Elementary arithmetic, with brief notices of its history... by Robert Potts.

FRACTIONS.

ART. 1.-DEF. 1. A fraction is an expression for a part or any
number of parts of anything considered as an integer or whole.
A fraction is expressed by two numbers, one placed above the
other with a line between them: the number placed below the line
is called the denominator, and denotes the number of equal parts into
which the integer is supposed to be divided; and the number above
the line is called the numerator, and shews how many of these equal
parts are expressed by the fraction.
Thus i is a fraction, of which the denominator 4 denotes that the
integer is to be divided into four equal parts, and the numerator 3
shews that three of these equal parts are expressed by the fraction.1
It follows from the definition of a fraction, that when the numerator is equal to the denominator, the value of the fraction is equal
to unity; but if the numerator be less than the denominator, the value
of the fraction is less than unity; and if the numerator be greater
than the denominator, the value of the fraction is greater than unity.2
DEF. 2. A fraction is called a proper fraction when the numerator
is less than the denominator, and an imnproper fraction when the
numerator is equal to or greater than the denominator.3
1 Fractions may be supposed to originate by considering what part one number is
of another. If parts of a number be taken, each part may be compared with the
number itself. Thus, if the numbers 1, 2, 3 be compared with the number 4,
1 is one part of 4, or ~ of 4,
2 is two parts or 2 of 4,
3 is three parts or a of 4.
The meaning of any fraction ~ may be exemplified by assuming any concrete number
whatever. Thus, if the integer or unit be ~1 or 20 shillings, and this unit be divided
into 4 equal parts, each part will consist of 5 shillings, and 3 of these parts will be
15 shillings, so that 3 of ~1 will be 15 shillings.
And again, if the integer be one lineal yard or 36 inches, when this integer is
divided into 4 equal parts, each part will consist of 9 inches, and 3 of these parts
will be 27 inches, wherefore 3 of one yard will be 27 inches.
2 In the series of fractions,,, -, ~, -7,,, &c., each is less than the one whichl
precedes it, and from the definition of a fraction it appears that the value of a fraction
is diminished by increasing the denominator, while the numerator remains unaltered.
And in the series, I, F, ], -, ], I, &c., each is greater than the one which precedes it, or that the value of a fraction is increased by increa6sing the numerator,
while the denominator remains constant.
3 The word fraction is used both for the expression of any part or parts less than the
whole, and for the whole itself, as well as for a number of parts greater than the
whole. There is nothing in the nature of the case, nor in the manner in which
fractions are written, to prevent the number of parts expressed being greater than the
number of parts into which the unit is divided. But the word fraction will clearly