arbitrary sized integers. Integer Operations The basic arithmetic operations, +, -, *, /, are available from the keyboard. Examples: (8 + 7) % 13 (returns an answer of 2) (5 * 4) % 13 (returns an answer of 7) 7 // 2 (returns an answer of 3) 3**2; (returns an answer of 9) pow(2,6,11); (returns an answer of 9) [There is a fine point with the division operation which may become important with Python 3.0. Number Theory Operations Beyond the basic arithmetic operations Python has few natively implemented number theory functions. The gcd function can be computed by hand as a succession of modular reductions. A simple recursive implementation of the gcd function is: def gcd(a,b): if a == 0: return b return gcd(b % a, a) For positive input values this can be implemented by: Appendices:
Basics of Computational Number Theory Robert Campbell Contents Introduction This document is a gentle introduction to computational number theory. Modular Arithmetic Modular arithmetic is arithmetic using integers modulo some fixed integer N. 7 + 7 = 14 = 2 (mod 12) 5 * 7 = 35 = 11 (mod 12) Further examples can be generated and checked out with the following short programs. Among the basic operations we have missed the division operator. 11 / 5 = 7 (mod 12) as 5 * 7 = 11 (mod 12) 5(-1) = 1 / 5 = 17 (mod 21) as 5 * 17 = 85 = 1 (mod 21) 48 / 31 = 72 (mod 91) as 31 * 72 = 48 (mod 91) 9 / 3 = 7 (mod 12) as 3 * 7 = 21 = 9 (mod 12) As the last example points out, modular division does not always produce a unique result, for other correct answers are 3 and 11 (as 3 * 3 = 9 (mod 12) and 3 * 11 = 33 = 9 (mod 12)). GCD - The Euclidean Algorithm The Euclidean Algorithm solves two problems we have posed: Common Factors - Given two numbers n and m, find any common factors they may have. The algorithm to find gcd(n, m) runs something like this: