Difference between revisions of "AoPS Wiki:Article of the Day/Archive"
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==11/30/07== | ==11/30/07== | ||
| − | < | + | <blockquote style="display:table;background:#eeeeee;padding:10px;" class="toccolours"> |
| + | ===Asymptote (geometry)=== | ||
| + | |||
| + | 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). | Linear asymptotes can be of three different kinds: horizontal, vertical or slanted (oblique). | ||
| − | The vertical asymptote can be found by finding values of <math>x</math> that make the function undefined, generally because it results in a division by zero, which is undefined... | + | The vertical asymptote can be found by finding values of <math>x</math> that make the function undefined, generally because it results in a division by zero, which is undefined... [[asymptote (geometry)|[more]]]</blockquote> |
==11/27/07== | ==11/27/07== | ||
| − | < | + | <blockquote style="display:table;background:#eeeeee;padding:10px;" class="toccolours"> |
| + | ===Complex number=== | ||
| + | The '''complex numbers''' arise when we try to solve [[equation]]s such as <math> x^2 = -1 </math>. | ||
| − | 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, <math> i </math>, such that <math> i = \sqrt{-1} </math>. If we add this new number to the reals, we will have solutions to <math> x^2 = -1 </math>. It turns out that in the system that results... | + | 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, <math> i </math>, such that <math> i = \sqrt{-1} </math>. If we add this new number to the reals, we will have solutions to <math> x^2 = -1 </math>. It turns out that in the system that results... [[complex number|[more]]]</blockquote> |
==11/25/07== | ==11/25/07== | ||
| − | < | + | <blockquote style="display:table;background:#eeeeee;padding:10px;" class="toccolours"> |
| + | ===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...[[set|[more]]] | ||
| + | </blockquote> | ||
Revision as of 21:15, 1 December 2007
11/30/07
Asymptote (geometry)
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]
