Generalized definition of all means of $n$ numbers?
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Following chapter 2 of Greek Means and the arithmetic-geometric mean by George and Silvia Toader, the Greek (or Pythagorean) means of two numbers have been studied for centuries originating in the Pythagorean School. The Greek means labeled $m$ of two numbers $a$ and $b$ are defined by equating one of
$$frac{a-m}{a-b},spacefrac{a-b}{m-b},spacetext{or}spacefrac{a-m}{m-b}$$
and
$$frac{a}{a},spacefrac{a}{b},spacefrac{b}{a},spacefrac{a}{m},spacefrac{m}{a},spacefrac{b}{m},spacetext{or}spacefrac{m}{b}$$
This yields 21 Greek means, but only 10 are unique and nontrivial.
This is a nice general definition for all means of two numbers, but what is a general definition for all means of $n$ numbers?
means
asked Dec 27 '18 at 23:38
tyobrientyobrien
1,185514
$endgroup$
add a comment |
$begingroup$
Following chapter 2 of Greek Means and the arithmetic-geometric mean by George and Silvia Toader, the Greek (or Pythagorean) means of two numbers have been studied for centuries originating in the Pythagorean School. The Greek means labeled $m$ of two numbers $a$ and $b$ are defined by equating one of
$$frac{a-m}{a-b},spacefrac{a-b}{m-b},spacetext{or}spacefrac{a-m}{m-b}$$
and
$$frac{a}{a},spacefrac{a}{b},spacefrac{b}{a},spacefrac{a}{m},spacefrac{m}{a},spacefrac{b}{m},spacetext{or}spacefrac{m}{b}$$
This yields 21 Greek means, but only 10 are unique and nontrivial.
This is a nice general definition for all means of two numbers, but what is a general definition for all means of $n$ numbers?
means
asked Dec 27 '18 at 23:38
tyobrientyobrien
1,185514
$endgroup$
add a comment |
$begingroup$
Following chapter 2 of Greek Means and the arithmetic-geometric mean by George and Silvia Toader, the Greek (or Pythagorean) means of two numbers have been studied for centuries originating in the Pythagorean School. The Greek means labeled $m$ of two numbers $a$ and $b$ are defined by equating one of
$$frac{a-m}{a-b},spacefrac{a-b}{m-b},spacetext{or}spacefrac{a-m}{m-b}$$
and
$$frac{a}{a},spacefrac{a}{b},spacefrac{b}{a},spacefrac{a}{m},spacefrac{m}{a},spacefrac{b}{m},spacetext{or}spacefrac{m}{b}$$
This yields 21 Greek means, but only 10 are unique and nontrivial.
This is a nice general definition for all means of two numbers, but what is a general definition for all means of $n$ numbers?
means
asked Dec 27 '18 at 23:38
tyobrientyobrien
1,185514
$endgroup$
Following chapter 2 of Greek Means and the arithmetic-geometric mean by George and Silvia Toader, the Greek (or Pythagorean) means of two numbers have been studied for centuries originating in the Pythagorean School. The Greek means labeled $m$ of two numbers $a$ and $b$ are defined by equating one of
$$frac{a-m}{a-b},spacefrac{a-b}{m-b},spacetext{or}spacefrac{a-m}{m-b}$$
and
$$frac{a}{a},spacefrac{a}{b},spacefrac{b}{a},spacefrac{a}{m},spacefrac{m}{a},spacefrac{b}{m},spacetext{or}spacefrac{m}{b}$$
This yields 21 Greek means, but only 10 are unique and nontrivial.
This is a nice general definition for all means of two numbers, but what is a general definition for all means of $n$ numbers?
means
means
asked Dec 27 '18 at 23:38
tyobrientyobrien
1,185514
asked Dec 27 '18 at 23:38
tyobrientyobrien
1,185514
edited Dec 28 '18 at 1:53
tyobrien
asked Dec 27 '18 at 23:38
tyobrientyobrien
1,185514
asked Dec 27 '18 at 23:38
tyobrientyobrien
1,185514
asked Dec 27 '18 at 23:38
tyobrientyobrien
1,185514
1,185514
add a comment |
add a comment |
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