Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/37

zero, and the quotient is indeterminate; that is, any number whatever may be considered as the quotient, a result which is of no value.

$\scriptstyle {\frac {a}{0}}$ has no meaning, $\scriptstyle {a}$ being different from zero, for there exists no number such that if it be multiplied by zero, the product will equal $\scriptstyle {a}$.

Therefore division by zero is not an admissible operation.

Care should be taken not to divide by zero inadvertently. The following fallacy is an illustration.

${\begin{array}{lcrcl}\scriptscriptstyle {\quad {\text{Assume that}}}&\qquad &\scriptscriptstyle {a}&\scriptscriptstyle {=}&\scriptscriptstyle {b.}\\\scriptscriptstyle {\quad {\text{Then evidently}}}&&\scriptscriptstyle {ab}&\scriptscriptstyle {=}&\scriptscriptstyle {a^{2}.}\\\scriptscriptstyle {\quad {\text{Subtracting }}b^{2},}&&\scriptscriptstyle {ab-b^{2}}&\scriptscriptstyle {=}&\scriptscriptstyle {a^{2}-b^{2}.}\\\scriptscriptstyle {\quad {\text{Factoring,}}}&&\scriptscriptstyle {b(a-b)}&\scriptscriptstyle {=}&\scriptscriptstyle {(a+b)(a-b).}\\\scriptscriptstyle {\quad {\text{Dividing by }}a-b,}&&\scriptscriptstyle {b}&\scriptscriptstyle {=}&\scriptscriptstyle {a+b.}\\\scriptscriptstyle {\quad {\text{But}}}&&\scriptscriptstyle {a}&\scriptscriptstyle {=}&\scriptscriptstyle {b,}\\\scriptscriptstyle {\text{therefore}}&&\scriptscriptstyle {b}&\scriptscriptstyle {=}&\scriptscriptstyle {2b,}\\\scriptscriptstyle {\text{or,}}&&\scriptscriptstyle {1}&\scriptscriptstyle {=}&\scriptscriptstyle {2.}\end{array}}$

The result is absurd, and is caused by the fact that we divided by $\scriptscriptstyle {a-b=0}$.

15. Infinitesimals A variable $\scriptstyle {v}$ whose limit is zero is called an infinitesimal.^[1] This is written

$\scriptstyle {\operatorname {limit} \;v=0}$, or, $\scriptstyle {v\doteq 0}$,

and means that the successive numerical values of $\scriptstyle {v}$ ultimately become and remain less than any positive number however small. Such a variable is said to become indefinitely small or to ultimately vanish.

that is, the difference between a variable and its limit is an infinitesimal.

Conversely, if the difference between a variable and a constant is an infinitesimal, then the variable approaches the constant as a limit.

16. The concept of infinity ($\scriptstyle {\infty }$). If a variable $\scriptstyle {v}$ ultimately becomes and remains greater than any assigned positive number however large, we say $\scriptstyle {v}$ increases without limit, and write

$\scriptstyle {\operatorname {limit\;} v=+\infty }$, or, $\scriptstyle {v\doteq +\infty }$.

If a variable $\scriptstyle {v}$ ultimately becomes and remains algebraically less than any assigned negative number, we say $\scriptstyle {v}$ decreases without limit, and write

$\scriptstyle {\operatorname {limit\;} v=-\infty }$, or, $\scriptstyle {v\doteq -\infty }$.

1. Hence a constant, no matter how small it may be, is not an infinitesimal.