harmonic mean

We are given two positive numbers a and b. We wish to show how to construct a length corresponding to the harmonic mean of a and b...

This article is the second in the ‘Inequalities’ series. We prove a very important inequality, the Arithmetic Mean-Geometric Mean inequality (‘AM-GM inequality’) which has a vast number of applications and generalizations. We prove it using algebra as well as geometry

For each of these three means (arithmetic mean, geometric mean and harmonic mean) , there is a simple and well-known geometric construction that illustrates it, but is there a single diagram that illustrated all three at the same time?
In the accompanying article Approximating Square Roots and Cube Roots, the author Ali Ibrahim Hussen has proposed
easy to use formulas for finding approximate values of the square root and cube root of an arbitrary positive number n. The formulas are found to give fairly satisfactory results, as measured by the low percentage error. In this article we explain mathematically why this is so.
The idea of equivalent geometric forms is used in this study to devise simple formulas to estimate the square root and the cube root of an arbitrary positive number. The resulting formulas are easy to use and they don’t take much time to calculate.
A “proof without words” sounds like a contradiction in terms! How can you prove something if you are not permitted the use of any words? In spite of the seeming absurdity of the idea, the notion of a proof without words — generally shortened to PWW — has acquired great popularity in mathematics in recent decades, and every now and then we come across new, elegant PWWs for old, familiar propositions. In this short article the seemingly contradictory nature of a PWW is discussed, and some examples of PWWs are presented.
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