Computer number format

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The computer number format is an internal representation of the numerical value in a digital computer and the hardware and software calculator. Typically, the numerical value is stored as a bit grouping, named for the number of bits that make up it. The encoding between the numeric value and the bit pattern is selected for ease of operation of the computer; bit formats used by computer instruction sets generally require conversions for external use such as printing and display. Different types of processors may have internal representations that differ from numerical values. Different conventions are used for integers and real numbers. Most calculations are performed with numeric formats corresponding to the processor registers, but some software systems allow random large numbers representation using multiple word memories.

Video Computer number format

Representasi angka biner

Computers represent data in sets of binary digits. Representations consist of bits, which in turn are grouped into larger sets like bytes.

A bit is a binary digit representing one of two conditions. A little concept can be understood as either a 1 or 0 , on or dead , yes > or no , true or false , or encoded by switches or switching from multiple types.

While one bit, by itself, is capable of representing only two values, a string of bits can be used to represent a larger value. For example, a three-bit string can represent up to eight different values ​​as illustrated in Table 1.

Since the number of bits that make up the strings increases, the number of possible combinations 0 and 1 increases exponentially. While one bit allows only two combinations of values ​​and two bits combined can create four separate values ​​and so on. The number of combinations that may double with each added binary digit as illustrated in Table 2.

Groupings with a certain number of bits are used to represent various things and have a specific name.

A byte is a bit string containing the number of bits required to represent a character. On most modern computers, this is an eight bit string. Because the byte definition is related to the number of bits that make up the characters, some older computers use different length bits for their bytes. In many computer architectures, bytes are used to overcome certain memory areas. For example, although 64-bit processors can overcome the memory of sixty-four bits at a time, they may still divide that memory into eight-bit pieces. This is called byte-addressed memory. Historically, many CPUs read data in several multiples of eight bits. Because the byte size of eight bits is very common, but the definition is not standard, the term octet is sometimes used to explicitly describe the sequence of eight bits.

A nibble (sometimes nybble ), is a number consisting of four bits. Being a half-byte, the nibble is named as a play of words. Someone may need some snack for one bite of something; Similarly, nybble is part of a byte. Because four bits allow for sixteen values, the nibble is sometimes known as the hexadecimal digit.

Maps Computer number format

Display octal and hex numbers

Octal and hex are easy ways to represent binary numbers, such as those used by computers. Computer engineers often need to write binary numbers, but in practice writing binary numbers like 1001001101010001 are boring and error-prone. Therefore, the binary number is written in base-8, or "octal", or, much more commonly, base-16, "hexadecimal" or "hex", number format. In the decimal system, there are 10 digits, 0 to 9, which are combined to form numbers. In the octal system, there are only 8 digits, 0 through 7. That is, the octal value "10" equals decimal "8", octal "20" is decimal "16", and so on. In a hexadecimal system, there are 16 digits, 0 to 9 followed, by convention, with A to F. That is, hex "10" equals decimal "16" and hex "20" equals decimal "32". Examples and comparison of numbers in different bases are described in the chart below.

When typing numbers, formatting characters are used to describe a number system, for example 000_0000B or 0b000_00000 for binary and 0F8H or 0xf8 for hexadecimal numbers.

Convert between bases

Each of these number systems is the positioning system, but while the decimal weights are power 10, the octal weight is power 8 and the hex weight is power 16. To convert from hex or octal to decimal, for each digit one multiplies the value of the digit by its position value and then add the result. As an example:

${\ displaystyle {\ text {oktal}} 756}  ÂÂ$

${\ displaystyle = (7 * 8 ^ {2}) (5 * 8 ^ {1}) (6 * 8 ^ {0})}  ÂÂ$

${\ displaystyle = (7 * 64) (5 * 8) (6 * 1)}  ÂÂ$

${\ displaystyle = 448 40 6}  ÂÂ$

${\ displaystyle = {\ text {decimal}} 494}  ÂÂ$

${\ displaystyle {\ teks {hex}} 3b2}  ÂÂ$

${\ displaystyle = (3 * 16 ^ {2}) (11 * 16 ^ {1}) (2 * 16 ^ {0})}  ÂÂ$

${\ displaystyle = (3 * 256) (11 * 16) (2 * 1)}  ÂÂ$

${\ displaystyle = 768 176 2}  ÂÂ$

${\ displaystyle = {\ text {decimal}} 946}  ÂÂ$

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Mewakili fraksi dalam biner

Nomor titik tetap

Fixed point formatting can be useful for representing fractions in binary.

The number of bits required for the desired precision and range must be selected to store the fractional and integer parts of a number. For example, using 32-bit format, 16 bits can be used for integers and 16 for fractions.

Eight bits followed by four bits, then second bits, then bits. The fractional bits continue the pattern set by the integer bits. The next bit is half a bit, then a little quarter, then 1/8, and so on. As an example:

Bentuk pengkodean ini tidak dapat mewakili beberapa nilai dalam biner. Misalnya, fraksi ${\ displaystyle {\ tfrac {1} {5}}}  ÂÂ$ , 0,2 dalam desimal, perkiraan terdekatnya adalah sebagai berikut:

Bahkan jika lebih banyak digit digunakan, representasi yang tepat tidak mungkin. Angka ${\ displaystyle {\ tfrac {1} {3}}}  ÂÂ$ , ditulis dalam desimal sebagai 0,333333333..., berlanjut tanpa batas. Jika diakhiri sebelum waktunya, nilainya tidak akan mewakili ${\ displaystyle {\ tfrac {1} {3}}}  ÂÂ$ secara tepat.

Nomor titik-mengambang

While both un signed and signed integers are used in digital systems, even a 32-bit integer is not sufficient to handle all ranges of numbers that the calculator can handle, and that does not even include fractions. To approach the range and the greater accuracy of the real numbers, we must ignore the integers and fixed point numbers that are signed and go to the "floating-point" format.

In decimal system, we know the floating-point number of the form (scientific notation):

1,1030402 ÃÆ'â € "10 ⁵ = 1,1030402 ÃÆ'â €" 100000 = 110304,02

or, more concise:

1.1030402E5

which means "1.1030402 times 1 followed by 5 zeros". We have a certain numerical value (1,1030402) known as "significand", multiplied by force 10 (E5, meaning 10 ⁵ or 100,000), known as "exponent". If we have negative exponents, that means the number is multiplied by 1 which is many places to the right of the decimal point. As an example:

2.3434E-6 = 2.3434 ÃÆ'â € "10 ^-6 = 2.3434 ÃÆ'â €" 0.000001 = 0.0000023434

The advantage of this scheme is that by using exponents we can get a much wider range of numbers, even if the number of digits in significand, or "numerical precision", is much smaller than the range. Similar floating-point forms can be defined for computers. There are a number of such schemes, the most popular being defined by the Institute of Electrical and Electronics Engineers (IEEE). The IEEE 754-2008 standard specification defines a 64 bit floating-point format with:

• 11-bit binary exponent, using the "over-1023" format. Excess-1023 means the exponent appears as an unsigned binary integer from 0 to 2047; subtract 1023 gives actual signed value
• 52-bit significand, also an unsigned binary number, which defines fractional values ​​with the implied instructions "1"
• small sign, ticking the number.

Let's see what this format looks like by showing how that number will be stored in 8 bytes of memory:

where "S" denotes a bit of a mark, "x" denotes the exponent bit, and "m" denotes the significand bits. Once the bits here have been extracted, they are converted by calculation:

& lt; sign & gt; ÃÆ'â € "(1 & lt; fractional significance & gt;) ÃÆ'â €" 2 ^{& lt; exponent & gt; - 1023}

This scheme delivers valid numbers up to about 15 decimal places, with the following numeric ranges:

The specification also defines some special values ​​that are not defined numbers, and are known as NaNs , for "Not A Number". This is used by programs to designate invalid operations and the like.

Some programs also use 32-bit floating-point numbers. The most common scheme uses a 23-bit significand with sign bit, plus an 8-bit exponent in "excess-127" format, giving valid decimal digits of seven digits.

The bits are converted to numerical values ​​by calculation:

& lt; sign & gt; ÃÆ'â € "(1 & lt; fractional significance & gt;) ÃÆ'â €" 2 ^{& lt; exponent & gt; - 127}

points to the following range of numbers:

These floating-point numbers are known as "real" or "floats" in general, but with a number of variations:

The 32-bit float value is sometimes called "real32" or "single", which means "single precision floating-point value".

A 64-bit float is sometimes called "real64" or "double", which means "double precision floating-point value".

The relationship between numbers and bit patterns is chosen for ease in computer manipulation; eight bytes stored in the computer's memory can represent 64-bit real, two 32-bit real, or four signed or not signed integers, or some other data type matching into eight bytes. The only difference is how the computer interprets it. If the computer stores four unassigned integers and then reads it back from memory as a real 64-bit, it will almost always be a valid real number, even though it will be garbage data.

Only a limited range of real numbers can be represented by a given number of bits. Arithmetic operations can be abundant or abundant, resulting in a value too large or too small to represent.

Representations have limited accuracy. For example, only 15 decimal places can be represented with 64-bit real. If a very small floating-point number is added to a large number, the result is only a large one. Small amounts are too small to appear in 15 or 16 digit resolutions, and the computer effectively discards them. Analyzing the effects of limited precision is a well-studied matter. The approximate magnitude of round-off errors and methods for limiting their effect on large calculations are part of large computing projects. The precision limit differs from the range limit, as it affects the significand, not the exponent.

Significand is a binary fraction that is not necessarily perfectly matched to decimal fractions. In many cases, the amount of reciprocal power 2 does not match a certain decimal fraction, and the calculation results will be slightly reduced. For example, a decimal fraction "0.1" is equivalent to an unlimited repeating binary fraction: 0,000110011...

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Numbers in programming languages ​​

Programming in assembly language requires the programmer to track the representation of numbers. Where the processor does not support the required mathematical operations, the programmer must create appropriate algorithms and sequence of instructions to carry out the operation; on some microprocessors, even integer multiplication must be done in the software.

High-level programming languages ​​such as LISP and Python offer abstract numbers that can be of extended type such as rational , bignum , or complex . Mathematical operations are performed by library routines provided by language implementations. The mathematical symbol given in the source code, by the redundant operator, will call the different object code according to the representation of the numeric type; mathematical operations on any number - whether signed, unsigned, rational, floating-point, fixed-point, integral, or complex - are written in exactly the same way.

Some languages, such as REXX and Java, provide decimal floating point operations, which provide different form rounding errors.

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See also

• binary-coded decimal
• Binary number system
• Gray code
• Hexadecimal
• Number system
• Octal

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Notes and references

The initial version of this article is based on a public domain article from Vectorsite owned by Greg Goebel.

Source of the article : Wikipedia