Chapter 3
Division and the Fraction

16. Numerical Division. The inverse operation to multiplication is division.

To divide a by b is to find a number which multiplied by b produces a. The result is called the quotient of a by b, and is written a b. By definition

a bb = a
Like subtraction, division cannot be always effected. Only in exceptional cases can the a-group be subdivided into b equal groups.

17. Determinateness of Numerical Division. When division can be effected at all, it can lead to but a single result; it is determinate.

For there can be but one number the product of which by b is a; in other words,

 Ifcb = db, c = d.5

For b groups each containing c individuals cannot be equal to b groups each containing d individuals unless c = d (§4).

This is a theorem of fundamental importance. It may be called the law of determinateness of division. It declares that if a product and one of its factors be determined, the remaining factor is definitely determined also; or that if one of the factors of a product changes while the other remains unchanged, the product changes. It alone makes division in the arithmetical sense possible. The fact that it does not hold for the symbol 0, but that rather a product remains unchanged (being always 0) when one of its factors is 0, however the other factor be changed, makes division by 0 impossible, rendering unjustifiable the conclusions which can be drawn in the case of other divisors.

The reasoning which proved law IX proves also that

 IX’.  Ascb >  or  < db, c >  or  < d.

18. Formal Rules of Division. The fundamental laws of the multiplication of numbers are

 III. ab = ba,  IV. a(bc) = abc,  V. a(b + c) = ab + ac.

Of these, the definition

 VIII. a bb = a,

the theorem

 IX.  If ac = bc, a = b, unless c = 0,

and the corresponding laws of addition and subtraction, the rules of division are purely formal consequences, deducible precisely as the rules of subtraction 1–5 of §10 in the preceding chapter. They follow without regard to the meaning of the symbols a, b, c, =, +, , ab, a b. Thus:


  1. a b c d = ac bd.   Fora b c d bd = a bb c dd,  Laws IV, III. = ac,  Def. VIII.  andac bd bd = ac.  Def VIII.

    The theorem follows by law IX.


  2. a b c d = dad bc .   For a b c d c d = a b,  Def. VIII.  andad bc c d = a b dc cd,  §18, 1; Law IV. = a b,  sincedc cd = dc = 1 × cd. Def. VIII, Law IX.

    The theorem follows by law IX.


  3. a b ± c d = ad ± bc bd .  For a b ± c dbd = a bb d ± c dd b,  Laws III–V: §10, 5. = ad ± bc,  Def. VIII.  and ad ± bc bd bd = ad ± bc.  Def. VIII.

    The theorem follows by law IX.

    By the same method it may be inferred that


  4. a b >, =,< c d,  asad >, =,< bc. Def. VIII, Laws III, IV, IX, IX’.

19. Limitations of Numerical Division. Symbolic Division. The Fraction. General as is the form of the preceding equations, they are capable of numerical interpretation only when a b, c d are numbers, a case of comparatively rare occurrence. The narrow limits set the quotient in the numerical definition render division an unimportant operation as compared with addition, multiplication, or the generalized subtraction discussed in the preceding chapter.

But the way which led to an unrestricted subtraction lies open also to the removal of this restriction; and the reasons for following it there are even more cogent here.

We accept as the quotient of a divided by any number b, which is not 0, the symbol a b defined by the equation

a bb = a,
regarding this equation merely as a declaration of the equivalence of the symbols (a b)b and a, of the right to substitute one for the other in any reckoning.

Whether a b be a number or not is to this definition irrelevant. When a mere symbol, a b is called a fraction, and in opposition to this a number is called an integer.

We then put ourselves in immediate possession of definitions of the addition, subtraction, multiplication, and division of this symbol, as well as of the relations of equality and greater and lesser inequality—definitions which are consistent with the corresponding numerical definitions and with one another—by assuming the permanence of form of the equations 1, 2, 3 and of the test 4 of § 18 as symbolic statements, when they cease to be interpretable as numerical statements.

The purely symbolic character of a b and its operations detracts nothing from their legitimacy, and they establish division on a footing of at least formal equality with the other three fundamental operations of arithmetic.6

20. Negative Fractions. Inasmuch as negatives conform to the laws and definitions I–IX, the equations 1, 2, 3 and the test 4 of §18 are valid when any of the numbers a, b, c, d are replaced by negatives. In particular, it follows from the definition of quotient and its determinateness, that

a b = a b ; a b = a b ; a b = a b.

It ought, perhaps, to be said that the determinateness of division of negatives has not been formally demonstrated. The theorem, however, that if (±a)(±c) = (±b)(±c),±a = ±b, follows for every selection of the signs ± from the one selection +, +, +, + by §14, 6, 8.

21. General Test of the Equality or Inequality of Fractions.

Given any two fractions ±a b,±c d.

±a b >, =   or  < ±c d, according as ± ad >,=  or  < ±bc.  Laws IX, IX’.  Compare §4, §14, 9.

22. Indeterminateness of Division by Zero. Division by 0 does not conform to the law of determinateness; the equations 1, 2, 3 and the test 4 of § 18 are, therefore, not valid when 0 is one of the divisors.

The symbols 0 0,a 0, of which some use is made in mathematics, are indeterminate.7

1. 0 0 is indeterminate. For 0 0 is completely defined by the equation 0 00 = 0; but since x x0 = 0, whatever the value of x, any number whatsoever will satisfy this equation.

2. a 0 is indeterminate. For, by definition, a 00 = a. Were a 0 determinate, therefore,—since then a 00 would, by § 18, 1, be equal to a x0 0 , or to 0 0,—the number a would be equal to 0 0, or indeterminate.

Division by 0 is not an admissible operation.

23. Determinateness of Symbolic Division. This exception to the determinateness of division may seem to raise an objection to the legitimacy of assuming—as is done when the demonstrations 1–4 of § 18 are made to apply to symbolic quotients—that symbolic division is determinate.

It must be observed, however, that 0 0, a 0 are indeterminate in the numerical sense, whereas by the determinateness of symbolic division is, of course, not meant actual numerical determinateness, but “symbolic determinateness,” conformity to law IX, taken merely as a symbolic statement. For, as has been already frequently said, from the present standpoint the fraction a b is a mere symbol, altogether without numerical meaning apart from the equation a bb = a, with which, therefore, the property of numerical determinateness has no possible connection. The same is true of the product, sum or difference of two fractions, and of the quotient of one fraction by another.

As for symbolic determinateness, it needs no justification when assumed, as in the case of the fraction and the demonstrations 1–4, of symbols whose definitions do not preclude it. The inference, for instance, that because

a b c dbd = ac bdbd, a b c d = ac bd,

which depends on this principle of symbolic determinateness, is of precisely the same character as the inference that

a b c d = a bb c dd,

which depends on the associative and commutative laws.

Both are pure assumptions made of the undefined symbol a b c d for the sake of securing it a definition identical in form with that of the product of two numerical quotients.8

24. The Vanishing of a Product. It has already been shown (§ 13, 3, § 14, 7, § 18, 1) that the sufficient condition for the vanishing of a product is the vanishing of one of its factors. From the determinateness of division it follows that this is also the necessary condition, that is to say:

If a product vanish, one of its factors must vanish.

Let xy = 0, where x, y may represent numbers or any of the symbols we have been considering.

   Since  xy = 0, xy + xz = xz, §13, 1.  or  x(y + z) = xz, Law V.  whence, if x be not 0y + z = z,   Law IX.  or  y = 0.   Law VII.

25. The System of Rational Numbers. Three symbols, 0, d, a b, have thus been found which can be reckoned with by the same rules as numbers, and in terms of which it is possible to express the result of every addition, subtraction, multiplication or division, whether performed on numbers or on these symbols themselves; therefore, also, the result of any complex operation which can be resolved into a finite combination of these four operations.

Inasmuch as these symbols play the same rôle as numbers in relation to the fundamental operations of arithmetic, it is natural to class them with numbers. The word “number,” originally applicable to the positive integer only, has come to apply to zero, the negative integer, the positive and negative fraction also, this entire group of symbols being called the system of rational numbers.9 This involves, of course, a radical change of the number concept, in consequence of which numbers become merely part of the symbolic equipment of certain operations, admitting, for the most part, of only such definitions as these operations lend them.

In accepting these symbols as its numbers, arithmetic ceases to be occupied exclusively or even principally with the properties of numbers in the strict sense. It becomes an algebra, whose immediate concern is with certain operations defined, as addition by the equations a + b = b + a, a + (b + c) = a + b + c, formally only, without reference to the meaning of the symbols operated on.10