Elements of Algebra

CHAP. VII. OF ALGEBRA. 23 78th article without interruption. This circumstance has introduced the expression, that the denominator must be infinite, or infinitely great, in order that the fraction may be reduced to $0$, or to nothing; hence the word infinite in reality signifies here, that we can never arrive at the end of the series of the above-mentioned fractions. 82. To express this idea, according to the sense of it above-mentioned, we make use of the sign $\infty$, which consequently indicates a number infinitely great; and we may therefore say, that this fraction $\frac{1}{\infty}$ is in reality nothing; because a fraction cannot be reduced to nothing, until the denominator has been increased to infinity. 83. It is the more necessary to pay attention to this idea of infinity, as it is derived from the first elements of our knowledge, and as it will be of the greatest importance in the following part of this treatise. We may here deduce from it a few consequences that are extremely curious, and worthy of attention. The fraction $\frac{1}{\infty}$ represents the quotient resulting from the division of the dividend $1$ by the divisor $\infty$. Now, we know, that if we divide the dividend $1$ by the quotient $\frac{1}{\infty}$, which is equal to nothing, we obtain again the divisor $\infty$: hence we acquire a new idea of infinity; and learn that it arises from the division of $1$ by $0$; so that we are thence authorised in saying, that $1$ divided by $0$ expresses a number infinitely great, or $\infty$. 84. It may be necessary also, in this place, to correct the mistake of those who assert, that a number infinitely great is not susceptible of increase. This opinion is inconsistent with the just principles which we have laid down; for $\frac{1}{0}$ signifying a number infinitely great, and $\frac{2}{0}$ being incontestably the double of $\frac{1}{0}$, it is evident that a number, though infinitely great, may still become twice, thrice, or any number of times greater *. * There appears to be a fallacy in this reasoning, which consists in taking the sign of infinity for infinity itself; and applying the property of fractions in general to a fractional expression, whose denominator bears no assignable relation to unity. It is certain, that infinity may be represented by a series of units (that is, by $\frac{1}{0} = \frac{1}{1-1} = 1+1+1, \&c.$) or by a series of numbers increasing in any given ratio. Now, though any definite part of one infinite series may be the half, the third, &c. of a definite part of another, yet still that part bears no proportion to the whole, and the series can only be said, in that case, to go on to infinity in a different ratio. But, farther, $\frac{a}{0}$, or any other nu-