суббота, 3 июля 2010 г.
суббота, 12 июня 2010 г.
Glossary
An algorithm is a detailed set of instructions for solving a problem. One example of an algorithm is the method of long division. Algorithms are named after the mathematician al-Khwarazmi.
The numerals 1,2,3,4,5,6,7,8,9, and 0 that we use today are often called Arabic numerals because al-Khwarazmi wrote a popular book about them. This number system probably comes from India.
Calculus is a branch of mathematics that involves rates of change, the slopes of curves, length, area, and volume. One important calculus technique is the careful study of very small changes. Sometimes we think of these changes as "infinitely small." Calculus is also concerned with limits and "infinitely large" numbers.
The coordinate plane is sometimes called the Cartesian plane in honor of René Descartes.
The distance around the outside of a shape is called its circumference. The word "circumference" is usually used for circles; for polygons like squares and triangles, we use the word "perimeter." Sometimes "circumference" is used for three-dimensional objects, like spheres. In this case, the circumference is the distance around the object at its widest point. You might think of this as the distance around the equator of a sphere.
Conic sections are the curves formed when a plane slices through a double cone.
Conic sections include hyperbolas, parabolas, and ellipses. They also include circles, because a circle is a special kind of ellipse. Lines and points can also be made by the intersection of a plane and a double cone, but they usually do not count as conic sections.
The coordinate plane is a plane that contains two lines called axes.
Coordinates based on distance from the axes can be used to locate points in the plane.
The diameter of a shape is the greatest possible distance between two points on that shape.
A directrix is a special line. A conic section such as a parabola can be defined by its distance from the directrix and a point called the focus.
Ellipses are one kind of conic section. They look like ovals, but they have a more precise definition. An ellipse can be described by the equation x2/a2 + y2/b2 = 1.
An ellipse can also be described using foci. An ellipse has two focus points, and the sum of the distances from a point on the ellipse to each focus is always constant. This fact can be written as the equation r1 + r2 = 2a.
A circle is a special kind of ellipse with only one focus.
The equator is the circle around the Earth exactly between the North Pole and the South Pole.
Fermat's Last Theorem states that the equation xn + yn = zn has no solutions when all of the variables are integers (numbers in the set . . . -2, -1, 0, 1, 2, 3, . . .), n is greater than 2, and x, y, and z are not all 0. Pierre de Fermat first stated this theorem, in the margin of one of his books, along with a note that the margin was not big enough to hold the proof. This theorem was not published until after his death. For hundreds of years, nobody could find a correct proof of the theorem. Andrew Wiles finally published a proof in 1995.
A focus is a special point. For example, the center of a circle can be considered its focus. A conic section such as a parabola can be defined by its distance from the focus and a line called the directrix.
The plural of "focus" is "focuses" or "foci."
Hyperbolas are one kind of conic section. A hyperbola can be described by the equation x2/a2 - y2/b2 = 1.
A hyperbola can also be described using foci. A hyperbola has two focus points. The difference of the distances from a point on the ellipse to each focus is always constant. This fact can be written as the equation r2 - r1 = 2a.
The latitude of a point on the Earth is its distance around the Earth from the equator. Since the Earth is a sphere, latitude is measured in degrees. Circles of points on the earth that all have the same latitude are called lines of latitude.
Nicole Oresme used "latitude" as a more general term for a type of coordinate.
The longitude of a point on the Earth is its distance around the Earth from the Prime Meridian. Since the Earth is a sphere, longitude is measured in degrees. Circles of points on the earth that all have the same longitude are called lines of longitude.
Nicole Oresme used "longitude" as a more general term for a type of coordinate.
A matrix is made up of numbers arranged in a rectangular shape of rows and columns.
Parabolas are one kind of conic section. A parabola that opens upward can be described by the equation y = ax2 + bx + c.
A parabola can also be described using a focus and a directrix. The distance from any point on the parabola to the focus is the same as the distance to the directrix.
Plane loci are curves that are defined by their distances from other objects in the plane. The conic sections are plane loci.
In polar coordinates, points are located by their distance from the origin, often labeled r, and their angle from the positive x-axis, often labeled with the Greek letter θ or theta.
The Prime Meridian is a line of longitude that travels from the North Pole to the South Pole through Greenwich, England.
The slope of a line is a number that tells how much it is slanted compared to the x-axis. A line with the equation y = mx + b has slope m. If we draw a tangent line to a curve at a particular point, then we can also define the slope of the curve at that point, by saying that the line and the curve have the same slope at the point where they touch.
A tangent line to a curve only touches the curve at one point.
There is also a function called the tangent function that is important in trigonometry.
Trigonometry is the study of angles.
The Witch of Agnesi is a curve. It can be described by the equation y = a3/(x2 + a2).
This curve is named after Maria Gaetana Agnesi, who included it in a calculus textbook she published in 1748. Agnesi called her curve the versiera, which means "turning" in Italian. When her book was translated into English, the translator confused the word "versiera" with the word "avversiera," which means witch, so he called the curve a witch.
пятница, 11 июня 2010 г.
среда, 9 июня 2010 г.
Basic Mathematical Symbols
can be read ``for all a and b in the set of real numbers, a less than b implies a cubed is less than b cubed'' (or ``for all real numbers a and b, if a is less than b, then a cubed is less than b cubed'').
| Written: | Spoken: | |
|---|---|---|
 | = | for all (or ``for every'') |
 | = | there exists |
 | = | such that |
 | = | is orthogonal to (perpendicular to) |
 | = | intersection |
 | = | union |
 | = | is contained in |
 | = | is an element of |
 | = | is not an element of |
 | = | implies |
 | = | if and only if (or ``is equivalent to'') |
 | = | if and only if (same as above) |
 | = | less than |
 | = | greater than |
 | = | less than or equal to |
 | = | greater than or equal to |
 | = | the set of real numbers |
 | = | the set of natural numbers |
 | = | the set of integers |
 | = | the set of rational numbers |
 | = | the set of complex numbers |
Diophantine Equation
Version 1.1
Introduction
The following diophantine equation was a topic problem from #math. The origin of the problem is unknown to me(landen). To explore ideas and cut down on mistakes I used the software package for number theory, PARI/GP, which is free for noncommercial use. This package is convenient for calculations in algebraic extensions of Q and Z.
2a + 3b = 5c (1)The problem can be split into four cases: a is odd with a ≥ 3, a=2, a=1, and a is even with a ≥ 4. Two of the cases can be solved by just trying all possibilities (mod 3) and (mod 8). The remaining cases are a=1, and a is even with a ≥ 4, which take more work.
Proof: Case 1, a is odd with a ≥ 3
2a + 3b = 52n+1 and then (mod 8)
2a = 0 (mod 8); 3b = 1 or 3 (mod 8); 52n+1 = 5 (mod 8)
So (1) fails (mod 8) for this case. This case gives no solutions.
Q.E.D.
Proof: Case 2, a = 2
Q.E.D.
Proof: Case 3
2 + 3b = 5c has only one solution, b = 1 and c = 1 (2)Assume b = 2m+1 and c = 2n+1 with n > 0. They must be odd by the argument in Case 1. Multiplying the equation by 3 and rearranging terms we get:
32(m+1) - 15 52n = -6 (3)In this form we see that any solutions to (2) must also be among solutions to:
x2 - 15 y2 = -6 (4)Of course (4) will have many more solutions than (3) but solutions to (4) are easy to study in Z[w], where w is sqrt(15). The only solutions to (4) which have a chance of solving (3) are the ones in which y is a power of 5. One way to show that (3) has no solutions is to show that when y is divisible by 3 it is also divisible by 11. This property of the solutions was first conjectured after using PARI/GP to experiment. It was then easy to prove that conjecture by considering the solutions to (4) (mod 5) and (mod 11). From the theory of binary quadratic forms we know that all the solutions to (4) can be written as:
s(k) = s0uk = x(k) + y(k)w (5)By factoring y(k) when it is a multiple of 5, it was conjected that y(k) is also divisible by 11 and thus is not a power of 5. When s(k) is computed (mod 5) and (mod 11) the results are periodic with a period of ten. Notice that
k is in Z
w = sqrt(11)
s0 = 3-w a solution of (4) and u = 4+w, the fundamental unit of Z[w].
norm(x(k) + y(k)w) = x(k)2 - 15 y(k)2 = -6
PARI/GP can compute solutions using (5) directly. Here are solutions for a few values of k:
k = -12; s(k) = (194888982963 - 50320119025*w); norm(s(k)) = -6
k = -11; s(k) = (24754146477 - 6391493137*w); norm(s(k)) = -6
k = -10; s(k) = (3144188853 - 811826071*w); norm(s(k)) = -6
k = -9; s(k) = (399364347 - 103115431*w); norm(s(k)) = -6
k = -8; s(k) = (50725923 - 13097377*w); norm(s(k)) = -6
k = -7; s(k) = (6443037 - 1663585*w); norm(s(k)) = -6
k = -6; s(k) = (818373 - 211303*w); norm(s(k)) = -6
k = -5; s(k) = (103947 - 26839*w); norm(s(k)) = -6
k = -4; s(k) = (13203 - 3409*w); norm(s(k)) = -6
k = -3; s(k) = (1677 - 433*w); norm(s(k)) = -6
k = -2; s(k) = (213 - 55*w); norm(s(k)) = -6
k = -1; s(k) = (27 - 7*w); norm(s(k)) = -6
k = 0; s(k) = (3 - w); norm(s(k)) = -6
k = 1; s(k) = (-3 - w); norm(s(k)) = -6
k = 2; s(k) = (-27 - 7*w); norm(s(k)) = -6
k = 3; s(k) = (-213 - 55*w); norm(s(k)) = -6
k = 4; s(k) = (-1677 - 433*w); norm(s(k)) = -6
k = 5; s(k) = (-13203 - 3409*w); norm(s(k)) = -6
k = 6; s(k) = (-103947 - 26839*w); norm(s(k)) = -6
k = 7; s(k) = (-818373 - 211303*w); norm(s(k)) = -6
k = 8; s(k) = (-6443037 - 1663585*w); norm(s(k)) = -6
k = 9; s(k) = (-50725923 - 13097377*w); norm(s(k)) = -6
k = 10; s(k) = (-399364347 - 103115431*w); norm(s(k)) = -6
k = 11; s(k) = (-3144188853 - 811826071*w); norm(s(k)) = -6
k = 12; s(k) = (-24754146477 - 6391493137*w); norm(s(k)) = -6
k = 13; s(k) = (-194888982963 - 50320119025*w); norm(s(k)) = -6
u5 = 15124 + 3905*wy(3) is 55 which is divisible by 5 and 11. This means that divisiblity by 5 and 11 occur together at y(3 + k*5). Thus y is never a power of 5.
u5 = 1 (mod 5) Proves solutions repeat with a period of 5 (mod 5).
u5 = -1 (mod 11) Proves solutions repeat with a period of 10 (mod 11).
But every 5th term (mod 11), the coefficient of w (mod 11) in u5 = 0, so is divisible by 11 also.
Q.E.D.
As a check for errors, PARI/GP calculated the solutions (mod 5) and (mod 11). Here are some results:
** means coefficient of w, y(k) is divisible by 5 and 11 both
k = -7; s(k) = 2 + 0*w (mod 5); s(k) = 7 + 0*w (mod 11) **
k = -6; s(k) = 3 + 2*w (mod 5); s(k) = 6 + 7*w (mod 11)
k = -5; s(k) = 2 + w (mod 5); s(k) = 8 + w (mod 11)
k = -4; s(k) = 3 + w (mod 5); s(k) = 3 + w (mod 11)
k = -3; s(k) = 2 + 2*w (mod 5); s(k) = 5 + 7*w (mod 11)
k = -2; s(k) = 3 + 0*w (mod 5); s(k) = 4 + 0*w (mod 11) **
k = -1; s(k) = 2 + 3*w (mod 5); s(k) = 5 + 4*w (mod 11)
k = 0; s(k) = 3 + 4*w (mod 5); s(k) = 3 + 10*w (mod 11)
k = 1; s(k) = 2 + 4*w (mod 5); s(k) = 8 + 10*w (mod 11)
k = 2; s(k) = 3 + 3*w (mod 5); s(k) = 6 + 4*w (mod 11)
k = 3; s(k) = 2 + 0*w (mod 5); s(k) = 7 + 0*w (mod 11) **
k = 4; s(k) = 3 + 2*w (mod 5); s(k) = 6 + 7*w (mod 11)
k = 5; s(k) = 2 + w (mod 5); s(k) = 8 + w (mod 11)
k = 6; s(k) = 3 + w (mod 5); s(k) = 3 + w (mod 11)
k = 7; s(k) = 2 + 2*w (mod 5); s(k) = 5 + 7*w (mod 11)
k = 8; s(k) = 3 + 0*w (mod 5); s(k) = 4 + 0*w (mod 11) **
Proof: Case 4
a is even with a ≥ 4 has only the solution a=4,b=2,c=2Assume a is even with a ≥ 4. Then b = 2m and c = 2n by considering equation (1) (mod 8). Rearranging (1) we have:
2a = 52n - 32m = (5n - 3m)(5n + 3m)By looking at the sum and difference of the two factors on the right, we see that they have 2 as their greatest common divisor. This means that
(5n - 3m) = 2, and therefore n=1 and m=1 by Case 3, so (a,b,c) = 4,2,2)Q.E.D.
http://efnet-math.org/math_tech/dioph2.htm
Positive Solutions to linear diophantine equation
Let a,b,m,n be integers > 0, let integer k >= 0. define g = gcd(a,b)
Theorem: am + bn = ab/g + g + kg has a solution m,n, given a,b,k. (eqn 1)
Proof: n must have a value such that ab/g + g + kg -bn is divisible by a. This can be accomplished by selecting n to be in the set {1,2,... a/g}. One of these numbers is sufficient because
ab/g + g + kg has only a/g possible residues (mod a/g).
We need to show that m>=1 is possible. The maximum value of bn is ba/g. So
am >= ab/g + g + kg -ab/g = g + kg (eqn 2)
m >= g/a + kg/a (eqn 3)
g/a > 0 is so m >= 1. Any k>0 can only help this.
Notice that if the +g term is ommited from (eqn 1), then (eqn 3) becomes m >= kg/a (eqn 4), thus m=0 cannot be ruled out. So (eqn 1) is the best we can do. Q.E.D
Special case, There are non-negative solutions starting at n=(a-1)(b-1)
x>=0,y>=0,g=1
applying the theorem with m=x+1, n=y+1;
a(x+1)+b(y+1)=ab+1+k
ax+by = (a-1)(b-1) + k
This is a common form of the problem.
LATIN TERMS USED IN MATHEMATICS
(AHD-in-fin-ITE-um) Literally, “to infinity,” indicates that a process or operation is to be carried out endlessly.a fortiori
(ah-FOR-tee-OHR-ee) “With stronger reason.” If every multiple of two is even, then a fortiori every multiple of four is even.a posteriori
(AH-paws-TEER-ee-OHR-ee) “From effect to cause.” A thing is known a posteriori if it is known from evidence or empirical reasoning.a priori
(AH-pree-OHR-ee) A thing is known a priori if it is evident by logic alone from what is already known.e.g.
See exempli gratia.exempli gratia
(ex-EMP-lee GRAH-tee-uh) “For example.” Usually abbreviated to ‘e.g.’ and often confused with ‘i.e.’ Example: “Many real numbers cannot be expressed as a ratio of integers, e.g., the square root of two.”id est
(id EST) Literally, “that is.” Usually abbreviated ‘i.e.’ and often confused with ‘e.g.’ Example: “She won the race, i.e., she crossed the finish line first.” The decision whether to use ‘i.e.,’ or ‘e.g.’ should be based on whether “that is” or “for example” is what is wanted in the sentence.i.e.
See id est.ipso facto
(IP-soh FAK-toh) Literally, “by that very fact.” Example: “Lie group representations are useful in characterizing quantum mechanical phenomena, and they are ipso facto an important part of a physicist’s mathematical training.”n.b.
See nota benenota bene
(NOH-tuh BAY-nay) Literally, “note well.” Usually abbreviated ‘n.b.’, this is a way of saying, “take note of this.”per impossibile
(pehr ihm-paws-SEE-bee-lay) “As is impossible.” Qualifies a proposition that cannot be true.QED
See quod erat demonstrandum.QEF
See quod erat faciendum.quod erat demonstrandum
(KWAWD eh-RAHT dem-on-STRAHND-um) “That which was to have been proved.” Traditionally placed at the end of proofs, the QED is now usually indicated by a small square. A few students have clung to use of the traditional letters, in the hope they might be interpreted as “quite elegantly done.”quod erat faciendum
(KWAWD eh-RAHT FAH-kee-END-um) “That which was to have been shown.” Abbreviated QEF, it was traditionally used to mark the end of a solution or calculation. It is rarely used now. (Impress your professor by putting it at the end of exam problems.)OTHER COMMON LATIN TERMS
ab initio
(AHB in-IT-ee-oh) From the beginning.accessit
(ahk-SESS-it) Honorable mention.ad hoc
(add-HOK) For the immediate purpose. An ad hoc committee is appointed for some specific purpose, after completing which it is dissolved.ad hominem
(add HOM-in-um) “To the man.” An argument is ad hominem when it attacks the opponent personally rather than addressing his arguments.ad nauseam
(add NAWS-ee-um) Something continues ad nauseam when it goes on so long you become sick of it.alma mater
(ALL-muh MAH-ter) Your alma mater is the university or college which granted your degree.alumnus/alumna
(a-LUM-nus/nuh) An alum, as it is sometimes shortly said, is a former member/student of a university or college. (The ‘us’ ending is masculine, the ‘a’ ending feminine. The plurals are alumni and alumnae, respectively.)a.d.
See anno domini.anno domini
(AN-noh DOM-in-ee) “In the year of Our Lord.” Indicates that a date is given in the western or Gregorian calendar, in which years are counted roughly from the birth of Christ.bona fide
(BONE-uh FIDE) “In good faith.” One’s bona fides are documents or testimonials establishing one’s credentials or honesty.carpe diem
(CAR-pay DEE-um) “Seize the day.” A motto which says to live in the now, and/or to not waste time or opportunity.cf.
See confer.circa
(SIR-kuh) Approximately. Used with dates, e.g., Euclid wrote the Elements circa 300 bce.confer
(KAWN-fehr) “Compare.” Usually abbreviated cf. and often used in footnotes, this indicates that one should compare the present passage or statement with the one referred to.cum laude
(coom LOUD-ay) “With praise.” Used on degree certificates to indicate exceptional academic standing.de facto
(day FAK-toh) “In reality.” Used to indicate that, whatever may be believed or legislated, the reality is as indicated here. E.g., she’s the de facto leader of the union.de jure
(day JHOOR-ay) “In law.” Contrast to de facto.dixi
(DIK-see) That settles it. Literally, “I have spoken.”emeritus
(ay-MARE-it-us) (feminine: emerita) Indicates someone who has served out his or her time and retired honorably. E.g., she is now professor emerita.ergo
(AIR-go) Therefore.erratum/errata
(air-AHT-um/uh) Literally, “error/errors,” this term in fact refers to the corrections included in a paper or book after it is published to correct minor errors in the text.et al.
(ETT ALL) Abbreviation of et alia, meaning “and others.” Used to indicate an unstated list of contributing authors following the main one, for instance.et cetera
(ETT SET-er-ah) And so forth. Note the pronunciation – there is no “eks” sound.ex post facto
(eks post FAK-toh) “From what is done afterward.”ibid.
See ibidem.ibidem
(ib-EED-em) “In the same place.” Used in footnotes to indicate that the reference is the same as the preceding one(s).in re
(IN RAY) “In regards to.” Often used to head formal correspondence. When only re is written, it should be translated as “regarding.”inter alia
(IN-ter ALL-ee-uh) Among other things.in toto
(in TOH-TOH) Entirely.in vacuo
(in VAK-yoo-oh) Literally, “in a vacuum.” Should be taken to mean “in the absence of other conditions or influences.” E.g., nobody achieves maturity in vacuo.magna cum laude
(MAG-nuh coom LOUD-ay) With great praise. See cum laude.modus operandi
(MODE-us op-ehr-AWN-dee) Manner or method of work characterizing a particular person’s professional habits.mutatis mutandis
(myoo-TAH-tis myoo-TAHN-dis) With necessary changes. “Mutatis mutandis, this proof applies in more general cases.”non sequitur
(nahn-SEK-wit-ter) “Not following.” Used to indicate a statement or conclusion that does not follow from what has gone before.per se
(per SAY) “In and of itself.” Example: “This argument does not force the conclusion per se, but with this added premise the result would follow.”post hoc, ergo propter hoc
(POST hawk air-go PROP-ter hawk) “After, therefore because of.” A common fallacy in reasoning, in which causality is ascribed to preceding conditions which were in fact irrelevent to the supposed effect.post scriptum
(post SKRIP-tum) “Written after.” Indicates an afterword or footnote to a main text, and is often used in written correspondence (where it is abbreviated p.s.).prima facie
(PRIME-uh FAYSH-uh) “On its face.” Indicates that a conclusion is indicated (but not necessarily proved) from the appearance of things.pro forma
(proh FOR-muh) “For form’s sake.” E.g., “It was a pro forma interview – the decision to hire her had already been made.”qua
(QWAH) “In the capacity of.” For example, “He is really very personable, but qua chairman he can be direct and even gruff.”quod vide
(kwawd VEE-day) Usually abbreviated q.v., this is a scholarly way of directing the reader to a reference.q.v.
See quod vide.sine qua non
(SIN-ay kwah NAHN) “That without which nothing.” Indicates an essential element or condition.summa cum laude
(SOOM-uh coom LOUD-ay) With greatest praise. See cum laude.tabula rasa
(TAB-yoo-lah RAH-sah) “Blank Slate.” Often refers to a person who has not yet formed prejudices or preconceptions on a given matter.verbatim
(ver-BATE-im) Word-for-word. Indicates a precise transmission of a phrase, discussion, or text.videlicet
(vee-DAY-lih-ket) Usually abbreviated viz., this is translated as “namely.” For example, “The math club picked a new president, viz., Carl.”http://www.mathacademy.com/pr/prime/articles/latin/index.asp