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Difference between revisions of "AoPS Wiki:Article of the Day/Archive"

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===Calculus===
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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 [[rate]]s of change, accumulations of change, [[volume]]s of irregular [[solid]]s, and... [[Calculus|[more]]]
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Revision as of 20:06, 7 December 2007

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]

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