Decimal and, only the best mathematicians were able to multiply or divide large numbers. These difficulties were completely solved with the introduction of the Hindu–Arabic numeral system for representing integers. This system has been extended to represent some non-integer numbers, called "decimal fractions" or "decimal numbers" for forming the "decimal numeral system". For writing numbers, the decimal system uses ten decimal digits, a decimal mark, and, for negative numbers, a minus sign "−". The decimal digits are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9; the decimal separator is the dot "" in many countries (including English speaking ones),
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