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Difference between revisions of "Inequality"

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== Motivation ==
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==Overview==
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We say that ''a>b'' (or, equivalently, ''b<a'') if ''a'' and ''b'' are [[real number]]s, and ''a-b'' is a [[positive number]]. However, there are many inequalities that are much more interesting and also very important, such as the ones listed below.
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Inequalities are arguably a branch of [[number theory]]. They deal with relations of variables denoted by four signs: <math>>,<,\ge,\le</math>.
  
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For two numbers <math>a</math> and <math>b</math>:
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*<math>a>b</math> if <math>a</math> is greater than <math>b</math>, that is, <math>a-b</math> is positive.
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*<math>a<b</math> if <math>a</math> is smaller than <math>b</math>, that is, <math>a-b</math> is negative.
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*<math>a\ge b</math> if <math>a</math> is greater than or equal to <math>b</math>, that is, <math>a-b</math> is either positive or <math>0</math>.
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*<math>a\le b</math> if <math>a</math> is less than or equal to <math>b</math>, that is, <math>a-b</math> is either negative or <math>0</math>.
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Note that if and only if <math>a>b</math>, <math>b<a</math>, and vice versa. The same applies to the latter two signs: if and only if <math>a\ge b</math>, <math>b\le a</math>, and vice versa.
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Some properties of inequalities are:
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*If <math>a>b</math>, then <math>a+c>b</math>, where <math>c\ge 0</math>.
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*If <math>a \ge b</math>, then <math>a+c\ge b</math>, where <math>c\ge 0</math>.
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*If <math>a \ge b</math>, then <math>a+c>b</math>, where <math>c>0</math>.
  
 
==Introductory==
 
==Introductory==

Revision as of 18:27, 25 October 2007

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.


Overview

Inequalities are arguably a branch of number theory. They deal with relations of variables denoted by four signs: $>,<,\ge,\le$.

For two numbers $a$ and $b$:

  • $a>b$ if $a$ is greater than $b$, that is, $a-b$ is positive.
  • $a<b$ if $a$ is smaller than $b$, that is, $a-b$ is negative.
  • $a\ge b$ if $a$ is greater than or equal to $b$, that is, $a-b$ is either positive or $0$.
  • $a\le b$ if $a$ is less than or equal to $b$, that is, $a-b$ is either negative or $0$.

Note that if and only if $a>b$, $b<a$, and vice versa. The same applies to the latter two signs: if and only if $a\ge b$, $b\le a$, and vice versa.

Some properties of inequalities are:

  • If $a>b$, then $a+c>b$, where $c\ge 0$.
  • If $a \ge b$, then $a+c\ge b$, where $c\ge 0$.
  • If $a \ge b$, then $a+c>b$, where $c>0$.

Introductory


Intermediate

Example Problems


Olympiad

See the list of famous inequalities below


Famous inequalities

Here are some of the more famous and useful inequalities, as well as general inequalities topics.

Problem solving tactics

substitution, telescoping, induction, etc. (write me please!)


Resources

Books

Intermediate

Olympiad

Articles

Olympiad


Classes

Olympiad


See also

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