AoPS Wiki:Article of the Day/Archive
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12/4/07
Logarithm
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
, then taking the logarithm with base
on both sides will give us
.
We would read this as "the logarithm of b, base a, is x". For example, we know that
. To express the same fact... [more]
, then taking the logarithm with base 
on both sides will give us 
.
. To express the same fact... 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)
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
that make the function undefined, generally because it results in a division by zero, which is undefined... [more]
that make the function undefined, generally because it results in a division by zero, which is undefined... 11/27/07
Complex number
The complex numbers arise when we try to solve equations such as
.
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,
, such that
. If we add this new number to the reals, we will have solutions to
.
, such that 
. If we add this new number to the reals, we will have solutions to 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]
