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decimal system[Lat.,=of tenths], numeration system based on powers of 10. A number is written as a row of digits, with each position in the row corresponding to a certain power of 10. A decimal point in the row divides it into those powers of 10 equal to or greater than 0 and those less than 0, i.e., negative powers of 10. Positions farther to the left of the decimal point correspond to increasing positive powers of 10 and those farther to the right to increasing negative powers, i.e., to division by higher positive powers of 10. For example, 4,309=(4×103)+(3x102)+(0×101)+(9×100)=4,000+300+0+9, and 4.309=(4×100)+(3×10−1)+(0×10−2)+(9×10−3)=4+ 3-10+ 0-100+ 9-1000. It is believed that the decimal system is based on 10 because humans have 10 fingers and so became used to counting by 10s early in the course of civilization. The decimal system was introduced into Europe c.1300. It greatly simplified arithmetic and was a much-needed improvement over the Roman numerals, which did not use a positional system. A number written in the decimal system is called a decimal, although sometimes this term is used to refer only to a proper fractionfraction
[Lat.,=breaking], in arithmetic, an expression representing a part, or several equal parts, of a unit. Notation for Fractions
In writing a fraction, e.g.
..... Click the link for more information. written in this system and not to a mixed number. Decimals are added and subtracted in the same way as are integers (whole numbers) except that when these operations are written in columnar form the decimal points in the column entries and in the answer must all be placed one under another. In multiplying two decimals the operation is the same as for integers except that the number of decimal places in the product, i.e., digits to the right of the decimal point, is equal to the sum of the decimal places in the factors; e.g., the factor 7.24 to two decimal places and the factor 6.3 to one decimal place have the product 45.612 to three decimal places. In division, e.g., 12.8 ÷ 4.32, where there is a decimal point in the divisor (4.32), the point is shifted to the extreme right (i.e., to 432.) and the decimal point in the dividend (12.8) is shifted the same number of places to the right (to 1280), with one or more zeros added before the decimal to make this possible. The decimal point in the quotient is then placed above that in the dividend, zeros are added to the right of the decimal point in the dividend as needed, and the division proceeds the same as for integers. The decimal system is widely used in various systems employing numbers. The metric system of weights and measures, used in most of the world, is based on the decimal system, as are most systems of national currency.
the most widespread system of numeration. The base of the decimal system is the number 10, which forms a unit of the second order. The unit of the third order is 100 = 102. In general, the unit of each subsequent order is 10 times greater than the unit of the preceding order (it has been suggested that the selection of the number 10 as the base of the decimal system is connected with counting on one’s fingers).
The decimal system is based on the positional principle, that is, the same symbol (digit) has different values, depending on the position in which it is placed. Thus, in order to write all numbers, only the first ten numbers require special symbols. These symbols, which are designated by 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9, are called digits. To write a number, one determines how many units of the highest order are contained in it: then in the remainder, one determines the number of units of the order, which is one unit smaller, and so forth. The resulting digits are written in a sequence: for example, 4 × 102 + 7 × 101 + 3 × 10° = 473. Operations on numbers are carried out according to the order, that is, independently for the numbers of each order. If in so doing one obtains a number greater than 10 (in addition and multiplication), one adds one or several units to the following, higher order. In division and subtraction, it is necessary to break the orders down into smaller orders.