ratio

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The ratio of width to height of typical computer displays.

A ratio is an expression that compares quantities relative to each other. The most common examples involve two quantities, but any number of quantities can be compared. Ratios are represented mathematically by separating each quantity with a colon – for example, the ratio 2:3, which is read as the ratio "two to three". The quantities separated by colons are sometimes called terms.

History and etymology

Look up ratio in Wiktionary, the free dictionary.

It would be impossible to trace the origin of the concept of ratio since the ideas from which it developed would have been familiar to preliterate cultures. For example the idea of one village being twice as large as another or a distance being half that of another are so basic that they would have been understood in prehistoric society.^[1] However, it is possible to trace the origin of the word ratio to the Ancient Greek λόγος (logos) appearing in Book V of Euclid's Elements. Early translators rendered this into Latin as ratio, meaning "reason". However a more modern interpretation of Euclid's meaning is more akin to computation or reckoning.^[2] Medieval writers used the word proportio ("proportion") to indicate ratio and proportionalitas ("proportionality") for the equality of ratios.^[3]

Euclid collected the results appearing the Elements from earlier sources. The Pythagoreans developed a theory of ratio and proportion as applied to numbers.^[4] The Pythagoreans' conception of number included only what would today be called rational numbers, casting doubt on the validity of the theory in geometry where, as the Pythagoreans also discovered, incommensurable ratios (corresponding to irrational numbers) exist. The discovery of a theory of ratios that does not assume commensurability is probably due to Eudoxus. The exposition of the theory of proportions that appears in Book VII of The Elements reflects the earlier theory of ratios of commensurables.^[5]

The existence of multiple theories seems unnecessarily complex to modern sensibility since ratios are, to a large extent, identified with quotients. This is a comparatively recent development however, as can be seem 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. 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.^[6]

Euclid's definitions

Book V of Euclid's Elements has 18 definitions, all of which relate to ratios. In addition, Euclid uses ideas that were in such common usage that he did not include definitions for them. The first two definitions say that a part of a quantity is another quantity which "measures" it and, conversely, a multiple of a quantity is another quantity which it measures. In modern terminology this means that a multiple of a quantity is that quantity multiplied by an integer greater than one and a part of a quantity (meaning aliquot part) is that which, when multiplied by an integer greater than one, gives the quantity. Euclid does not define the term "measure" as used here but one may infer that a quantity is taken as a unit of measurement, and a second quantity is can given as an integral number of these units, then the first quantity measures the second. Note that these definitions are repeated, nearly word for word, as definitions 3 and 5 in book VII.

Definition 3 describes what a ratio is in a general way. It is not rigorous in a mathematical sense and some have ascribed it to Euclid's editors rather than Euclid himself.^[7] Euclid defines a ratio to be between two quantities of the same type, so by this definition the ratios of two lengths or of two areas are defined, but not the ratio of a length and an area. Definition 4 makes this more rigorous. It states that a ratio of two quantities exists when there is a multiple of each which exceeds the other. In modern notation, a ratio exists between quantities p and q if there exist integers m and n so that mp>q and nq>m. This condition is known as the Archimedean property.

Definition 5 is the most complex and difficult; it defines what it means for two ratios to be equal. Today, this can be done by simply stating that ratios are equal when the quotients of the terms are equal, but Euclid did not accept the existence of the quotients of incommensurables, so such a definition would have been meaningless to him. Thus, a more subtle definition is needed where quantities involved are not measured directly to one another. Though it may not be possible to assign a rational value to a ratio, it is possible to compare a ratio with a rational number. Specifically, given two quantities, p and q, and a rational number m/n we can say that the ratio of p to q is less than, equal to, or greater than m/n when np is less than, equal to, or greater than mq respectively. Euclid's definition of equality can be stated as that two ratios are equal when they behave identically with respect to being less than, equal to, or greater than any rational number. In modern notation this says that given quantities p, q, r and s, then p:q::r:s if for any positive integers m and n, np<mq, np=mq, np>mq according as nr<ms, nr=ms, nr>ms respectively. There is a remarkable similarity between this definition and the theory of Dedekind cuts used in the modern definition of irrational numbers.

Definition 6 says that quantities that have the same ratio are proportional or in proportion. Euclid uses the Greek ἀναλόγον (analogon), this has the same root as λόγος and is related to the English word "analog".

Definition 7 defines what it means for one ratio to be less than or greater than another and is based on the ideas present in definition 5. In modern notation it says that given quantities p, q, r and s, then p:q>r:s if there are positive integers m and n so that np>mq and nr≤ms.

As with definition 3, definition 8 is regarded by some as being a later insertion by Euclid's editors. It defines three terms p, q and r to be in proportion when p:q::q:r. This is extended to 4 terms p, q, r and s as p:q::q:r::r:s, and so on. Sequences which have the property that the ratios of consecutive terms are equal are called Geometric progressions. Definitions 9 and 10 apply this, saying that if p, q and r are in proportion then p:r is the duplicate ratio of p:q and p, q, r and s are in proportion then p:s is the triplicate ratio of p:q. If p, q and r are in proportion then q is called a mean proportional to p and r. Similarly, if p, q, r and s are in proportion then q and r are called two mean proportionals to p and s.

The remaining definitions are less important and need not be covered here.

Examples

The quantities being compared in a ratio might be physical quantities such as speed or temperature, or may simply refer to amounts of particular objects. A common example of the latter case is the weight ratio of water to cement used in concrete, which is commonly stated as 1:4. This means that the weight of cement used is four times the weight of water used. It does not say anything about the total amounts of cement and water used, nor the amount of concrete being made.

Ratios are often used for simple dilutions applied in biology. A simple dilution is one in which a unit volume of a liquid material of interest is combined with an appropriate volume of a solvent liquid to achieve the desired concentration. The dilution factor is the total number of unit volumes in which your material will be dissolved. The diluted material must then be thoroughly mixed to achieve the true dilution. For example, a 1:5 dilution (verbalize as "1 to 5" dilution) entails combining 1 unit volume of diluent (the material to be diluted) + 4 unit volumes of the solvent medium (hence, 1 + 4 = 5 = dilution factor). The dilution factor is frequently expressed using exponents: 1:5 would be 5e-1; 1:100 would be 10e-2, and so on.

Special notice should be made to ratios (usually) comparing one item to another item, as opposed to the whole. If there are 2 oranges and 3 apples, the ratio of oranges to apples is shown as 2:3 or 2/3, however the fraction of oranges to total fruit is 2/5.

Number of terms

In general, a ratio of 2:3 means that the amount of the first quantity is (two thirds) of the amount of the second quantity. This pattern works with ratios with more than two terms. However, a ratio with more than two terms cannot be completely converted into a single fraction; a single fraction represents only one part of the ratio. If the ratio deals with objects or amounts of objects, this is often expressed as "for every two parts of the first quantity there are three parts of the second quantity".

Proportions

If the two or more ratio quantities encompass all of the quantities in a particular situation, for example two apples and three oranges in a fruit basket containing no other types of fruit, it could be said that "the whole" contains five parts, made up of two parts apples and three parts oranges. In this case, , or 40% of the whole are apples and , or 60% of the whole are oranges. This comparison of a specific quantity to "the whole" is sometimes called a proportion. Proportions are sometimes expressed as percentages as demonstrated above.

Reduction

Note that ratios can be reduced as fractions are, by dividing each quantity by the common factors of all the quantities. Thus the ratio 4:6 may be considered equivalent in meaning to the ratio 2:3 within contexts concerned only with relative quantities.

Different units

Ratios are unit-less when they relate quantities which have the same units. When the two quantities being compared are of different types, the units are the first quantity "per" unit of the second – for example, a speed or velocity can be expressed in "miles per hour". A ratio for which the second unit is a measure of time is called a rate.

Ratios are used frequently throughout the physical sciences, and in many cases a ratio is thought of as a single value. For example, the ratio 60 metres to 1 second, or 60:1 is written as 60 m/s, or 60 ms⁻¹, "60 metres per second" and is thought of as a measurement of velocity. In this case, the measurement is actually a ratio between two quantities with different units.

See also

• Proportionality (mathematics)
• Rule of three (mathematics)

References

1. ^ Smith, p. 477
2. ^ Penny Cyclopedia, p. 307
3. ^ Smith, p. 478
4. ^ Heath, p. 112
5. ^ Heath, p. 113
6. ^ Smith, p. 480
7. ^ "Geometry, Euclidean" Encyclopædia Britannica Eleventh Edition p682.

• "Ratio" The Penny Cyclopædia vol. 19, The Society for the Diffusion of Useful Knowledge (1841) Charles Knight and Co., London pp. 307ff
• The thirteen books of Euclid's Elements, vol 2. trans. Sir Thomas Little Heath (1908). Cambridge Univ. Press. pp. 112ff. http://books.google.com/books?id=lxkPAAAAIAAJ&pg=RA3-PA112. 
• D.E. Smith, History of Mathematics, vol 2 Dover (1958) pp. 477ff