Mediant (math)

The median of two fractions ${\frac {a}{b}}$ ${\ displaystyle {\ frac {a} {b}}}$ ${\ displaystyle {\ frac {a} {b}}}$ and ${\frac {c}{d}}$ ${\ displaystyle {\ frac {c} {d}}}$ ${\ displaystyle {\ frac {c} {d}}}$ with positive denominators is called a fraction whose numerator is equal to the sum of the numerators, and the denominator is the sum of the denominators of these two fractions:

{\frac {a+c}{b+d}}.

{\ displaystyle {\ frac {a + c} {b + d}}.}

{\ displaystyle {\ frac {a + c} {b + d}}.}

Content

Properties

The median of two fractions is enclosed between them, i.e.

if a

{\frac {a}{b}}<{\frac {c}{d}}

{\ displaystyle {\ frac {a} {b}} <{\ frac {c} {d}}}

then

{\frac {a}{b}}<{\frac {a+c}{b+d}}<{\frac {c}{d}}

{\ displaystyle {\ frac {a} {b}} <{\ frac {a + c} {b + d}} <{\ frac {c} {d}}}

.

Evidence

This property is a consequence of the relations

{\frac {a+c}{b+d}}-{\frac {a}{b}}={{bc-ad} \over {b(b+d)}}={d \over {b+d}}\left({\frac {c}{d}}-{\frac {a}{b}}\right)>0

{\ displaystyle {\ frac {a + c} {b + d}} - {\ frac {a} {b}} = {{bc-ad} \ over {b (b + d)}} = {d \ over {b + d}} \ left ({\ frac {c} {d}} - {\ frac {a} {b}} \ right)> 0}

and

{\frac {c}{d}}-{\frac {a+c}{b+d}}={{bc-ad} \over {d(b+d)}}={b \over {b+d}}\left({\frac {c}{d}}-{\frac {a}{b}}\right)>0.

{\ displaystyle {\ frac {c} {d}} - {\ frac {a + c} {b + d}} = {{bc-ad} \ over {d (b + d)}} = {b \ over {b + d}} \ left ({\ frac {c} {d}} - {\ frac {a} {b}} \ right)> 0.}

If we write down 2 fractions, and then several times between each 2 adjacent fractions of their median, we get the Farey series .

History

The concept of the median of two fractions was introduced by A. Ya. Khinchin ^[1] in the theory of continued fractions for the purpose of better understanding the relative position and the law of successive formation of intermediate fractions. However, in the theory of continued fractions, for the study of intermediate fractions, the term “median” did not take root ^[2] . In other mathematical sciences, for example, in mathematical analysis ^[3] and in the theory of ordinary differential equations ^{[4], the} properties of the median n of the relations of real numbers were used in the proof of some propositions, although the definition of the concept of a median was not given. Indirectly, the broadest use of the median of n relations of real numbers was found in applied mathematics, in particular in mathematical statistics. ^[5] ^[6] ^[7] But the definition of the median in these works was also not given. Maurice Kline ^[8] , in fact, “rediscovered” the median by proposing “football arithmetic” for adding fractions. This addition M. Kline used to determine the average performance of a football player striker in two games. He also examined cases of determining the effectiveness of trade and the average vehicle speed based on speeds on two sections of the path.

Currently, the median is used in demography ^[9] and biology ^[10] .

Use

Stern Binary Tree - Broco
Minkowski function

Literature and notes

↑ Khinchin A.Ya. Chain fractions. - M .: Fizmatlit, 1961. 112 p.
↑ Leng S. Introduction to the theory of Diophantine approximations. - M.: Mir, 1970 .-- 104 p.
↑ Fichtenholtz G.M. The course of differential and integral calculus. T.1. - M.-L .: Gostekhlit, 1947 .-- 680 p.
↑ Stepanov V.V. The course of differential equations. - M .: Fizmatlit, 1959.- 468s.
↑ Salton G.A. Automatic processing, storage and retrieval of information. - M .: Owls. Radio, 1973. - 560 p.
↑ Schwartz G. The selective method. Guidance on the application of statistical estimation methods. - M.: Statistics, 1978. - 213 p.
↑ Crane M., Lemoine O. Introduction to the regenerative method of model analysis. - M .: Nauka, 1982. - 104 p.
↑ Kline M. Mathematics. Loss of certainty. - M .: Mir, 1984. - 434 p.
↑ Syomkin B.I., Soboleva T.A. Estimation of the rate of change in the total population of the cities of Primorsky Krai // Geography and Natural Resources. Number 4. 2005.S. 118-123.
↑ Syomkin B.I., Gorshkov M.V., Varchenko L.I. About changes in water content in annual shoots of coniferous woody plants in the temperate climatic zone // Siberian Ecol. journal 2008. No4. T. 15. P. 537-544.

[1] Khinchin A.Ya. Chain fractions. - M .: Fizmatlit, 1961. 112 p.

[2] Leng S. Introduction to the theory of Diophantine approximations. - M.: Mir, 1970 .-- 104 p.

[3] Fichtenholtz G.M. The course of differential and integral calculus. T.1. - M.-L .: Gostekhlit, 1947 .-- 680 p.

[4] Stepanov V.V. The course of differential equations. - M .: Fizmatlit, 1959.- 468s.

[5] Salton G.A. Automatic processing, storage and retrieval of information. - M .: Owls. Radio, 1973. - 560 p.

[6] Schwartz G. The selective method. Guidance on the application of statistical estimation methods. - M.: Statistics, 1978. - 213 p.

[7] Crane M., Lemoine O. Introduction to the regenerative method of model analysis. - M .: Nauka, 1982. - 104 p.

[8] Kline M. Mathematics. Loss of certainty. - M .: Mir, 1984. - 434 p.

[9] Syomkin B.I., Soboleva T.A. Estimation of the rate of change in the total population of the cities of Primorsky Krai // Geography and Natural Resources. Number 4. 2005.S. 118-123.

[10] Syomkin B.I., Gorshkov M.V., Varchenko L.I. About changes in water content in annual shoots of coniferous woody plants in the temperate climatic zone // Siberian Ecol. journal 2008. No4. T. 15. P. 537-544.

Mediant (math)

Content

Properties

History

Use

Literature and notes

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