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QM-AM-GM-HM Inequalities
In mathematics, the QM-AM-GM-HM inequalities, also known as the mean inequality chain, state the relationship between the harmonic mean (HM), geometric mean (GM), arithmetic mean (AM), and Root mean square, quadratic mean (QM; also known as root mean square). Suppose that x_1, x_2, \ldots, x_n are positive real numbers. Then : 0 0 implies the inequality: : \frac \leq \sqrt[n]. The ''n'' = 2 case When ''n'' = 2, the inequalities become :\frac \leq \sqrt \leq \frac\leq\sqrt for all x_1, x_2 > 0, which can be visualized in a semi-circle whose diameter is [''AB''] and center ''D''. Suppose ''AC'' = ''x''1 and ''BC'' = ''x''2. Construct perpendiculars to [''AB''] at ''D'' and ''C'' respectively. Join [''CE''] and [''DF''] and further construct a perpendicular [''CG''] to [''DF''] at ''G''. Then the length of ''GF'' can be calculated to be the harmonic mean, ''CF'' to be the geometric mean, ''DE'' to be the arithmetic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
