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 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).

Generally, to find the sum or difference of two or more fractional numbers, we must replace them by other fractional numbers having the same **denominator**; it is usually most convenient to take as this **denominator** the L.C.M.

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.