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In mathematics, a ratio is a relationship between two numbers indicating how many times the first number contains the second.〔Penny Cyclopedia, p. 307〕 For example, if a bowl of fruit contains eight oranges and six lemons, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ratio 4:3). Thus, a ratio can be a fraction as opposed to a whole number. Also, in this example the ratio of lemons to oranges is 6:8 (or 3:4), and the ratio of oranges to the total amount of fruit is 8:14 (or 4:7). The numbers compared in a ratio can be any quantities of a comparable kind, such as objects, persons, lengths, or spoonfuls. A ratio is written "''a'' to ''b''" or ''a'':''b'', or sometimes expressed arithmetically as a quotient of the two.〔New International Encyclopedia〕 When the two quantities have the same units, as is often the case, their ratio is a dimensionless number. A rate is a quotient of variables having different units. But in many applications, the word ''ratio'' is often used instead for this more general notion as well.〔''"The quotient of two numbers (or quantities); the relative sizes of two numbers (or quantities)"'', "The Mathematics Dictionary" ()〕 ==Notation and terminology== The ratio of numbers ''A'' and ''B'' can be expressed as:〔New International Encyclopedia〕 *the ratio of A to B *A is to B '' (followed by "as ''C'' is to ''D''")'' *A:B *A fraction that is the quotient: A divided by B: , which can be expressed as either a simple or a decimal fraction.〔Decimal fractions are frequently used in technological areas where ratio comparisons are important, such as aspect ratios (imaging), compression ratios (engines or data storage), etc.〕 The numbers ''A'' and ''B'' are sometimes called ''terms'' with ''A'' being the ''antecedent'' and ''B'' being the ''consequent''.〔(from the Encyclopedia Britannica )〕 The proportion expressing the equality of the ratios ''A'':''B'' and ''C'':''D'' is written ''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'' is to ''B'' as ''C'' is to ''D''. ''A'', ''B'', ''C'' and ''D'' are called the terms of the proportion. ''A'' and ''D'' are called the ''extremes'', and ''B'' and ''C'' are called the ''means''. The equality of three or more proportions is called a continued proportion.〔New International Encyclopedia〕 Ratios are sometimes used with three or more terms. The ratio of the dimensions of a "two by four" that is ten inches long is 2:4:10. A good concrete mix is sometimes quoted as 1:2:4 for the ratio of cement to sand to gravel.〔(Belle Group concrete mixing hints )〕 For a mixture of 4/1 cement to water, it could be said that the ratio of cement to water is 4:1, that there is 4 times as much cement as water, or that there is a quarter (1/4) as much water as cement. ==History and etymology== It is impossible to trace the origin of the ''concept'' of ratio, because 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 is so basic that it would have been understood in prehistoric society.〔Smith, p. 477〕 However, 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 rational number may be expressed as the quotient of two integers.) A more modern interpretation of Euclid's meaning is more akin to computation or reckoning.〔Penny Cyclopedia, p. 307〕 Medieval writers used the word ''proportio'' ("proportion") to indicate ratio and ''proportionalitas'' ("proportionality") for the equality of ratios.〔Smith, p. 478〕 Euclid collected the results appearing in the Elements from earlier sources. The Pythagoreans developed a theory of ratio and proportion as applied to numbers.〔Heath, p. 112〕 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 of Cnidus. The exposition of the theory of proportions that appears in Book VII of The Elements reflects the earlier theory of ratios of commensurables.〔Heath, p. 113〕 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 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. 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.〔Smith, p. 480〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「ratio」の詳細全文を読む スポンサード リンク
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