Arithmetic - Decimal Arithmetic

Decimal Arithmetic

Decimal representation refers exclusively, in common use, to the written numeral system employing arabic numerals as the digits for a radix 10 ("decimal)" positional notation; however, any numeral system based on powers of ten, e.g., Greek, Cyrillic, roman, or Chinese numerals may conceptually be described as "decimal notation" or "decimal representation".

Modern methods for four fundamental operations (addition, subtraction. multiplication and division) were first devised by Brahmagupta of India. This was known during medieval Europe as "Modus Indoram" or Method of the Indians. Positional notation (also known as "place-value notation") refers to the representation or encoding of numbers using the same symbol for the different orders of magnitude (e.g., the "ones place", "tens place", "hundreds place") and, with a radix point, using those same symbols to represent fractions (e.g., the "tenths place", "hundredths place"). For example, 507.36 denotes 5 hundreds (102), plus 0 tens (101), plus 7 units (100), plus 3 tenths (10−1) plus 6 hundredths (10−2).

Zero as a number comparable to the other basic digits is a concept that is essential to this notation, as is the concept of zero's use as a placeholder, and as is the definition of multiplication and addition with zero. The use of zero as a placeholder and, therefore, the use of a positional notation is first attested to in the Jain text from India entitled the Lokavibhâga, dated 458 AD and it was only in the early 13th century that these concepts, transmitted via the scholarship of the Arabic world, were introduced into Europe by Fibonacci using the Hindu-Arabic numeral system.

Algorism comprises all of the rules for performing arithmetic computations using this type of written numeral. For example, addition produces the sum of two arbitrary numbers. The result is calculated by the repeated addition of single digits from each number that occupies the same position, proceeding from right to left. An addition table with ten rows and ten columns displays all possible values for each sum. If an individual sum exceeds the value nine, the result is represented with two digits. The rightmost digit is the value for the current position, and the result for the subsequent addition of the digits to the left increases by the value of the second (leftmost) digit, which is always one. This adjustment is termed a carry of the value one.

The process for multiplying two arbitrary numbers is similar to the process for addition. A multiplication table with ten rows and ten columns lists the results for each pair of digits. If an individual product of a pair of digits exceeds nine, the carry adjustment increases the result of any subsequent multiplication from digits to the left by a value equal to the second (leftmost) digit, which is any value from one to eight (9 × 9 = 81). Additional steps define the final result.

Similar techniques exist for subtraction and division.

The creation of a correct process for multiplication relies on the relationship between values of adjacent digits. The value for any single digit in a numeral depends on its position. Also, each position to the left represents a value ten times larger than the position to the right. In mathematical terms, the exponent for the radix (base) of ten increases by one (to the left) or decreases by one (to the right). Therefore, the value for any arbitrary digit is multiplied by a value of the form 10n with integer n. The list of values corresponding to all possible positions for a single digit is written as {..., 102, 10, 1, 10−1, 10−2, ...}.

Repeated multiplication of any value in this list by ten produces another value in the list. In mathematical terminology, this characteristic is defined as closure, and the previous list is described as closed under multiplication. It is the basis for correctly finding the results of multiplication using the previous technique. This outcome is one example of the uses of number theory.