Ratio
The numbers in a ratio may be quantities of any kind, such as counts of people or objects, or such as measurements of lengths, weights, time, etc.
[10] A statement expressing the equality of two ratios A:B and C:D is called a proportion,[11] written as A:B = C:D or A:B∷C:D. This latter form, when spoken or written in the English language, is often expressed as A, B, C and D are called the terms of the proportion.
It is possible to trace the origin of the word "ratio" to the Ancient Greek λόγος (logos).
Early translators rendered this into Latin as ratio ("reason"; as in the word "rational").
A more modern interpretation of Euclid's meaning is more akin to computation or reckoning.
[15] Euclid collected the results appearing in the Elements from earlier sources.
The Pythagoreans developed a theory of ratio and proportion as applied to numbers.
The discovery of a theory of ratios that does not assume commensurability is probably due to Eudoxus of Cnidus.
[17] The existence of multiple theories seems unnecessarily complex since ratios are, to a large extent, identified with quotients and their prospective values.
However, this is a comparatively recent development, as can be seen from the fact that modern geometry textbooks still use distinct terminology and notation for ratios and quotients.
The reasons for this are twofold: first, there was the previously mentioned reluctance to accept irrational numbers as true numbers, and second, the lack of a widely used symbolism to replace the already established terminology of ratios delayed the full acceptance of fractions as alternative until the 16th century.
[19] In addition, Euclid uses ideas that were in such common usage that he did not include definitions for them.
It states that a ratio of two quantities exists, when there is a multiple of each that exceeds the other.
In modern notation, Euclid's definition of equality is that given quantities p, q, r and s, p:q∷r :s if and only if, for any positive integers m and n, np
Euclid uses the Greek ἀναλόγον (analogon), this has the same root as λόγος and is related to the English word "analog".
In modern notation it says that given quantities p, q, r and s, p:q>r:s if there are positive integers m and n so that np>mq and nr≤ms.
Sequences that have the property that the ratios of consecutive terms are equal are called geometric progressions.
In both ratios and fractions, it is important to be clear what is being compared to what, and beginners often make mistakes for this reason.
It is usual either to reduce terms to the lowest common denominator, or to express them in parts per hundred (percent).
For example, older televisions have a 4:3 aspect ratio, which means that the width is 4/3 of the height (this can also be expressed as 1.33:1 or just 1.33 rounded to two decimal places).
More recent widescreen TVs have a 16:9 aspect ratio, or 1.78 rounded to two decimal places.
Representing ratios as decimal fractions simplifies their comparison.
Such a comparison works only when values being compared are consistent, like always expressing width in relation to height.
As for fractions, the simplest form is considered that in which the numbers in the ratio are the smallest possible integers.
Similarly, the silver ratio of a and b is defined by the proportion This equation has the positive, irrational solution
[23][24] In chemistry, mass concentration ratios are usually expressed as weight/volume fractions.
The locations of points relative to a triangle with vertices A, B, and C and sides AB, BC, and CA are often expressed in extended ratio form as triangular coordinates.
In barycentric coordinates, a point with coordinates α, β, γ is the point upon which a weightless sheet of metal in the shape and size of the triangle would exactly balance if weights were put on the vertices, with the ratio of the weights at A and B being α : β, the ratio of the weights at B and C being β : γ, and therefore the ratio of weights at A and C being α : γ.
Since all information is expressed in terms of ratios (the individual numbers denoted by α, β, γ, x, y, and z have no meaning by themselves), a triangle analysis using barycentric or trilinear coordinates applies regardless of the size of the triangle.
