Proportion (mathematics)

Properties of proportions

• Fundamental rule of proportion. This rule is sometimes called Means‐Extremes Property. If the ratios are expressed as fractions, then the same rule can be phrased in terms of the equality of "cross-products" and is called Cross‐Products Property. :If \ \frac ab=\frac cd, then \ ad=bc
• If \ \frac ab=\frac cd, then \ \frac ba=\frac dc
• If \ \frac ab=\frac cd, then : \ \frac ac=\frac bd, : \ \frac db=\frac ca.
• If \ \frac ab=\frac cd, then : \ \dfrac{a+b}{b}=\dfrac{c+d}{d}, : \ \dfrac{a-b}{b}=\dfrac{c-d}{d}.
• If \ \frac ab=\frac cd, then : \ \dfrac{a+c}{b+d}=\frac ab =\frac cd, : \ \dfrac{a-c}{b-d}=\frac ab =\frac cd.

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 a-b = c-d is called arithmetic proportion or difference proportion.

Harmonic proportion

Main article: Golden ratio

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: a : b = b : (a - b) . In this case the ratio a : b is called golden ratio.

References

References

1. Stapel, Elizabeth. "Proportions: Introduction".
2. (January 2012). "Intermediate Algebra: Identify Ratios, Rates, and Proportions". Cengage Learning.
3. "Geometrical proportion".
4. "Properties of Proportions".
5. "Arithmetic proportion".
6. "Harmonic Proportion in Architecture: Definition & Form".