My Study In Computer science : Day 17

Hallo every body today I go to speak about The Desimal sistem

The decimal numeral system (also called the base-ten positional numeral system and denary /ˈdiːnəri/^[1] or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers (decimal fractions) of the Hindu–Arabic numeral system. The way of denoting numbers in the decimal system is often referred to as decimal notation.^[2]

A decimal numeral (also often just decimal or, less correctly, decimal number), refers generally to the notation of a number in the decimal numeral system. Decimals may sometimes be identified by a decimal separator (usually "." or "," as in $25.9703$ or $3,1415$ ).^[3] Decimal may also refer specifically to the digits after the decimal separator, such as in " $3.14$ is the approximation of $π$ to two decimals". Zero-digits after a decimal separator serve the purpose of signifying the precision of a value.

The numbers that may be represented in the decimal system are the decimal fractions. That is, fractions of the form $a /10 n$ , where $a$ is an integer, and $n$ is a non-negative integer. Decimal fractions also result from the addition of an integer and a fractional part; the resulting sum sometimes is called a fractional number.

Decimals are commonly used to approximate real numbers. By increasing the number of digits after the decimal separator, one can make the approximation errors as small as one wants, when one has a method for computing the new digits.

Originally and in most uses, a decimal has only a finite number of digits after the decimal separator. However, the decimal system has been extended to infinite decimals for representing any real number, by using an infinite sequence of digits after the decimal separator (see decimal representation). In this context, the usual decimals, with a finite number of non-zero digits after the decimal separator, are sometimes called terminating decimals. A repeating decimal is an infinite decimal that, after some place, repeats indefinitely the same sequence of digits (e.g., $5.123144144144144... = 5.123 144$ ).^[4] An infinite decimal represents a rational number, the quotient of two integers, if and only if it is a repeating decimal or has a finite number of non-zero digits.

Decimal notation

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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;^[6] the decimal separator is the dot " $.$ " in many countries (mostly English-speaking),^[7] and a comma " $,$ " in other countries.^[3]

For representing a non-negative number, a decimal numeral consists of

either a (finite) sequence of digits (such as "2017"), where the entire sequence represents an integer:
$a_{m}a_{m-1}\ldots a_{0}$
or a decimal mark separating two sequences of digits (such as "20.70828")

a_{m}a_{m-1}\ldots a_{0}.b_{1}b_{2}\ldots b_{n}

If $m > 0$ , that is, if the first sequence contains at least two digits, it is generally assumed that the first digit $a m$ is not zero. 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 the final digit on the right of the decimal mark is zero—that is, if $b n = 0$ —it may be removed; conversely, trailing zeros may be added after the decimal mark without changing the represented number; ^{[note 1]} for example, $15 = 15.0 = 15.00$ and $5.2 = 5.20 = 5.200$ .

For representing a negative number, a minus sign is placed before $a m$ .

The numeral $a_{m}a_{m-1}\ldots a_{0}.b_{1}b_{2}\ldots b_{n}$ represents the number

a_{m}10^{m}+a_{m-1}10^{m-1}+\cdots +a_{0}10^{0}+{\frac {b_{1}}{10^{1}}}+{\frac {b_{2}}{10^{2}}}+\cdots +{\frac {b_{n}}{10^{n}}}

The integer part or integral part of a decimal numeral is the integer written to the left of the decimal separator (see also truncation). For a non-negative decimal numeral, it is the largest integer that is not greater than the decimal. The part from the decimal separator to the right is the fractional part, which equals the difference between the numeral and its integer part.

When the integral part of a numeral is zero, it may occur, typically in computing, that the integer part is not written (for example, $.1234$ , instead of $0.1234$ ). In normal writing, this is generally avoided, because of the risk of confusion between the decimal mark and other punctuation.

In brief, 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 sys tem.

Infinite decimal expansion

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For a real number $x$ and an integer $n \geq 0$ , let $[x] n$ denote the (finite) decimal expansion of the greatest number that is not greater than $x$ that has exactly $n$ digits after the decimal mark. Let $d i$ denote the last digit of $[x] i$ . It is straightforward to see that $[x] n$ may be obtained by appending $d n$ to the right of $[x] n -1$ . This way one has

[x] n = [x] 0 . d 1 d 2 ... d n -1 d n

and the difference of $[x] n -1$ and $[x] n$ amounts to

\left\vert \left[x\right]_{n}-\left[x\right]_{n-1}\right\vert =d_{n}\cdot 10^{-n}<10^{-n+1}

which is either 0, if $d n = 0$ , or gets arbitrarily small as $n$ tends to infinity. According to the definition of a limit, $x$ is the limit of $[x] n$ when $n$ tends to infinity. This is written as ${\textstyle \;x=\lim _{n\rightarrow \infty }[x]_{n}\;}$ or

x = [x] 0 . d 1 d 2 ... d n ...

which is called an infinite decimal expansion of $x$ .

Conversely, for any integer $[x] 0$ and any sequence of digits ${\textstyle \;(d_{n})_{n=1}^{\infty }}$ the (infinite) expression $[x] 0 . d 1 d 2 ... d n ...$ is an infinite decimal expansion of a real number $x$ . This expansion is unique if neither all $d n$ are equal to 9 nor all $d n$ are equal to 0 for $n$ large enough (for all $n$ greater than some natural number $N$ ).

If all $d n$ for $n > N$ equal to 9 and $[x] n = [x] 0 . d 1 d 2 ... d n$ , the limit of the sequence ${\textstyle \;([x]_{n})_{n=1}^{\infty }}$ is the decimal fraction obtained by replacing the last digit that is not a 9, i.e.: $d N$ , by $d N + 1$ , and replacing all subsequent 9s by 0s (see 0.999...).

Any such decimal fraction, i.e.: $d n = 0$ for $n > N$ , may be converted to its equivalent infinite decimal expansion by replacing $d N$ by $d N - 1$ and replacing all subsequent 0s by 9s (see 0.999...).

In summary, every real number that is not a decimal fraction has a unique infinite decimal expansion. Each decimal fraction has exactly two infinite decimal expansions, one containing only 0s after some place, which is obtained by the above definition of $[x] n$ , and the other containing only 9s after some place, which is obtained by defining $[x] n$ as the greatest number that is less than $x$ , having exactly $n$ digits after the decimal mark

My Study In Computer science

الجمعة، 25 أكتوبر 2024

Day 17

Decimal notation

Infinite decimal expansion

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