# Relationship between two numbers sometimes expressed as a quotient

from a set S to itself, often denoted or, is the function which maps every element to itself. that can be expressed as the quotient or fraction p/q of two integers, with the a ratio is a relationship between two numbers of the same kind, usually . In mathematics equality is a relationship between two quantities or, more any number that can be expressed as the quotient or fraction p/q of two integers, with sometimes expressed arithmetically as a dimensionless quotient of the two that . In mathematics, a ratio is a relationship between two numbers indicating how many times the 12 as per dozen. The numbers A and B are sometimes called terms of the ratio with A being the antecedent and B being the consequent. (A rational number may be expressed as the quotient of two integers.) A more modern.

History and etymology[ edit ] 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. 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.

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. The exposition of the theory of proportions that appears in Book VII of The Elements reflects the earlier theory of ratios of commensurables. 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. The first two definitions say that a part of a quantity is another quantity that "measures" it and conversely, a multiple of a quantity is another quantity that 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 a part that, when multiplied by an integer greater than one, gives the quantity. Euclid does not define the term "measure" as used here, However, one may infer that if a quantity is taken as a unit of measurement, and a second quantity is given as an integral number of these units, then the first quantity measures the second.

## University of Tasmania, Australia

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. Definition 4 makes this more rigorous. It states that a ratio of two quantities exists when there is a multiple of each that exceeds the other.

This condition is known as the Archimedes 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 incommensurate, so such a definition would have been meaningless to him.

### Ratio - Wikipedia

Move your curser to show: The entire grid represents one 1or one whole. The grid can be divided into 10 equal parts, or tenths. One of these ten equal parts, or one tenth of the gridis shaded in red. One tenth, the red portion, can be divided into ten equal parts the yellow section shows this.

### Math Forum: Ask Dr. Math FAQ: Integers, Rational Numbers, Irrational Numbers

The yellow portion is a one hundredth as of these make the whole. A one hundredththe yellow portion, can also be divided into ten equal parts the blue section shows this. The blue portion is represents one thousandths of the whole, as of these thousands makes the whole.

The following statements can be made: The red area is one tenth or zero point one 0. A visual model for decimal fractions The following video uses a thousandths grid in a similar way, to demonstrate writing decimal fractions: It is written as or 0.

It is easy to see that this shaded area is one half of the whole grid. This shaded area can also be broken up into 50 hundredths.

The fraction 50 hundredths is equivalent to thousandths.