Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

AoPS Wiki:Article of the Day/Archive

< AoPS Wiki:Article of the Day
Revision as of 11:49, 22 December 2007 by Temperal (talk | contribs) (add to archive)

12/21/07

Zermelo-Fraenkel Axioms

The Zermelo-Fraenkel Axioms are a set of axioms that compiled by Ernst Zermelo and Abraham Fraenkel that make it very convenient for set theorists to determine whether a given collection of objects with a given property describable by the language of set theory could be called a set. As shown by paradoxes such as Russell's Paradox, some restrictions must be put on which collections to call sets.

This axiom establishes the... [more]

12/20/07

Joining an ARML team

Team selection for the American Regions Mathematics League varies from team to team.

Florida ARML sends two teams to ARML each year. The selection criteria for the Florida ARML team takes into consideration several factors:

  • AMC and AIME performance
  • Past AMC, AIME, and USAMO scores
  • Past ARML performance
  • FAMAT-designated competitions
  • An annual statewide tryout test

The organizers are Jason Wiggins of... [more]

12/19/07

Zorn's Lemma

Zorn's Lemma is a set theoretic result which is equivalent to the Axiom of Choice.

Let $A$ be a partially ordered set.

We say that $A$ is inductively ordered if every totally ordered subset $T$ of $A$ has an upper bound, i.e., an element $a \in A$ such that for all $x\in T$, $x \le a$. We say that $A$ is strictly inductively ordered if every totally ordered subset $T$ of $A$ has a least upper bound, i.e., an upper bound $a$ so that if $b$ is an upper bound of $T$, then $a \le b$.

An element $m \in A$ is maximal if the relation $a \ge m$ implies $a=m$. (Note that a set may have several maximal... [more]

12/18/07

There was no AotD for December eighteenth.

12/17/07

Diophantine equation

Awards.gif This article was also a AoPSWiki word of the week

A Diophantine equation is an multi-variable equation for which integer solutions (or sometimes natural number or whole number solutions) are to be found.

Finding the solution or solutions to a Diophantine equation is closely tied to modular arithmetic and number theory. Often, when a Diophantine equation has infinitely many solutions, parametric form is used to express the relation between the variables of the equation.

Diophantine equations are named for the ancient Greek/Alexandrian mathematician Diophantus.

A Diophantine equation in the form $ax+by=c$ is known as a linear combination. If two relatively prime integers $a$ and $b$ are written in this form with $c=1$, the equation will have an infinite number of solutions. More generally, there will always be an... [more]

12/16/07

Fibonacci sequence

Awards.gif This article was also a AoPSWiki word of the week

The Fibonacci sequence is a sequence of integers in which the first and second terms are both equal to 1 and each subsequent term is the sum of the two preceding it. The first few terms are $1, 1, 2, 3, 5, 8, 13, 21, 34, 55,...$.

The Fibonacci sequence can be written recursively as $F_1 = F_2 = 1$ and $F_n=F_{n-1}+F_{n-2}$ for $n \geq 3$. This is the simplest nontrivial... [more]

12/15/07

Cauchy-Schwarz inequality

The Cauchy-Schwarz Inequality (which is known by other names, including Cauchy's Inequality, Schwarz's Inequality, and the Cauchy-Bunyakovsky-Schwarz Inequality) is a well-known inequality with many elegant applications... [more]

12/14/07

Rearrangement inequality

The Rearrangement Inequality states that, if $A=\{a_1,a_2,\cdots,a_n\}$ is a permutation of a finite set (in fact, multiset) of real numbers and $B=\{b_1,b_2,\cdots,b_n\}$ is a permutation of another finite set of real numbers, the quantity $a_1b_1+a_2b_2+\cdots+a_nb_n$ is maximized when ${A}$ and ${B}$ are similarly sorted (that is, if $a_k$ is greater than or equal to exactly ${i}$ of the other members of $A$, then ${b_k}$ is also greater than or equal to exactly ${i}$ of the other members of $B$). Conversely, $a_1b_1+a_2b_2+\cdots+a_nb_n$ is minimized when $A$ and $B$ are oppositely sorted (that is, if $a_k$ is less than or equal

12/13/07

There was no AotD for December thirteenth.

12/12/07

Trigonometric identities

Trigonometric identities are used to manipulate trigonometry equations in certain ways. Here is a list of them:

The six basic trigonometric functions can be defined using [more]

12/11/07

MATHCOUNTS

MATHCOUNTS is a large national mathematics competition and mathematics coaching program that has served millions of middle school students since 1984. Sponsored by the CNA Foundation, National Society of Professional Engineers, the National Council of Teachers of Mathematics, and others, the focus of MATHCOUNTS is on mathematical problem solving. Students are eligible for up to three years, but cannot compete beyond their eighth grade year.

MATHCOUNTS curriculum... [more]

12/10/07

Polynomial

A polynomial is a function in one or more variables that consists of a sum of variables raised to nonnegative, integral powers and multiplied by coefficients (usually integral, rational, real or complex, but in abstract algebra often coming from an arbitrary field).

For example, these are... [more]

12/9/07

Pascal's identity

Pascal's identity is a common and useful theorem in the realm of combinatorics dealing with combinations (also known as binomial coefficients), and is often used to reduce large, complicated combinations.

Pascal's identity is also known as Pascal's rule, Pascal's formula, and occasionally... [more]

12/8/07

Canada/USA MathCamp

Awards.gif This article was also a AoPSWiki word of the week

The Canada/USA Mathcamp is an intensive 5-week-long summer program for mathematically talented high school students, designed to inspire and motivate mathematically talented high school students by exposing them to the beauty and variety of mathematics, to impart valuable knowledge and skills for the pursuit of mathematics in high school, university, and beyond, and to provide a supportive and fun environment for interaction among students who love mathematics.

The environment of Mathcamp tends to be relaxed in terms of rules; in fact it officially only has four rules (generally involving common sense and respect). Originally it had been stricter, though by 1997 most of the mentors found the structure too... [more]

12/7/07

Prime number

A prime number (or simply prime) is a positive integer $p>1$ whose only positive divisors are 1 and itself. Note that $1$ is usually defined as being neither prime nor composite because it is its only factor among the natural numbers. The Sieve of Eratosthenes is a relatively quick method for... [more]

12/6/07

Calculus

The discovery of the branch of mathematics known as calculus was motivated by two classical problems: how to find the slope of the tangent line to a curve at a point and how to find the area bounded by a curve. What is surprising is that these two problems are fundamentally connected and, together with the notion of limits, can be used to analyse instantaneous rates of change, accumulations of change, volumes of irregular solids, and... [more]

12/5/07

Pi

Pi is an irrational number (in fact, transcendental number, as proved by Lindeman in 1882) denoted by the greek letter $\pi$.

Pi is the ratio of the circumference (perimeter) of a given circle to its diameter. It is approximately equal to 3.141592653. The number pi is one of the most important constants in all of mathematics and appears in some of the most surprising places, such as in the sum $\sum_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6}$. Some common... [more]

12/4/07

Isaac Newton

Isaac Newton (1643 – 1727) was a famous British physicist and mathematician. His most famous work in mathematics was the compilation of calculus.

Isaac Newton was born on January 4, 1643 in Lincolnshire, England. Newton was born very shortly after the death of his father. He did very well... [more]

12/3/07

Logarithm

Awards.gif This article was also a AoPSWiki word of the week

Logarithms and exponents are very closely related. In fact, they are inverse functions. This means that logarithms can be used to reverse the result of exponentiation and vice versa, just as addition can be used to reverse the result of subtraction. Thus, if we have $a^x=b$, then taking the logarithm with base $a$ on both sides will give us $x=\log_{a}b$.

We would read this as "the logarithm of b, base a, is x". For example, we know that $3^4=81$. To express the same fact... [more]

12/2/07

American Invitational Mathematics Examination

The American Invitational Mathematics Examination (AIME) is the second exam in the series of exams used to challenge bright students on the path toward choosing the team that represents the United States at the International Mathematics Olympiad (IMO). While most AIME participants are high school students, some bright middle school students also qualify each year.

High scoring AIME students are invited to take the prestigious United States of America Mathematics Olympiad (USAMO).

The AIME is administered by... [more]

12/1/07

Inequality

The subject of mathematical inequalities is tied closely with optimization methods. While most of the subject of inequalities is often left out of the ordinary educational track, they are common in mathematics Olympiads.

Inequalities are arguably a branch of... [more]

11/30/07

Asymptote (geometry)

Awards.gif This article was also a AoPSWiki word of the week

An asymptote is a line or curve that a certain function approaches.

Linear asymptotes can be of three different kinds: horizontal, vertical or slanted (oblique).

The vertical asymptote can be found by finding values of $x$ that make the function undefined, generally because it results in a division by zero, which is undefined... [more]

11/27/07

Complex number

The complex numbers arise when we try to solve equations such as $x^2 = -1$.

We know (from the trivial inequality) that the square of a real number cannot be negative, so this equation has no solutions in the real numbers. However, it is possible to define a number, $i$, such that $i = \sqrt{-1}$. If we add this new number to the reals, we will have solutions to $x^2 = -1$. It turns out that in the system that results... [more]

11/25/07

Set

The notion of a set is one of the fundamental notions in mathematics that is difficult to precisely define. Of course, we have plenty of synonyms for the word "set," like collection, ensemble, group, etc., but those names really do not define the meaning of the word set; all they can do is replace it in various sentences. So, instead of defining what sets are, one has to define what can be done with them or, in other words, what axioms the sets satisfy. These axioms are chosen to agree with our intuitive concept of a set, on one hand, and to allow various, sometimes quite sophisticated, mathematical constructions on the other hand. For the full collection...[more]

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=AoPS_Wiki:Article_of_the_Day/Archive&oldid=21452"