Proof Without Words: Ordering Arithmetic,Geometric, and Harmonic Means

C. L. Frenzen

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Proof Without Words: Ordering Arithmetic,Geometric, and Harmonic Means

机译：没有单词的证明：算术，几何和调和方法的排序

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The arithmetic mean A(a, b) = (a + b)/2, the geometric mean G(a, b) =Jab, andthe harmonic mean H (a , b) = 2ab I (a + b) can be seen from the picture to satisfyH (a , b) < G (a , b) < A (a , b), with equality if and only if a = b.

机译：算术平均值A（a，b）=（a + b）/ 2，几何平均值G（a，b）= Jab，谐波平均值H（a，b）= 2ab I（a + b）从图片满足H（a，b）

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《Mathematics Magazine》 |2009年第2期|共1页
作者
C. L. Frenzen;
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原文格式 PDF
正文语种 eng
中图分类数学;
关键词
arithmetic; harmonic; geometric;

机译：算术;谐波;几何;

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专利

1. Proof Without Words: Ordering Arithmetic,Geometric, and Harmonic Means [J] . C. L. Frenzen Mathematics Magazine . 2009,第2期

机译：没有单词的证明：算术，几何和调和方法的排序
2. Proof Without Words: Limit of a Recursive Arithmetic Mean [J] . ángel Plaza Mathematics Magazine . 2016,第3期

机译：缺勤没有词：递归算术平均值的限制
3. 102.42 Proof without Words: An alternating geometrical series [J] . áNGEL PLAZA The Mathematical gazette . 2018,第555期

机译：102.42缺乏单词的证明：交替的几何系列
4. Isometry group, words and proofs of geometric theorems [C] . Dominique Michelucci ACM symposium on Applied computing . 2008

机译：等距组，单词和几何定理的证明
5. Reducing the Costs of Proof Assistant Based Formal Verification or: Conviction without the Burden of Proof. [D] . Tatlock, Zachary Lee. 2014

机译：降低基于证明助手的形式验证的成本或：无需证明负担的定罪。
6. Harmonic publication and citation counting: sharing authorship credit equitably – not equally geometrically or arithmetically [O] . Nils T. Hagen -1

机译：谐波发布和引文计数：平等地分享作者的著作权-几何算术不均等
7. Proof Without Words: Limit of a Recursive Arithmetic Mean [O] . Ángel Plaza 2016

机译：证明没有词：递归算术平均值的限制
8. History of Arithmetic. Search for Traces of the Abacus System after This Method Had Taken the Name Algorithm. Proofs That During All Times Up Until the 16th Century, It Was Known That Common Arithmetic Originated in This Ancient Method [R] . Chasles, M. 1943

机译：算术史。在此方法采用名称算法后搜索算盘系统的痕迹。在16世纪以来的所有时间的证据，已知这种古老方法的共同算术

Proof Without Words: Ordering Arithmetic,Geometric, and Harmonic Means

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