Proportion (mathematics)
Template:Short description A proportion is a mathematical statement expressing equality of two ratios.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref name="fundrule">Template:Cite book</ref>
<math>a:b=c:d</math>
a and d are called extremes, b and c are called means.
Proportion can be written as <math>\frac{a}{b}=\frac{c}{d}</math>, where ratios are expressed as fractions.
Such a proportion is known as geometrical proportion,<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> not to be confused with arithmetical proportion and harmonic proportion.
Properties of proportions
- Fundamental rule of proportion. This rule is sometimes called Means‐Extremes Property.<ref name="proportionrule">{{#invoke:citation/CS1|citation
|CitationClass=web }}</ref> If the ratios are expressed as fractions, then the same rule can be phrased in terms of the equality of "cross-products"<ref name="fundrule"/> and is called Cross‐Products Property.<ref name="proportionrule"/>
- If <math>\ \frac ab=\frac cd</math>, then <math>\ ad=bc</math>
- If <math>\ \frac ab=\frac cd</math>, then <math>\ \frac ba=\frac dc</math>
- If <math>\ \frac ab=\frac cd</math>, then
- <math>\ \frac ac=\frac bd</math>,
- <math>\ \frac db=\frac ca</math>.
- If <math>\ \frac ab=\frac cd</math>, then
- <math>\ \dfrac{a+b}{b}=\dfrac{c+d}{d}</math>,
- <math>\ \dfrac{a-b}{b}=\dfrac{c-d}{d}</math>.
- If <math>\ \frac ab=\frac cd</math>, then
- <math>\ \dfrac{a+c}{b+d}=\frac ab =\frac cd</math>,
- <math>\ \dfrac{a-c}{b-d}=\frac ab =\frac cd</math>.
History
A Greek mathematician Eudoxus provided a definition for the meaning of the equality between two ratios. This definition of proportion forms the subject of Euclid's Book V, where we can read:
Later, the realization that ratios are numbers allowed to switch from solving proportions to equations, and from transformation of proportions to algebraic transformations.
Related concepts
Arithmetic proportion
An equation of the form <math>a-b = c-d</math> is called arithmetic proportion or difference proportion.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>
Harmonic proportion
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} If the means of the geometric proportion are equal, and the rightmost extreme is equal to the difference between the leftmost extreme and a mean, then such a proportion is called harmonic:<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> <math> a : b = b : (a - b) </math>. In this case the ratio <math> a : b </math> is called golden ratio.