Decimals

decimal numeral, or just decimal, or, improperly decimal number, refers generally to the notation of a number in the decimal system, which contains a decimal separator: for example 3.14; the decimal separator is the dot “.” in many countries, but may be a comma “,” in other countries (mainly in Europe). The numbers that may be represented in the decimal system are the decimal fractions.

The decimal numeral system (decimal notation) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral system.

Decimal fraction

The numbers that are represented by decimal numeral are the decimal fractions (sometimes called decimal numbers), that is, the rational numbers that may be expressed as a fraction, the denominator of which is a power of ten. The following notation has been accepted for a decimal fraction:

\[a_ma_{m-1}\ldots a_0.b_1b_2\ldots b_n\]

For representing a negative number, a minus sign “-” is placed before \(a_m\). It is generally assumed that, if \(m > 0\), the first digit am is not zero, but, in some circumstances, it may be useful to have one or more 0’s on the left. This does not change the value represented by the decimal. For example: 3.14 = 03.14 = 003.14. Similarly, if \(b_n =0\), it may be removed, and conversely, trailing zeros may be added without changing the represented number. For example: 14 = 14.0 = 14.00 and 4.3 = 4.30 = 4.300.

The numeral \(a_ma_{m-1}\ldots a_0.b_1b_2\ldots b_n\) represents the number:

\[a_m10^m+a_{m-1}10^{m-1}+\cdots +a_010^0+\dfrac{b_1}{10^1}+\dfrac{b_2}{10^2}+\cdots + \dfrac{b_n}{10^n}\]

Therefore, the contribution of each digit to the value of a number depends on its position in the numeral. That is, the decimal system is a positional numeral system.

For example:

\[\dfrac{3}{10}=0.3\]

\[\dfrac{3471}{100}=34.71\]

The digits to the right of the decimal point are known as the decimal digits. If a decimal fraction contains no integer part, i.e. its absolute value is smaller than one, a zero is placed to the left of the decimal point.


References

  1. Wikipedia. Decimal. https://en.wikipedia.org/wiki/Decimal

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