A fractional number is called a proper fraction or an improper fraction according as the numerator is or is not 3 less than the denominator; and an expression 4 such as 24 is called a mixed number.
Hence the value of a fraction is not altered by substituting for the numerator and denominator the corresponding numbers in any other column of a multiple-table (§ 36).
Hence we can treat the fractional numbers which have any one denominator as 0 o constituting a number-series, as shown in the 2 adjoining diagram.
Hence, so long as the denominator remains unaltered, we can deal with, exactly as if they were numbers, any operations being performed on the numerators.
- A fraction (or fractional number), the numerator or denominator of which is a fractional number, is called a complex fraction (or fractional number), to distinguish it from a simple fraction, which is a fraction having integers for numerator and denominator.
A fraction written in this way is called a decimal fraction; or we might define a decimal fraction as a fraction having a power of To for its denominator, there being a special notation for writing such fractions.
The modern system of placing the numerator above the denominator is due to the Hindus; but the dividing line is a later invention.
The pth root of a number (§43) may, if the number is an integer, be found by expressing it in terms of its prime factors; or, if it is not an integer, by expressing it as a fraction in its lowest terms, and finding the pth roots of the numerator and of the denominator separately.
To add or subtract fractional numbers, we must reduce them to a common denominator; and similarly, to multiply or divide surds, we must express them as power-numbers with the same index.
Fraction in its Lowest Terms.-A fraction is said to be in its lowest terms when its numerator and denominator have no common the more correct method is to write it a: b.
The modern method is to deal with fractions which have ioo as denominator; such fractions are called percentages.
In the sexagesimal system the numerators of the successive fractions (the denominators of which were the successive powers of 60) were followed by', ", "', ", the denominator not being written.
Then the denominator of the fraction, the numerical aperture, must be correspondingly increased, in order to ascertain the real resolving power.
Thus to divide by a fractional number we must multiply by the number obtained by interchanging the numerator and the denominator, i.e.
In the case of fractions of the more general kind, the numerator was written first with ', and then the denominator, followed by ", was written twice.
B,, Y the numerator (or denominator) of the last preceding term by the corresponding quotient and adding the numerator (or denominator) of the term before that.
(iv) Each convergent is nearer to the true value than any other fraction whose denominator is less than that of the convergent.
A simple fraction with ioo for denominator, can be expressed by writing the two figures of the numerator (or, if there is only one figure, this figure preceded by o) with a dot or " point " before them; thus 76 means 76%, or 17 -6 6 o.
By means of the present and the preceding sections the rule given in § 63 can be extended to the statement that a fractional number is equal to the number obtained by multiplying its numerator and its denominator by any fractional number.