Decimal numbers that may be expressed as a fraction, the denominator of which is a power of ten. For example, the numerals formula_5 represent the fractions , , . More generally, a decimal with digits after the separator represents the fraction with denominator , whose numerator is the integer obtained by removing the separator. Expressed as a fully reduced fraction, the decimal numbers are those, whose denominator is a product of a power of 2 by a power of 5. Thus the smallest denominators of decimal numbers are The "integer part", or "integral part" of a decimal is the integer written
Decimal representation unique. The number defined by a decimal representation is often written more briefly as That is to say, "a" is the integer part of "r", not necessarily between 0 and 9, and "a", "a", "a", ... are the digits forming the fractional part of "r". Both notations above are, by definition, the following limit of a sequence: Any real number can be approximated to any desired degree of accuracy by rational numbers with finite decimal representations. Assume formula_4. Then for every integer formula_5 there is a finite decimal formula_6 such that Proof: Let formula_8, where formula_9. Then formula_10, and the