95 5

— =

–

19 1

98 8

— =

–

49 4

64 4

— =

–

16 1

56

C

calculus (infinitesimal calculus) The branch of math-

ematics that deals with the notion of continuous

growth and change is called calculus. It is based on the

concept of

INFINITESIMAL

s, exceedingly small quanti-

ties, and on the concept of a

LIMIT

, quantities that can

be approached more and more closely but never

reached. By treating continuous changes as if they con-

sisted of infinitely small increments,

DIFFERENTIAL CAL

-

CULUS

can be used, for example, to find the

VELOCITY

of a moving object at any particular instant.

INTEGRAL

CALCULUS

represents the reverse process, finding the

aggregate end-result if the continuous change is already

known. For example, by integrating the instantaneous

velocity of an object over a given time period, one finds

the total distance the object moved during this time.

The word calculus comes from the Latin word calx

for “pebble,” which in turn is derived from the Greek

word chalis for “limestone.” Small beads or stones

arranged in a counting board or on an

ABACUS

were

often used to aid mathematical calculations, and the

word calculus came to refer to all mathematical activ-

ity. Today, however, the word is used almost exclusively

to denote the study of continuous change.

See also

HISTORY OF CALCULUS

(essay).

cancellation In

ARITHMETIC

, the process of dividing

the numerator and the denominator of a

FRACTION

by

a common factor to produce a simpler fraction is called

cancellation. For example, the fraction 18/12, which is

(6 ×3)/(6 ×2), can be simplified to 3/2 by canceling

“6” from the fraction.

An amusing activity in mathematics searches for

fractions that remain unchanged by the action of

“anomalous cancellation.” For example, deleting the

digit 6 from each of the numerator and denominator of

the fraction 65/26 produces the number 5/2. Curiously,

65/26 does equal five halves. Other fractions with this

property include: , , and . Of

course deleting digits this way is an invalid mathemati-

cal operation, and the equalities obtained here are

purely coincidental.

In

ALGEBRA

the word cancellation is used in two

settings. Akin to the process of cancellation for frac-

tions, one may simplify a rational expression of the

form xy/xz by canceling the factor xfrom both the

numerator and denominator. This gives xy/xz as equal

to y/z. Thus, for instance, the expression

simplifies to .

The process of removing two equal quantities from

an equation via subtraction is also called cancellation.

For example, in the equation x+ 2y= 5 + 2y, the term

2ycan be cancelled to leave x= 5.

Both actions fall under the umbrella of a general

cancellation law. In an abstract setting, a

BINARY OPER

-

ATION

“*” is said to satisfy such a law if whenever a*x

= b*x holds, a= bfollows. For the operation of addi-

tion, this reads:

a+ x= b+ x implies a = b