Computer numbering formats

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The term computer numbering formats refers to the schemes implemented in digital computer and calculator hardware and software to represent numbers. A common mistake made by non-specialist computer users is a certain misplaced faith in the infallibility of numerical computations.

For example, if one multiplies: $3 \times \frac13$ one might perhaps expect to get a result of exactly 1, which is the correct answer when applying an exact rational number or algebraic model. In practice, however, the result on a digital computer or calculator may prove to be something such as 0.9999999999999999 (as one might find when doing the calculation on paper) or, in certain cases, perhaps 0.99999999923475.

The latter result seems to indicate a bug, but it is actually an unavoidable consequence of the use of a binary floating-point approximation. Decimal floating-point, computer algebra systems, and certain bignum systems would give either the answer of 1 or 0.9999999999999999...

1 Bits, bytes, nibbles, and unsigned integers
2 Octal and hex number display
- 2.1 Converting between bases
3 Representing signed integers in binary
4 Representing fractions in binary
- 4.1 Fixed-point numbers
- 4.2 Floating-point numbers
5 Numbers in programming languages
6 Footnotes
7 See also
8 Resources

[edit] Bits, bytes, nibbles, and unsigned integers

[edit] Bits

The concept of a bit can be understood as a value of either 1 or 0, on or off, yes or no, or encoded by a switch or toggle of some kind. A single bit must represent one of two states:

  one-digit binary value:       decimal value:
  -----------------------       --------------
  0                             0
  1                             1                  two distinct values

While a single bit, on its own, is able to represent only two values, a string of two bits together are able to represent increasingly many unique values:

  two-digit binary value:       decimal value:
  -----------------------       --------------
  00                            0
  01                            1
  10                            2
  11                            3                  four distinct values

A series of three binary digits can likewise designate twice as many distinct values as the two-bit string.

  three-digit binary value:     decimal value:
  -------------------------     --------------
  000                           0
  001                           1
  010                           2
  011                           3
  100                           4
  101                           5
  110                           6
  111                           7                   eight distinct values

As the number of bits within a sequence goes up, the number of possible 0 and 1 combinations increases exponentially. The examples above show that a single bit allows only two value-combinations, while two bits combined can make four separate values; three bits yield eight possibilities, and the amount of possible combinations doubles with each binary digit added:

  bits in series (b):           number of possible values (N):
  -------------------------     ------------------------------
  1                             2
  2                             4
  3                             8
  4                             16
  5                             32
  6                             64
  7                             128
  8                             256
  
  ...                                                $2 b = N$

[edit] Bytes

A byte is a sequence of eight bits or binary digits that can represent one of 256 possible values. Modern computers process information in 8-bit units, or some other multiple thereof (such as 16, 32, or 64 bits) at a time. A group of 8 bits is now widely used as a fundamental unit, and has been given the name of octet. A computer's smallest addressable memory unit (a byte) is typically an octet, so the word byte is now generally understood to mean an octet.

[edit] Nibbles

A unit of four bits, or half an octet, is often called a nibble (or nybble). It can encode 16 different values, such as the numbers 0 to 15. Any arbitrary sequence of bits could be used in principle, but in practice the most common scheme is:

  0000  =  decimal 00           1000  =  decimal 08
  0001  =  decimal 01           1001  =  decimal 09
  0010  =  decimal 02           1010  =  decimal 10
  0011  =  decimal 03           1011  =  decimal 11
  0100  =  decimal 04           1100  =  decimal 12
  0101  =  decimal 05           1101  =  decimal 13
  0110  =  decimal 06           1110  =  decimal 14
  0111  =  decimal 07           1111  =  decimal 15

This order (rather than gray code) is used because it is a positional notation, like the decimal notation that humans are more used to. For example, given the decimal number:

7531

is commonly interpreted as:

(7 × 1000) + (5 × 100) + (3 × 10) + (1 × 1)

or, using powers-of-10 notation:

(7 × 10³) + (5 × 10²) + (3 × 10¹) + (1 × 10⁰)

(Note that any non-zero number to the zero power is 1.)

Each digit in the number represents a value from 0 to 9 (hence ten different possible values) which is why this is called a decimal or base-10 number. Each digit also has a weight of a power of ten associated with its position.

Similarly, in the binary number encoding scheme mentioned above, the (decimal) value 13 is encoded as:

Each bit can only have a value of 1 or 0 (hence only two possible values) so this is a binary, or base-2 number. Accordingly, the positional weighting is as follows:

 1101
 = (1 × 2³) + (1 × 2²) + (0 × 2¹) + (1 × 2⁰)
 = (1 × 8) + (1 × 4) + (0 × 2) + (1 × 1)
 = 13 decimal

Notice the values of powers of 2 used here: 1, 2, 4, 8. Experienced computer programmers generally know the powers of 2 up to the 16^th power because they use them often:

  2⁰  =   1        2⁸  =    256
  2¹  =   2        2⁹  =    512
  2²  =   4        2¹⁰  =  1,024
  2³  =   8        2¹¹  =  2,048
  2⁴  =  16        2¹²  =  4,096
  2⁵  =  32        2¹³  =  8,192
  2⁶  =  64        2¹⁴  = 16,384
  2⁷  = 128        2¹⁵  = 32,768
                  2¹⁶  = 65,536

Sometimes, in this context (and unlike the SI International System of Units) the value 2¹⁰ = 1,024 is referred to as Kilo, or simply K (sometimes referred to as Kibibyte), so any higher powers of 2 are often conveniently referred to as multiples of that value:

  2¹¹  =  2 K  =  2,048    2¹⁴  =  16 K  =  16,384
  2¹²  =  4 K  =  4,096    2¹⁵  =  32 K  =  32,768
  2¹³  =  8 K  =  8,192    2¹⁶  =  64 K  =  65,536

Similarly, the value 2²⁰ = 1,024 × 1,024 = 1,048,576 is referred to as a Meg, or simply M (sometimes referred to as Mebibyte):

  2²¹  =  2 M
  2²²  =  4 M

and the value 2³⁰ is referred to as a Gig, or simply G (sometimes referred to as Gibibyte).

However, in December 1998 the International Electrotechnical Commission produced new units for these power-of-two values, in order to bring prefixes such as kilo- and mega- back to their SI definitions. (See Binary prefix.)

(There is another subtlety in this discussion. If we use 16 bits, we can have 65,536 different values, but the values are from 0 to 65,535. Humans start counting at one, machines start counting from zero, since it is easier to program them this way. This detail often confuses.)

The binary scheme just outlined defines a simple way to count with bits, but it has a few restrictions:

You can only perform simple arithmetic within the bounds of the number of bits that you have. That is, if you are working with 16 bits at a time, you can't perform arithmetic that gives a result of 65,536 or more.
There is no way to represent fractions with this scheme. You can only work with non-fractional (integer) quantities.
There is no way to represent negative numbers with this scheme. All the numbers are zero or positive (unsigned).

Despite these limitations, such unsigned integer numbers are very useful in computers for counting things one-by-one. They are very simple for the computer to manipulate.

[edit] Why binary?

The logic that computers use is Boolean logic which is a two-valued logic, and thus the two states of a binary system can relate directly to the two states of a Boolean logical system.
It was easier to make hardware which can distinguish between two values than multiple values. Imagine a light switch compared to a clock.
Binary is slightly more efficient than decimal. Many early computers used decimal (usually in binary-coded decimal representation). This seemingly-natural approach was eventually largely abandoned due to the increase in processing circuitry (as compared to binary) which reduced reliability.
Other bases have been tried. A few experimental computers have been built with ternary (base 3) representation, as it was thought it might be more efficient than binary.^[1]

[edit] Octal and hex number display

[edit] Converting between bases

Each of these number systems are positional systems, but while decimal weights are powers of 10, the octal weights are powers of 8 and the hex weights are powers of 16. To convert from hex or octal to decimal, for each digit one multiplies the value of the digit by the value of its position and then adds the results. For example:

octal 756
= (7 × 8²) + (5 × 8¹) + (6 × 8⁰) 
= (7 × 64) + (5 × 8)  + (6 × 1)  
=    448   +  40      +  6   = decimal 494

hex 3b2
= (3 × 16²) + (11 × 16¹) + (2 × 16⁰)
= (3 × 256) + (11 × 16)  + (2 × 1)
=    768    + 176        +  2 = decimal 946

Thus, an octal digit has a perfect correspondence to a 3-bit binary value number:

  000  =  octal 0
  001  =  octal 1
  010  =  octal 2
  011  =  octal 3
  100  =  octal 4
  101  =  octal 5
  110  =  octal 6
  111  =  octal 7

Similarly, a hex digit has a perfect correspondence to a 4-bit binary number:

  0000  =  hex 0       1000  =  hex 8
  0001  =  hex 1       1001  =  hex 9
  0010  =  hex 2       1010  =  hex a
  0011  =  hex 3       1011  =  hex b
  0100  =  hex 4       1100  =  hex c
  0101  =  hex 5       1101  =  hex d
  0110  =  hex 6       1110  =  hex e
  0111  =  hex 7       1111  =  hex f

So it is easy to convert a long binary number, such as 1001001101010001, to octal:

  001 001 001 101 010 001 binary = 
    1   1   1   5   2   1          111521 octal

and easier to convert that number to hex:

  1001 0011 0101 0001 binary =
     9    3    5    1          9351 hexadecimal

but it is harder to convert it to decimal (37713).

Conversion of numbers from hex or octal to decimal can also be done by using the following pattern.

(d1 * base + d2) * base + dn........

Where the first digit in the number is multiplied by the numbers base and added to the second digit. To convert numbers with three digits or more the pattern is just continued.

Examples of this are shown below.

hex A1

d1=A (or decimal 10)
d2=1
base=16

d1 * base + d2 =
10 * 16   + 1  = decimal 161

hex 129

d1=1
d2=2
d3=9
base=16

(d1 * base + d2) * base + d3 =
( 1 * 16   + 2)  * 16   + 9  = decimal 297

The same method can be applied to conversion of octal and binary numbers:

binary 1011

d1=1
d2=0
d3=1
d4=1
base=2

((d1 * base + d2) * base + d3) * base + d4 =
((1  * 2    + 0)  * 2    + 1)  * 2    + 1  = decimal 11

octal 1232

d1=1
d2=2
d3=3
d4=2
base=8

((d1 * base + d2) * base + d3) * base + d4 =
((1  * 8    + 2)  * 8    + 3)  * 8    + 2  = decimal 666

[edit] Representing signed integers in binary

Main article: Signed number representations

Binary numbers have no inherent way to representing negative numbers in a computer. In order to create these "signed integers" a few different systems have been developed. In each, a special bit is set aside as the "sign bit", which is usually the leftmost (most significant) bit. If the sign bit is 1 the number is negative; if 0, positive.

[edit] Sign-magnitude

Sign-magnitude representation is possibly the simplest way to represent a signed number. The representation consists of one sign bit and other bits denoting the magnitude, or absolute value, of the number. For example, using 4 bits:

0101 = +5
1101 = -5

[edit] Ones' complement

In ones' complement, the inverse of a number is formed by complementing each bit — that is, performing a bitwise NOT operation. For example:

0101 = +5
1010 = -5

A side effect of both this and the previous system, one of the reasons these systems are not often used for computing, is that there are two representations for zero. In ones' complement:

0000 = +0
1111 = -0

In sign-magnitude:

0000 = +0
1000 = -0

[edit] Two's complement

Two's complement is the most widely used system in modern computing. To form the two's complement, take the bitwise NOT of the number and add 1. For example:

0101  =  +5
1011  =  -5

Thus:

  0000  =  decimal 0    1000  =  decimal -8
  0001  =  decimal 1    1001  =  decimal -7
  0010  =  decimal 2    1010  =  decimal -6
  0011  =  decimal 3    1011  =  decimal -5
  0100  =  decimal 4    1100  =  decimal -4
  0101  =  decimal 5    1101  =  decimal -3
  0110  =  decimal 6    1110  =  decimal -2
  0111  =  decimal 7    1111  =  decimal -1

Using this system, 16 bits will encode numbers from −32,768 to 32,767, while 32 bits will encode −2,147,483,648 to 2,147,483,647.

The great advantage of the two's complement system is that most operations are not dependent on the sign of the operands and furthermore are identical to operations on unsigned binary integers.

For instance, 5 + (-5) would be performed like this:

  0101
 +1011
 10000

However, seeing as we have taken the numbers as 4 bits long, the leading 1 is discarded and we have the expected result of 0.

The fact that most operations work no matter the sign of the operands can be explained through the duplicity of numbers modulo 2ⁿ; e.g. 15 ≡ -1 (mod 16). Computers generally use a fixed number of bits for binary numbers and thus such a system is ideal. Essentially the only difference between two's complement numbers and unsigned numbers is how they are displayed and compared.

One quirk of two's complement is that the lowest encodable number (e.g. -32768 for 16 bit numbers) appears to be its own negative. However, this rarely causes problems.

As a number composed entirely of 1s (such as 11111111) equates to -1 in two's complement notation, many programming languages use -1 for true and 0 for false.

[edit] Representing fractions in binary

[edit] Fixed-point numbers

Fixed-point formats are often used in business calculations (such as with spreadsheets or COBOL), where floating-point with insufficient precision is unacceptable when dealing with money. It is helpful to study it to see how fractions can be stored in binary.

A number of bits sufficient for the precision and range required must be chosen to store the fractional and integer parts of a number. For example, using a 32-bit format, 16 bits might be used for the integer and 16 for the fraction.

The fractional bits continue the pattern set by the integer bits: if the eight's bit is followed by the four's bit, then the two's bit, then the one's bit, then of course the next bit is the half's bit, then the quarter's bit, then the ⅛'s bit, et cetera.

Examples:

                            integer bits   fractional bits
   0.5    =   ½  =  00000000 00000000.10000000 00000000
   1.25   =  1¼  =  00000000 00000001.01000000 00000000
   7.375  =  7⅜  =  00000000 00000111.01100000 00000000

However, using this form of encoding means that some numbers cannot be represented in binary. For example, for the fraction 1/5 (in decimal, this is 0.2), the closest one can get is:

  13107 / 65536  =  00000000 00000000.00110011 00110011  =  0.1999969... in decimal
  13108 / 65536  =  00000000 00000000.00110011 00110100  =  0.2000122... in decimal

And even with more digits, an exact representation is impossible. Consider the number ⅓. If you were to write the number out as a decimal (0.333333...) it would continue indefinitely. If you were to stop at any point, the number written would not exactly represent the number ⅓.

The point is: some fractions cannot be expressed exactly in binary notation... not unless you use a special trick. The trick is, to store a fraction as two numbers, one for the numerator and one for the denominator, and then use arithmetic to add, subtract, multiply, and divide them. However, arithmetic will not let you do higher math (such as square roots) with fractions, nor will it help you if the lowest common denominator of two fractions is too big a number to handle. This is why there are advantages to using the fixed-point notation for fractional numbers.

[edit] Floating-point numbers

While both unsigned and signed integers are used in digital systems, even a 32-bit integer is not enough to handle all the range of numbers a calculator can handle, and that's not even including fractions. To approximate the greater range and precision of real numbers we have to abandon signed integers and fixed-point numbers and go to a "floating-point" format.

In the decimal system, we are familiar with floating-point numbers of the form:

1.1030402 × 10⁵ = 1.1030402 × 100000 = 110304.02

or, more compactly:

  1.1030402E5

which means "1.103402 times 1 followed by 5 zeroes". We have a certain numeric value (1.1030402) known as a "significand", multiplied by a power of 10 (E5, meaning 10⁵ or 100,000), known as an "exponent". If we have a negative exponent, that means the number is multiplied by a 1 that many places to the right of the decimal point. For example:

2.3434E-6 = 2.3434 × 10^-6 = 2.3434 × 0.000001 = 0.0000023434

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

an 11-bit binary exponent, using "excess-1023" format. Excess-1023 means the exponent appears as an unsigned binary integer from 0 to 2047, and you have to subtract 1023 from it to get the actual signed value
a 52-bit significand, also an unsigned binary number, defining a fractional value with a leading implied "1"
a sign bit, giving the sign of the number.

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

  byte 0:         S    x10 x9   x8   x7   x6   x5   x4
  byte 1:         x3   x2   x1   x0   m51 m50 m49 m48
  byte 2:         m47 m46 m45 m44 m43 m42 m41 m40
  byte 3:         m39 m38 m37 m36 m35 m34 m33 m32
  byte 4:         m31 m30 m29 m28 m27 m26 m25 m24
  byte 5:         m23 m22 m21 m20 m19 m18 m17 m16
  byte 6:         m15 m14 m13 m12 m11 m10 m9   m8
  byte 7:         m7   m6   m5   m4   m3   m2   m1   m0

where "S" denotes the sign bit, "x" denotes an exponent bit, and "m" denotes a significand bit. Once the bits here have been extracted, they are converted with the computation:

<sign> × (1 + <fractional significand>) × 2^{<exponent> - 1023}

This scheme provides numbers valid out to about 15 decimal digits, with the following range of numbers:

	maximum	minimum
positive	1.797693134862231E+308	4.940656458412465E-324
negative	-4.940656458412465E-324	-1.797693134862231E+308

The spec also defines several special values that are not defined numbers, and are known as NaNs, for ‘Not A Number’. These are used by programs to designate invalid operations and the like. You will rarely encounter them and NaNs will not be discussed further here. Some programs also use 32-bit floating-point numbers. The most common scheme uses a 23-bit significand with a sign bit, plus an 8-bit exponent in "excess-127" format, giving 7 valid decimal digits.

  byte 0:         S    x7   x6   x5   x4   x3   x2   x1   
  byte 1:         x0   m22 m21 m20 m19 m18 m17 m16  
  byte 2:         m15 m14 m13 m12 m11 m10 m9   m8   
  byte 3:         m7   m6   m5   m4   m3   m2   m1   m0

The bits are converted to a numeric value with the computation:

<sign> × (1 + <fractional significand>) × 2^{<exponent> - 127}

leading to the following range of numbers:

	maximum	minimum
positive	3.402823E+38	2.802597E-45
negative	-2.802597E-45	-3.402823E+38

Such floating-point numbers are known as "reals" or "floats" in general, but with a number of inconsistent variations, depending on context:

A 32-bit float value is sometimes called a "real32" or a "single", meaning "single-precision floating-point value".

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

The term "real" without any elaboration generally means a 64-bit value, while the term "float" similarly generally means a 32-bit value.

Once again, remember that bits are bits. If you have 8 bytes stored in computer memory, it might be a 64-bit real, two 32-bit reals, or 4 signed or unsigned integers, or some other kind of data that fits into 8 bytes.

The only difference is how the computer interprets them. If the computer stored four unsigned integers and then read them back from memory as a 64-bit real, it almost always would be a perfectly valid real number, though it would be junk data.

So now our computer can handle positive and negative numbers with fractional parts. However, even with floating-point numbers you run into some of the same problems that you did with integers:

As with integers, you only have a finite range of values to deal with. Granted, it is a much bigger range of values than even a 32-bit integer, but if you keep multiplying numbers you'll eventually get one bigger than the real value can hold and have a numeric overflow.

If you keep dividing you'll eventually get one with a negative exponent too big for the real value to hold and have a numeric underflow. Remember that a negative exponent gives the number of places to the right of the decimal point and means a really small number. The maximum real value is sometimes called "machine infinity", since that's the biggest value the computer can wrap its little silicon brain around.

A related problem is that you have only limited "precision" as well. That is, you can only represent 15 decimal digits with a 64-bit real. If the result of a multiply or a divide has more digits than that, they're just dropped and the computer doesn't inform you of an error.

This means that if you add a very small number to a very large one, the result is just the large one. The small number was too small to even show up in 15 or 16 digits of resolution, and the computer effectively discards it. If you are performing computations and you start getting really insane answers from things that normally work, you may need to check the range of your data. It's possible to "scale" the values to get more accurate results. It also means that if you do floating-point computations, there's likely to be a small error in the result since some lower digits have been dropped. This effect is unnoticeable in most cases, but if you do some math analysis that requires lots of computations, the errors tend to build up and can throw off the results. The fraction of people who use computers for doing math understand these errors very well, and have methods for minimizing the effects of such errors, as well as for estimating how big the errors are. By the way, this "precision" problem is not the same as the "range" problem at the top of this list. The range issue deals with the maximum size of the exponent, while the resolution issue deals with the number of digits that can fit into the significand.

Another more obscure error that creeps in with floating-point numbers is the fact that the significand is expressed as binary fraction that doesn't necessarily perfectly match a decimal fraction.

That is, if you want to do a computation on a decimal fraction that is a neat sum of reciprocal powers of two, such as 0.75, the binary number that represents this fraction will be 0.11, or ½ + ¼, and all will be fine. Unfortunately, in many cases you can't get a sum of these "reciprocal powers of 2" that precisely matches a specific decimal fraction, and the results of computations will be very slightly off, way down in the very small parts of a fraction. For example, the decimal fraction "0.1" is equivalent to an infinitely repeating binary fraction: 0.000110011 ...

[edit] Numbers in programming languages

Low-level programmers have to worry about unsigned and signed, fixed and floating-point numbers. They have to write extremely different code, with different opcodes and operands, to add two floating point numbers compared to the code to add two integers.

However, high-level programming languages such as LISP and Python offer an abstract number that may be an expanded type such as rational, bignum, or complex. Programmers in LISP or Python (among others) have some assurance that their program code will Do The Right Thing with mathematical operations. Due to operator overloading, mathematical operations on any number—whether signed, unsigned, rational, floating-point, fixed-point, integral, or complex—are written exactly the same way. Others languages such as REXX and Java provide decimal floating-point which avoids many 'unexpected' results. One drawback in Java is though its lack of native support for unsigned integer types.

[edit] Footnotes

^ The number of possible symbols needed to represent a range of numbers is most often minimal for base 3, although binary is close in this measure. Since no substantial advantage emerged from this fact, use of base 3 for computers has not been actively pursued. See Brian Hayes, "Third Base", American Scientist 89(6): 490-494 (2001), [1].

[edit] See also

[edit] Resources

The initial version of this article was based on a public domain article from Greg Goebel's Vectorsite.

Categories: Computer arithmetic | Numeration

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