Difference between revisions of "AoPS Wiki:Article of the Day/Archive"
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| + | <font color="#B2B7F2" style="font-size:40px;">“</font>The '''complex numbers''' arise when we try to solve [[equation]]s such as <math> x^2 = -1 </math>. | ||
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| + | 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... <font color="#B2B7F2" style="font-size:40px">”</font> [[complex number|[more]]] | ||
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| + | ==11/25/07== | ||
<font color="#B2B7F2" style="font-size:40px">“</font>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...<font color="#B2B7F2" style="font-size:40px">”</font> [[set|[more]]] | <font color="#B2B7F2" style="font-size:40px">“</font>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...<font color="#B2B7F2" style="font-size:40px">”</font> [[set|[more]]] | ||
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Revision as of 23:28, 27 November 2007
11/27/07“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, 11/25/07“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] |
.
, such that 
. If we add this new number to the reals, we will have solutions to 
