Part I. Section I. Chapter 8. “Of the Properties of Fractions.”

85 We have already seen, that each of the fractions,

²⁄₂, ³⁄₃, ⁴⁄₄, ⁵⁄₅, ⁶⁄₆, ⁷⁄₇, ⁸⁄₈, etc.

makes an integer, and that consequently they are all equal to one another. The same equality prevails in the following fractions,

²⁄₁, ⁴⁄₂, ⁶⁄₃, ⁸⁄₄, ¹⁰⁄₅, ¹²⁄₆, etc.

each of them making two integers; for the numerator of each, divided by its denominator, gives 2. So all the fractions

³⁄₁, ⁶⁄₂, ⁹⁄₃, ¹²⁄₄, ¹⁵⁄₅, ¹⁸⁄₆, etc.

are equal to one another, since 3 is their common value.

86 We may likewise represent the value of any fraction in an infinite variety of ways. For if we multiply both the numerator and the denominator of a fraction by the same number, which may be assumed at pleasure, this fraction will still preserve the same value. For this reason, all the fractions

½, ²⁄₄, ³⁄₆, ⁴⁄₈, ⁵⁄₁₀, ⁶⁄₁₂, ⁷⁄₁₄, ⁸⁄₁₆, ⁹⁄₁₈, ¹⁰⁄₂₀, etc.

are equal, the value of each being ½. Also,

⅓, ²⁄₆, ³⁄₉, ⁴⁄₁₂, ⁵⁄₁₅, ⁶⁄₁₈, ⁷⁄₂₁, ⁸⁄₂₄, ⁹⁄₂₇, ¹⁰⁄₃₀, etc.

are equal fractions, the value of each being ⅓. The fractions

⅔, ⁴⁄₆, ⁶⁄₉, ⁸⁄₁₂, ¹⁰⁄₁₅, ¹²⁄₁₈, ¹⁴⁄₂₁, ¹⁶⁄₂₄, etc.

have likewise all the same value. Hence we may conclude, in general, that the fraction \(\frac{a}{b}\) may be represented by any of the following expressions, each of which is equal to \(\frac{a}{b}\); namely

\[\frac{a}{b},\frac{2a}{2b},\frac{3a}{3b},\frac{4a}{4b},\frac{5a}{5b},\frac{6a}{6b}, \frac{7a}{7b},\textrm{etc.}\]

87 To be convinced of this, we have only to write for the value of the fraction \(\frac{a}{b}\) a certain letter \(c\), representing by this letter \(c\) the quotient of the division of \(a\) by \(b\); and to recollect that the multiplication of the quotient \(c\) by the divisor \(b\) must give the dividend. For since \(c\) multiplied by \(b\) gives \(a\), it is evident that \(c\) multiplied by \(2b\) will give \(2a\), that \(c\) multiplied by \(3b\) will give \(3a\), and that, in general, \(c\) multiplied by \(mb\) will give \(ma\). Now, changing this into an example of division, and dividing the product \(ma\) by \(mb\), one of the factors, the quotient must be equal to the other factor \(c\); but \(ma\) divided by \(mb\) gives also the fraction \(\frac{ma}{mb}\), which is consequently equal to \(c\); which is what was to be proved; for \(c\) having been assumed as the value of the fraction \(\frac{a}{b}\), it is evident that this fraction is equal to the fraction \(\frac{ma}{mb}\), whatever be the value of \(m\).

88 We have seen that every fraction may be represented in an infinite number of forms, each of which contains the same value; and it is evident that of all these forms, that which is composed of the least numbers, will be most easily miderstood. For example, we might substitute, instead of ⅔, the following fractions,

⁴⁄₆, ⁶⁄₉, ⁸⁄₁₂, ¹⁰⁄₁₅, ¹²⁄₁₈, etc.

but of all these expressions ⅔ is that of which it is easiest to form an idea. Here therefore a problem arises, how a fraction, such as ⁸⁄₁₂, which is not expressed by the least possible numbers, may be reduced to its simplest form, or to its least terms; that is to say, in our present example, to ⅔.

89 It will be easy to resolve this problem, if we consider that a fraction still preserves its value, when we multiply both its terms, or its numerator and denominator, by the same number. For from this it also follows, that if we divide the numerator and denominator of a fraction by the same number, the fraction will still preserve the same value. This is made more evident by means of the general expression \(\frac{ma}{mb}\); for if we divide both the numerator \(ma\) and the denominator \(mb\) by the number \(m\), we obtain the fraction \(\frac{a}{b}\), which, as was before proved, is equal to \(\frac{ma}{mb}\).

90 In order therefore to reduce a given fraction to its least terms, it is required to find a number, by which both the numerator and denominator may be divided. Such a number is called a common divisor; and as long as we can find a common divisor to the numerator and the denominator, it is certain that the fraction may be reduced to a lower form; but, on the contrary, when we see that, except unity, no other common divisor can be found, this shows that the fraction is already in its simplest form.

91 To make this more clear, let us consider the fraction ⁴⁸⁄₁₂₀. We see immediately that both the terms are divisible by 2, and that there results the fraction ²⁴⁄₆₀; which may also be divided by 2, and reduced to ¹²⁄₃₀; and as this likewise has 2 for a common divisor, it is evident that it may be reduced to ⁶⁄₁₅. But now we easily perceive, that the numerator and denominator are still divisible by 3; performing this division, therefore, we obtain the fraction ⅖, which is equal to the fraction proposed, and gives the simplest expression to which it can be reduced; for 2 and 5 have no common divisor but 1, which cannot diminish these numbers any farther.

92 This property of fractions preserving an invariable value, whether we divide or multiply the numerator and denominator by the same number, is of the greatest importance, and is the principal foundation of the doctrine of fractions. For example, we can seldom add together two fractions, or subtract the one from the other, before we have, by means of this property, reduced them to other forms; that is to say, to expressions whose denominators are equal. Of this we shall treat in the following chapter.

93 We will conclude the present, however, by remarking, that all whole numbers may also be represented by fractions. For example, 6 is the same as ⁶⁄₁, because 6 divided by 1 makes 6; we may also, in the same manner, express the number 6 by the fractions ¹²⁄₂, ¹⁸⁄₃, ²⁴⁄₄, ³⁶⁄₆, and an infinite number of others, which have the same value.


  1. Leonhard Euler. Elements of Algebra. Translated by Rev. John Hewlett. Third Edition. Longmans, Hurst, Rees, Orme, and Co. London. 1822.
  2. Leonhard Euler. Vollständige Anleitung zur Algebra. Mit den Zusätzen von Joseph Louis Lagrange. Herausgegeben von Heinrich Weber. B. G. Teubner. Leipzig and Berlin. 1911. Leonhardi Euleri Opera omnia. Series prima. Opera mathematica. Volumen primum.