### Part I. Section I. Chapter 10. “Of the Multiplication and Division of Fractions.”

101 The rule for the multiplication of a fraction by an integer, or whole number, is to multiply the numerator only by the given number, and not to change the denominator: thus,

2 times ½ makes ²⁄₂, or one whole;

2 times ⅓ makes ⅔; and

3 times ⅙ makes ³⁄₆, or ½;

4 times ⁵⁄₁₂ makes ²⁰⁄₁₂, or 1⁸⁄₁₂, or 1⅔.

But, instead of this rule, we may use that of dividing the denominator by the given integer, which is preferable, when it can be done, because it shortens the operation. Let it be required, for example, to multiply ⁸⁄₉ by 3; if we multiply the numerator by the given integer we obtain ²⁴⁄₉, which product we must reduce to ⁸⁄₃. But if we do not change the numerator, and divide the denominator by the integer, we find immediately ⁸⁄₃, or 2⅔, for the given product; and, in the same manner, ¹³⁄₂₄ multiphed by 6 gives ¹³⁄₄, or 3¼.

102 In general, therefore, the product of the multiplication of a fraction $$\frac{a}{b}$$ by $$c$$ is $$\frac{ac}{b}$$; and here it may be remarked, when the integer is exactly equal to the denominator, that the product must be equal to the numerator.

So that

½ taken 2 times, gives 1;

⅔ taken three times, gives 1;

¾ taken four times, gives 1.

And, in general, if we multiply the fraction $$\frac{a}{b}$$ by the number $$b$$, the product must be $$a$$, as we have already shown; for since $$\frac{a}{b}$$ expresses the quotient resulting from the division of the dividend $$a$$ by the divisor $$b$$, and because it has been demonstrated that the quotient multiplied by the divisor will give the dividend, it is evident that $$\frac{a}{b}$$ multiplied by $$b$$ must produce $$a$$.

103 Having thus shown how a fraction is to be multiplied by an integer; let us now consider also how a fraction is to be divided by an integer. This inquiry is necessary, before we proceed to the multiplication of fractions by fractions. It is evident, if we have to divide the fraction ⅔ by 2, that the result must be ⅓; and that the quotient of ⁶⁄₇ divided by 3 is ²⁄₇. The rule therefore is, to divide the numerator by the integer without changing the denominator. Thus

¹²⁄₂₅ divided by 2 gives ⁶⁄₂₅;

¹²⁄₂₅ divided by 3 gives ⁴⁄₂₅;

¹²⁄₂₅ divided by 4 gives ³⁄₂₅; etc.

104 This rule may be easily practised, provided the numerator be divisible by the number proposed; but very often it is not: it must therefore be observed, that a fraction may be transformed into an infinite number of other expressions, and in that number there must be some, by which the numerator might be divided by the given integer. If it were required, for example, to divide ¾ by 2, we should change the fraction into ⁶⁄₈, and then dividing the numerator by 2, we should immediately have ⅜ for the quotient sought.

In general, if it be proposed to divide the fraction $$\frac{a}{b}$$ by $$c$$, we change it into $$\frac{ac}{bc}$$, and then dividing the numerator $$ac$$ by $$c$$, write $$\frac{a}{bc}$$ for the quotient sought.

105 When therefore a fraction $$\frac{a}{b}$$ is to be divided by an integer $$c$$, we have only to multiply the denominator by that number, and leave the numerator as it is. Thus ⅝ divided by 3 gives ⁵⁄₂₄, and ⁹⁄₁₆ divided by 5 gives ⁹⁄₈₀.

This operation becomes easier, when the numerator itself is divisible by the integer, as we have supposed in Art. 103. For example, ⁹⁄₁₆ divided by 3 would give, according to our last rule, ⁹⁄₄₈; but by the first rule, which is applicable here, we obtain ³⁄₁₆, an expression equivalent to ⁹⁄₄₈, but more simple.

106 We shall now be able to understand how one fraction $$\frac{a}{b}$$ may be multiplied by another fraction $$\frac{c}{d}$$. For this purpose, we have only to consider that $$\frac{c}{d}$$ means that $$c$$ is divided by $$d$$; and on this principle we shall first multiply the fraction $$\frac{a}{b}$$ by $$c$$, which produces the result $$\frac{ac}{b}$$; after which we shall divide by $$d$$, which gives $$\frac{ac}{bd}$$.

Hence the following rule for multiplying fractions. Multiply the numerators together for a numerator, and the denominators together for a denominator. Thus,

½ by ⅔ gives the product ²⁄₆, or ⅓;

⅔ by ⅘ gives ⁸⁄₁₅;

¾ by ⁵⁄₁₂ produces ¹⁵⁄₄₈, or ⁵⁄₁₆; etc.

107 It now remains to show how one fraction may be divided by another. Here we remark first, that if the two fractions have the same number for a denominator, the division takes place only with respect to the numerators; for it is evident, that ³⁄₁₂ are contained as many times in ⁹⁄₁₂ as 3 is contained in 9, that is to say, three times; and, in the same manner, in order to divide ⁸⁄₁₂ by ⁹⁄₁₂, we have only to divide 8 by 9, which gives ⁸⁄₉. We shall also have ⁶⁄₂₀ in ¹⁸⁄₂₀, 3 times; ⁷⁄₁₀₀ in ⁴⁹⁄₁₀₀, 7 times; ⁷⁄₂₅ in ⁶⁄₂₅, ⁶⁄₇ times; etc.

108 But when the fractions have not equal denominators, we must have recourse to the method already mentioned for reducing them to a common denominator. Let there be, for example, the fraction $$\frac{a}{b}$$ to be divided by the fraction $$\frac{c}{d}$$. We first reduce them to the same denominator, and there results $$\frac{ad}{bd}$$ to be divided by $$\frac{cb}{bd}$$; it is now evident that the quotient must be represented simply by the division of $$ad$$ by $$bc$$; which gives $$\frac{ad}{bc}$$.

Hence the following rule: Multiply the numerator of the dividend by the denominator of the divisor, and the denominator of the dividend by the numerator of the divisor; then the first product will be the numerator of the quotient, and the second will be its denominator.

109 Applying this rule to the division of ⅝ by ⅔, we shall have the quotient ¹⁵⁄₁₆; also the division of ¾ by ½ will give ⁶⁄₄, or ³⁄₂, or 1½; and ²⁵⁄₄₈ by ⅚ will give ¹⁵⁰⁄₂₄₀, or ⅝.

110 This rule for division is often expressed in a manner that is more easily remembered, as follows: Invert the terms of the divisor, so that the denominator may be in the place of the numerator, and the latter be written under the line; then multiply the fraction, which is the dividend by this inverted fraction, and the product will be the quotient sought. Thus, ¾ divided by ½ is the same as ¾ multiplied by ²⁄₁, which makes ⁶⁄₄, or 1½. Also ⅝ divided by ⅔ is the same as ⅝ multiplied by ³⁄₂, which is ¹⁵⁄₁₆; or ²⁵⁄₄₈ divided by ⅚ gives the same as ²⁵⁄₄₈ multiplied by ⁶⁄₅, the product of which is ¹⁵⁰⁄₂₄₀, or ⅝.

We see then, in general, that to divide by the fraction ½ is the same as to multiply by ²⁄₁, or 2; and that dividing by ⅓ amounts to multiplying by ³⁄₁, or by 3, etc.

111 The number 100 divided by ½ will give 200; and 1000 divided by ⅓ will give 3000. Farther, if it were required to divide 1 by ¹⁄₁₀₀₀, the quotient would be 1000; and dividing 1 by ¹⁄₁₀₀₀₀₀, the quotient is 100000. This enables us to conceive that, when any number is divided by 0, the result must be a number indefinitely great; for even the division of 1 by the small fraction ¹⁄₁₀₀₀₀₀₀₀₀₀ gives for the quotient the very great number 1000000000.

112 Every number, when divided by itself, producing unity, it is evident that a fraction divided by itself must also give 1 for the quotient; and the same follows from our rule: for, in order to divide ¾ by ¾, we must multiply ¾ by ⁴⁄₃, in which case we obtain ¹²⁄₁₂, or 1; and if it be required to divide $$\frac{a}{b}$$ by $$\frac{a}{b}$$, we multiply $$\frac{a}{b}$$ by $$\frac{b}{a}$$; where the product $$\frac{ab}{ab}$$ is also equal to 1.

113 We have still to explain an expression which is frequently used. It may be asked, for example, what is the half of ¾? This means, that we must multiply ¾ by ½. So likewise, if the value of ⅔ of ⅝ were required, we should multiply ⅝ by ⅔, which produces ¹⁰⁄₂₄; and ¾ of ⁹⁄₁₆ is the same as ⁹⁄₁₆ multiplied by ¾, which produces ²⁷⁄₆₄.

114 Lastly, we must here observe, with respect to the signs + and -, the same rules that we before laid down for integers. Thus +½ multiplied by -⅓, makes -⅙; and -⅔ multiplied by -⅘, gives +⁸⁄₁₅. Farther -⅝ divided by ⅔, gives -¹⁵⁄₁₆; and -¾ divided by -¾, gives +¹²⁄₁₂, or +1.

#### Editions

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.