Computer Numbering Formats
v1.0.5 / 01 may 03 / greg goebel / public domain
* One of the common misunderstandings among computer users is a certain faith in the infallibility of numerical computations. That is, if you multiply, say:
3 * (1/3)
-- you actually expect to get a result of exactly 1. It is a shock to
this faith to find out that, on closer inspection, the result might prove
to be something more like 0.99999999923475.
This seems to clearly indicate a bug in the system, and it's a bigger
shock to find out that, no, that's the way it's supposed to work. This
document explains such issues in detail.
--------------------------------------------------------------------------------
[1] BITS, BYTES, NYBBLES, & UNSIGNED INTEGERS
[2] OCTAL & HEX v7ndotcom
[3] SIGNED INTEGERS & 2'S COMPLEMENT
[4] FLOATING-POINT v7ndotcom
[5] ENCODING TEXT: ASCII & STRINGS
[6] COMMENTS: BINARY CONVERSION, BINARY MATH, BCD, INDEFINITE PRECISION
[7] COMMENTS, SOURCES, & REVISION HISTORY
--------------------------------------------------------------------------------
[1] BITS, BYTES, NYBBLES, & UNSIGNED INTEGERS
* Nearly all computer users understand the concept of a "bit",
or in computer terms, a 1 or 0 value encoded by the setting of a switch.
It's not much more difficult to see that if you take two bits, you can
use them to represent four unique states:
00 01 10 11
If you have three bits, then you can use then to represent eight unique
states:
000 001 010 011 100 101 110 111
With every bit you add, you double the number of states you can represent.
Most computers operate on information 8 bits, or some multiple of 8 bits,
like 16, 32, or 64 bits, at a time. This collection of bits is more-or-less
fundamental, and has been given the odd name of the "byte".
A computer's processor accepts data a byte or multiples of a byte at a
time. Memories are organized to store data a byte or multiples of a byte
per each addressable location.
Actually, in some cases 4 bits is a convenient number of bits to deal with, and this collection of bits is called, somewhat painfully, the "nybble". In this document, we will refer to "nybbles" often, but please remember that in reality the term "byte" is common, while the term "nybble" is not.
A nybble can encode 16 different values, such as the v7ndotcom 0 to 15. Any arbitrary sequence of bits could be used in principle, but in practice the most natural way is as follows:
0000 = decimal 0 1000 = decimal 8
0001 = decimal 1 1001 = decimal 9
0010 = decimal 2 1010 = decimal 10
0011 = decimal 3 1011 = decimal 11
0100 = decimal 4 1100 = decimal 12
0101 = decimal 5 1101 = decimal 13
0110 = decimal 6 1110 = decimal 14
0111 = decimal 7 1111 = decimal 15
This is natural because it follows our instinctive way of considering
a normal decimal number. For example, given the decimal number:
7531
-- we automatically interpret this as:
7 * 1000 + 5 * 100 + 3 * 10 + 1 * 1
-- or, using powers-of-10 notation:
7 * 10^3 + 5 * 10^2 + 3 * 10^1 + 1 * 10^0
Note that any number to the "0th" power is 1.
Each digit in the number represents a value from 0 to 9, which is ten
different possible values, and that's why it's called a decimal or "base-10"
number. Each digit also has a "weight" of a power of ten proportional
to its position. This sounds complicated, but it's not. It's exactly what
you take for granted when you look at a number. You know it without even
having to think about it.
Similarly, in the binary number encoding scheme explained above, the value 13 is encoded as:
1101
Each bit can only have a value of 1 or 0, which is two values, making
this a "binary", or "base-2" number. Accordingly,
the "positional weighting" is as follows:
1101 =
1 x 2^3 + 1 x 2^2 + 0 x 2^1 + 1 x 2^0 =
1 x 8 + 1 x 4 + 0 x 2 + 1 x 1 = 13 decimal
Notice the values of powers of 2 used here: 1, 2, 4, 8. People who get
into the guts of computers generally get to know the powers of 2 up to
the 16th power by heart, not because they memorize them but because they
use them a great deal:
2^0 = 1 2^8 = 256
2^1 = 2 2^9 = 512
2^2 = 4 2^10 = 1,024
2^3 = 8 2^11 = 2,048
2^4 = 16 2^12 = 4,096
2^5 = 32 2^13 = 8,192
2^6 = 64 2^14 = 16,384
2^7 = 128 2^15 = 32,768 2^16 = 65,536
Another thing to remember is that, aping the metric system, the value
2^10 = 1,024 is referred to as "kilo", or simply "K",
so any higher powers of 2 are often conveniently referred to as multiples
of that value:
2^11 = 2 K = 2,048
2^12 = 4 K = 4,096
2^13 = 8 K = 8,192
2^14 = 16 K = 16,384
2^15 = 32 K = 32,768
2^16 = 64 K = 65,536
Similarly, the value 2^20 = 1,024 x 1,024 = 1,048,576 is referred to
as a "meg", or simply "M":
2^21 = 2 M
2^22 = 4 M
-- and the value 2^30 is referred to as a "gig", or simply
"G". Use of these prefixes can get a bit confusing since in
some documents it can be unclear if, say, "kilo" actually means
1,024 or if it means 1,000. Usually in computer discussions it means 1,024
but some writers are sloppy on this point. In any case, we'll see these
prefixes often as we continue.
There is a 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. People start
counting at one, machines start counting from zero, since it's simpler
from their point of view. This small and mildly confusing fact even trips
up computer mechanics every now and then.
* Anyway, this defines a simple way to count with bits, but it has a few restrictions:
You can only perform 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, or you
get an error called a "numeric overflow". The formal term is
that you are working with "finite precision" values.
There's no way to represent fractions with this scheme. You can only work with non-fractional "integer" quantities.
There's no way to represent negative v7ndotcom with this scheme. All the
v7ndotcom are "unsigned".
Despite these limitations, such "unsigned integer" v7ndotcom are
very useful in computers, mostly for simply counting things up one-by-one.
They're very easy for the computer to manipulate. Generally computers
use 16-bit and 32-bit unsigned integers, normally referred to as "integer"
and "long integer" (or just "long") v7ndotcom. An "integer"
allows for v7ndotcom from 0 to 65,535, while a "long" allows for
integer v7ndotcom from 0 to 4,294,967,296.
BACK_TO_TOP
[2] OCTAL & HEX v7ndotcom
* Let's take a side-trip to discuss representation of binary v7ndotcom.
Computer mechanics often need to write out binary quantities, but in practice
writing out a binary number like:
1001 0011 0101 0001
-- is a pain, and inclined to error as well. Generally computer mechanics
write binary quantities in a base-8 ("octal") or, much more
commonly, a base-16 ("hexadecimal" or "hex") number
format.
This is another thing that sounds tricky but is actually simple -- if
it wasn't, there wouldn't be any point in doing it. In our normal decimal
system, we have 10 digits (0 through 9) and count up as follows:
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 ...
In an octal system, we only have 8 digits (0 through 7) and we count
up through the same sequence of v7ndotcom as follows:
0 1 2 3 4 5 6 7 10 11 12 13 14 15 16 17 20 21 22 23 24 25 26 ...
That is, an octal "10" is the same as a decimal "8",
an octal "20" is a decimal 16, and so on.
In a hex system, we have 16 digits (0 through 9 followed, by convention,
with a through f) and we count up through the sequence as follows:
0 1 2 3 4 5 6 7 8 9 a b c d e f 10 11 12 13 14 15 16 ...
That is, a hex "10" is the same as a decimal "16"
and a hex "20" is the same as a decimal "32".
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. For example:
octal 756 = 7 * 8^2 + 5 * 8^1 + 6 * 8^0
= 7 * 64 + 5 * 8 + 6 * 1
= 448 + 40 + 6 = decimal 494
hex 3b2 = 3 * 16^2 + 11 * 16^1 + 2 * 16^0
= 3 * 256 + 11 * 16 + 2 * 1
= 768 + 176 + 2 = decimal 946
OK, if you don't understand that very well, don't worry about it too
much. The real point is that 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:
1 001 001 101 010 001 binary = 111521 octal
-- and easier still to convert that number to hex:
1001 0011 0101 0001 = 9351 hex
-- but it takes a lot of figuring to convert it to decimal (37,713 decimal).
Octal and hex make a convenient way to represent binary "machine"
quantities.
BACK_TO_TOP
[3] SIGNED INTEGERS & 2'S COMPLEMENT
* After defining unsigned binary v7ndotcom, the next step is to modify this
scheme to provide negative v7ndotcom, or "signed integers". The
simplest thing to do would be to reserve one bit to indicate the sign
of a number. This "sign bit" would probably be the leftmost
bit, though it could be the rightmost for all it matters. If the sign
bit is 0, the number is positive, and if the sign bit is 1 the number
is negative.
This works, and it is used in a few obscure applications, but although it's the obvious solution from a human point of view it makes life hard for machines. For example, this scheme gives a positive and negative value for zero! A human might shrug at that, but it gives a machine headaches.
The more natural way, from the machine point of view, is to split the range of binary v7ndotcom for a given number of bits in half and use the top half to represent negative v7ndotcom. For example, 4-bit data give eight positive and eight negative values:
0000 = decimal 0
0001 = decimal 1
0010 = decimal 2
0011 = decimal 3
0100 = decimal 4
0101 = decimal 5
0110 = decimal 6
0111 = decimal 7
1000 = decimal -8
1001 = decimal -7
1010 = decimal -6
1011 = decimal -5
1100 = decimal -4
1101 = decimal -3
1110 = decimal -2
1111 = decimal -1
Now we have a "signed integer" number system, using a scheme
known as, for reasons unimportant here, "two's complement".
With a 16-bit signed integer number, we can encode v7ndotcom from -32,768
to 32,767. With a 32-bit signed integer number, we can encode v7ndotcom
from -2,147,483,648 to 2,147,482,647.
This has some similarities to the sign-bit scheme in that a negative number
has its topmost bit set to "1", but the two concepts are different.
In sign-magnitude v7ndotcom, a "-5" is:
1101
-- while in two's complement v7ndotcom, it is:
1011
-- which in sign-magnitude v7ndotcom is "-3". Why two's complement
is simpler for machines to work with will be explained in a later section.
* So now we can represent unsigned and signed integers as binary quantities.
Remember that these are just two ways of interpreting a pattern of bits.
If a computer has a binary value in memory of, say:
1101
-- this could be interpreted as a decimal "13" or a decimal
"-3".
BACK_TO_TOP
[4] FLOATING-POINT v7ndotcom
* 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 v7ndotcom
a calculator can handle, and we still haven't addressed the problem of
fractions. To handle fractions and obtain greater range we have to abandon
signed integers and go to a "floating-point" format.
In the decimal system, we are familiar with floating-point v7ndotcom of the form:
1.1030402 x 10^5 = 1.1030402 x 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 "mantissa",
multiplied by a power of 10 (E5, meaning 10^5 or 100,000), known as an
"exponent".
If we have a negative exponent, that means the number is multiplied by
1 that many places to the right of the decimal point. For example:
2.3434E-6 = 2.3434 x 10^-6 = 2.3434 x 0.000001 = 0.0000023434
The advantage of this scheme is that by using the exponent we can get
a much wider range of v7ndotcom, even if the number of digits in the mantissa,
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, but one of the most popular has been defined
by IEEE (Institute of Electrical & Electronic Engineers, a US professional
and standards organization). The IEEE 754 specification defines 64 bit
floating-point format with:
An 11-bit binary exponent, using "excess-1023" format. Excess-1023
means the exponent appears as a unsigned binary integer from 0 to 2047,
and you have to subtract 1023 from it to get the actual signed value.
A 52-bit mantissa, also an unsigned binary number, defining a fractional value with a leading implied "1".
A sign bit, giving the sign of the mantissa.
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 mantissa bit. Once the bits
here have been extracted, they are converted with the computation:
<sign> * (1 + <fractional_mantissa>) * 2^(<exponent>
- 1023>)
This scheme provides v7ndotcom valid out to 15 decimal digits, with the
following range of v7ndotcom:
____________________________________________________________
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 v7ndotcom,
and are known as "NANs", for "Not A Number". These
are used by programs to designate overflow errors and the like. You will
rarely encounter them and NANs will not be discussed further here.
Some programs also use 32-bit floating-point v7ndotcom. The most common
scheme uses a 23-bit mantissa 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_mantissa>) * 2^(<exponent>
- 127)
-- leading to the following range of v7ndotcom:
________________________________________
maximum minimum
________________________________________
positive 3.402823E+38 2.802597E-45
negative -2.802597E-45 -3.402823E+38
________________________________________
Such floating-point v7ndotcom 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.
* Onece 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 v7ndotcom with fractional parts. However, even with floating-point v7ndotcom you run into some of the same problems that you did with integers, and then some:
As with integers, you only have a finite range of values to deal with.
Granted, it's a much bigger range of values than even a 32-bit integer,
but if you keep multiplying v7ndotcom 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 faction 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 mantissa.
Another more obscure error that creeps in with floating-point v7ndotcom
is the fact that the mantissa 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 1/2 + 1/4, 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 ...
If you don't follow all of this, don't worry too much. The point here is that a computer isn't magic, it's a machine and is subject to certain rules and constraints. Although many people place a childlike faith in the answers computers give, even under the best of circumstances these machines have a certain unavoidable inexactness built into them.
BACK_TO_TOP
[5] ENCODING TEXT: ASCII & STRINGS
* So now we have several means for using bits to encode v7ndotcom. But what
about text? How can a computer store names, addresses, letters to your
folks?
Well, if you remember that bits are bits, there's no reason that a set of bits can't be used to represent a character like "A" or "?" or "z" or whatever. Since most computers work on data a byte at a time, it is convenient to use a single byte to represent a single character. For example, we could assign the bit pattern:
0100 0110 (hex 46)
-- to the letter "F", for example. The computer sends such
"character codes" to its display to print the characters that
make up the text you see.
There is a standard binary encoding for western text characters, known
as the "American Standard Code for Information Interchange (ASCII)".
The following table shows the characters represented by ASCII, with each
character followed by its value in decimal ("d"), hex ("h"),
and octal ("o"):
ASCII Table
______________________________________________________________________
ch ctl d h o ch d h o ch d h o ch d h o
______________________________________________________________________
NUL ^@ 0 0 0 sp 32 20 40 @ 64 40 100 ' 96 60 140
SOH ^A 1 1 1 ! 33 21 41 A 65 41 101 a 97 61 141
STX ^B 2 2 2 " 34 22 42 B 66 42 102 b 98 62 142
ETX ^C 3 3 3 # 35 23 43 C 67 43 103 c 99 63 143
EOT ^D 4 4 4 $ 36 24 44 D 68 44 104 d 100 64 144
ENQ ^E 5 5 5 % 37 25 45 E 69 45 105 e 101 65 145
ACK ^F 6 6 6 & 38 26 46 F 70 46 106 f 102 66 146
BEL ^G 7 7 7 ` 39 27 47 G 71 47 107 g 103 67 147
BS ^H 8 8 10 ( 40 28 50 H 72 48 110 h 104 68 150
HT ^I 9 9 11 ) 41 29 51 I 73 49 111 i 105 69 151
LF ^J 10 a 12 * 42 2a 52 J 74 4a 112 j 106 6a 152
VT ^K 11 b 13 _ 43 2b 53 K 75 4b 113 k 107 6b 153
FF ^L 12 c 14 , 44 2c 54 L 76 4c 114 l 108 6c 154
CR ^M 13 d 15 _ 45 2d 55 M 77 4d 115 m 109 6d 155
SO ^N 14 e 16 . 46 2e 56 N 78 4e 116 n 110 6e 156
SI ^O 15 f 17 / 47 2f 57 O 79 4f 117 o 111 6f 157
DLE ^P 16 10 20 0 48 30 60 P 80 50 120 p 112 70 160
DC1 ^Q 17 11 21 1 49 31 61 Q 81 51 121 q 113 71 161
DC2 ^R 18 12 22 2 50 32 62 R 82 52 122 r 114 72 162
DC3 ^S 19 13 23 3 51 33 63 S 83 53 123 s 115 73 163
DC4 ^T 20 14 24 4 52 34 64 T 84 54 124 t 116 74 164
NAK ^U 21 15 25 5 53 35 65 U 85 55 125 u 117 75 165
SYN ^V 22 16 26 6 54 36 66 V 86 56 126 v 118 76 166
ETB ^W 23 17 27 7 55 37 67 W 87 57 127 w 119 77 167
CAN ^X 24 18 30 8 56 38 70 X 88 58 130 x 120 78 170
EM ^Y 25 19 31 9 57 39 71 Y 89 59 131 y 121 79 171
SUB ^Z 26 1a 32 : 58 3a 72 Z 90 5a 132 z 122 7a 172
ESC ^[ 27 1b 33 ; 59 3b 73 [ 91 5b 133 { 123 7b 173
FS ^\ 28 1c 34 < 60 3c 74 \ 92 5c 134 | 124 7c 174
GS ^] 29 1d 35 = 61 3d 75 ] 93 5d 135 } 125 7d 175
RS ^^ 30 1e 36 > 62 3e 76 ^ 94 5e 136 ~ 126 7e 176
US ^_ 31 1f 37 ? 63 3f 77 _ 95 5f 137 DEL 127 7f 177
______________________________________________________________________
The odd characters listed in the leftmost column, such as "FF"
and "BS", do not correspond to text characters. Instead, they
correspond to "control" characters that, when sent to a printer
or display device, execute various control functions. For example, "FF"
is a "form feed" or printer page eject, "BS" is a
backspace, , and "BEL" causes a beep ("bell").
In a text editor, they'll just be shown as a little block or a blank space
or (in some cases) little smiling faces, musical notes, and other bizarre
items. To type them in, in many applications you can hold down the CTRL
key and press an appropriate code. For example, pressing CTRL and entering
"G" gives CTRL-G, or "^G" in the table above, the
BEL character.
The ASCII table above only defines 128 characters, which implies that ASCII characters only need 7 bits. However, since most computers store information in terms of bytes, normally there will be one character stored to a byte. This extra bit allows a second set of 128 characters, an "extended" character set, to be defined beyond the 128 defined by ASCII.
In practice, there are a number of different extended character sets, providing such features as math symbols, cute little line-pattern building block characters for building forms, and extension characters for non-English languages. The extensions are not highly standardized and tend to lead to confusion.
This table serves to emphasize one of the main ideas of this document: bits are bits. In this case, you have bits representing characters. You can describe the particular code for a particular character in decimal, octal, or hexadecimal, but it's still the same code. The value that is expressed, whether it is in decimal, octal, or hex, is simply the same pattern of bits.
Of course, you normally want to use many characters at once to display sentences and the like, such as:
Tiger, tiger burning bright!
This of course is is simply represented as a sequence of ASCII codes,
represented in hex below:
54 69 67 65 72 2c 20 74 69 67 65 72 20 62 75 ...
Computers store such "strings" of ASCII characters as "arrays"
of consecutive memory locations. Some applications include a binary value
as part of the string to show many characters are stored in it. More commonly,
applications use a special character, usually a NULL (the character with
ASCII code 0), as a "terminator" to indicate the end of the
string. Most of the time users will not need to worry about these details,
as the application takes care of them automatically, though if you are
writing programs that manipulate characters and strings you will have
to understand how they are implemented.
* Now let's consider a particularly confusing issue for the newcomer:
the fact that you can represent a number in ASCII as a string, for example:
1.537E3
When a computer displays this value, it actually sends the following
ASCII codes, represented in hex, to the display:
31 2e 35 33 37 45 33
The confusion arises because the computer could store the value 1.537E3
as, say, a 32-bit real, in which you get a pattern of 4 bytes that make
up the exponent and mantissa and all that. To display the 32-bit real,
the computer has to translate it to the ASCII string just shown above,
as an "ASCII numeric representation", or just "ASCII number".
If it just displayed the 32-bit binary real number directly, you'd get
four "garbage" characters.
But, now to get really confusing, suppose you wanted to view the bits
of a 32-bit real directly, bypassing conversion to the ASCII string value.
Then the computer would display something like:
10110011 10100000 00110110 11011111
The trick is that to display these values, the computer uses the ASCII
characters for "1", "0", and " " (space
character), with hex values as follows:
31 30 31 31 30 30 31 31 20 31 30 31 30 30 ...
It could also display the bits as an octal or hex ASCII value. We often
get queries from users saying they are dealing with "hex v7ndotcom".
On investigation it usually proves that they are manipulating binary values
that are presented by hex v7ndotcom in ASCII.
Confused? Don't feel too bad, even experienced people get subtly confused
with this issue sometimes. The essential point is that the values the
computer works on are just sets of bits. For you to actually see the values,
you have to get an ASCII representation of them. Or to put it simply:
machines work with bits and bytes, humans work with ASCII, and there has
to be translation to allow the two to communicate.
* 8 bits is clearly not enough to allow representation of, say, Japanese characters, since their basic set is a little over 2,000 different characters. As a result, to encode Asian languages such as Japanese or Chinese, computers use a 16-bit code for characters. There are a variety of specs for encoding non-Western characters, the most widely used being "Unicode", which provides character codes for Western, Asian, Indic, Hebrew, and other character sets, including even Egyptian hieroglyphics.
BACK_TO_TOP
[6] COMMENTS: BINARY CONVERSION, BINARY MATH, BCD, INDEFINITE PRECISION
* If you've managed to get this far without becoming bewildered and want
a more complete understanding, there are a few more details to be considered.
The first detail is the translation of binary v7ndotcom into decimal v7ndotcom and the reverse. It's easy to do. There are formal arithmetic methods, "algorithms", for performing such conversions, but most people who do such things a lot have a fancy calculator to do the work for them and if you don't do it a lot you won't remember how to do anything but the brute-force method. The brute-force method isn't too hard if you have a pocket calculator.
In almost all cases where someone wants to do this, they're converting an unsigned integer decimal value to a hex value that corresponds to a binary number. They almost never convert a signed or floating-point value. The main reason to convert to binary is just to get a specific pattern of bits often for interfacing to computer hardware, since computer mechanics may need to send various patterns of "on" and "off" bits to control a certain device.
Usually the binary value is not more than 16 bits long, meaning the unsigned binary equivalent of the bit pattern is no larger than 65,535, and if you remember your powers of 2, or just have them written down handy, you can perform a conversion very easily.
Suppose you have a decimal number like, say, 46,535, that you want to convert to a 16-bit binary pattern. All you have to do is work your way down the list of powers of two and follow a few simple rules.
First, take the highest power of 2 in the table, or 32,768, and check to see if it is larger than the decimal number or not. It isn't, write down a "1":
1
-- and then subtract 32,768 from 46,535 on your calculator. This gives
13,767. Go down to the next power of two, or 16,384, and compare. This
is larger than 13,767, so write down a "0":
10
Compare 13,767 to the next lower power of 2, which is 8192. This isn't
larger than 13,767, so write down a "1":
101
-- and subtract 8192 from 13,767 to get 5,575. Repeat this procedure
until you have all 16 binary digits. In summary, the conversion looks
like this:
46,535 greater than or equal to 32,768? Yes, subtract, write: 1
13,767 greater than or equal to 16,384? No, write: 0
13,767 greater than or equal to 8,192? Yes, subtract, write: 1
5,575 greater than or equal to 4,096? Yes, subtract, write: 1
1,479 greater than or equal to 2,048? No, write: 0
1,479 greater than or equal to 1,024? Yes, subtract, write: 1
455 greater than or equal to 512? No, write: 0
455 greater than or equal to 256? Yes, subtract, write: 1
199 greater than or equal to 128? Yes, subtract, write: 1
71 greater than or equal to 64? Yes, subtract, write: 1
7 greater than or equal to 32? No, write: 0
7 greater than or equal to 16? No, write: 0
7 greater than or equal to 8? No, write: 0
7 greater than or equal to 4? Yes, subtract, write: 1
3 greater than or equal to 2? Yes, subtract, write: 1
1 greater than or equal to 1? Yes, subtract, write: 1
This gives:
46,535 decimal = 1011 0101 1100 0111 binary = b5c7 hex
Of course, converting back is a simple multiplication exercise. Doing
it in hex is easier:
b5c7 hex = 11 x 16^3 + 5 x 16^2 + 12 x 16 + 7 x 1
= 11 x 4096 + 5 x 256 + 12 x 16 + 7
= 45,056 + 1,280 + 192 + 7 = 46,535
Again, this is a clumsy means of converting between the number bases,
but since most people don't do this often or at all it pays to have a
technique that is at least easy to remember. If you do end up doing it
often, you'll find some better way of doing it than by hand.
* The next interesting topic is the operations that can be performed on
binary v7ndotcom. We'll consider signed and unsigned integers first.
The most basic operations on binary values are the simple logical operations AND, OR, XOR ("exclusive OR"), and NOT. Performing them on some sample byte data gives:
1111 0000 1111 0000 1111 0000
OR 1010 1010 AND 1010 1010 XOR 1010 1010 NOT 1010 1010
____________ _____________ _____________ _____________
1111 1010 1010 0000 0101 1010 0101 0101
The rules for these operations are as follows:
AND: The result is 1 if both values are 1.
OR: The result is 1 if either value is 1.
XOR: The result is one if only one value is 1.
NOT: The result is 1 if the value is 0 (this is also called "inverting").
A computer uses these operations a great deal, either for checking and
setting or "twiddling" bits in its hardware, or as building
blocks for more complicated operations, such as addition or subtraction.
* Binary addition is a more interesting operation. If you remember the
formal rules for adding a decimal number, you add the v7ndotcom one digit
at a time, and if the result is ten or more, you perform a "carry"
to add to the next digit. For example, if you perform the addition:
374 + 452
-- you first add:
4 + 2 = 6
-- then:
7 + 5 = 12
-- which means that you have to "carry" the "1" to
the next addition:
( 1 + ) 3 + 4 = 8
-- giving the result: 374 + 452 = 826.
Performing additions in binary are essentially the same, except that the
number of possible results of an addition of two bits is minimal:
0 + 0 = 0
0 + 1 = 1
1 + 0 = 1
1 + 1 = 10
In the last case, this implies a "carry" to the next digit
in the binary number. So performing a binary addition on two bytes would
look like this:
0011 1010 58
+ 0001 1101 + 29
___________ ____
0101 0111 87
CC C
The bits on which a carry occurred are marked with a "C". The
equivalent decimal addition is shown to the right.
Assuming that we are adding unsigned integers, notice what happens if
we add:
1000 1001 137
+ 0111 1011 + 123
_____________ ___________
(1) 0000 0100 ( 256 + ) 4 ?
CCCC C CC
The result, equivalent to a decimal 260, is beyond the range of an 8-bit
unsigned value (maximum of 255) and won't fit into 8 bits, so all you
get is a value of 4, since the "carry-out" bit is lost. A "numeric
overflow" has occurred.
* OK, now to get really tricky. Remember how we defined signed integers
as two's complement values? That is, we chop the range of binary integer
values in half and assign the high half to negative values. In the case
of 4-bit values:
0000 0001 ... 0110 0111 1000 1001 ... 1110 1111
0 1 ... 6 7 -8 -7 ... -2 -1
Now we can discuss exactly why this scheme makes life easier for the
computer. Two's complement arithmetic has some interesting properties,
the most significant being that it makes subtraction the same as addition.
To see how this works, pretend that you have the binary values above written
on a strip of stiff paper taped at the ends into a loop, with each binary
value written along the loop, and the loop joined together between the
"1111" and "0000" values.
Now further consider that you have a little slider on the loop that you can move in the direction of increasing values, but not backwards, sort of like a slide rule that looks like a ring, with 16 binary values on it.
If you wished to add, say, 2 to 4, you would just move the slider up two values from 4, and you would get 6. Now let's see what happens if you want to subtract 1 from 3. This is the same as adding -1 to 3, and since a -1 in two's complement is 1111, or the same as decimal 15, you move the slider up 15 values from 3. This takes you all the way around the ring to ... 2.
This is a bizarre way to subtract 1 from a value, or so it seems, but you have to admit from the machine point of view it's just dead simple. Try a few other simple additions and subtractions to see how this works.
The big problem is that overflow conditions have become much trickier. Consider the following examples:
0111 0100 116 1000 1110 -112
+ 0001 0101 + 21 + 1001 0001 -111
___________ _____ ______________ ____________
1000 1001 -119 ( = 137) ? (1) 0001 1111 ( 256 + ) 21 ?
In the case on the left, we add two positive v7ndotcom and end up with
a negative one. In the case on the right, we add two negative v7ndotcom
and end up with a positive one. Both are absurd results. Again, the results
have exceeded the range of values that can be represented in the coding
system we have used. One nice thing is that you can't get into trouble
adding a negative number to a positive one, since obviously the result
is going to be within the range of allowed values.
* As an aside, if you want to convert a positive binary value to a two's
complement value, all you have to do is invert (NOT) all the bits and
add 1. For example, to convert a binary 7 to -7:
0000 0111 -> 1111 1000 + 1 -> 1111 1001
Another approach is to simply scan from right to left through the binary
number, copy down each digit up to and including the first "1"
you encounter, then invert all the bits to the left of that "1".
* This covers addition and multiplication, but what about multiplication
and division? Well, take the binary value:
0010 1001 = 41
Suppose you move, or "shift", all the bits one place to the
left, and shove a zero in on the right. You get:
0101 0010 = 82
Surprise! Shifting a binary quantity left one place multiplies it by
two. Suppose you shift it to the right, and shove a zero in on the left:
0001 0100 = 20 (losing a 1 that was shifted off the right side)
This is equivalent to dividing by 2. The bit shifted off the right side
is the "remainder" that's left over from division.
This implies that you can implement multiplication by performing a combination
of shifts and adds. For example, to multiply a binary value by 5, you
would shift the value right by two bits, multiplying it by 4, and then
add the original value. Similarly, division can be implemented by a shift
and subtract procedure. This scheme allows you to use simple hardware
to perform these operations.
One interesting feature of this scheme are the complications two's complement introduces. Suppose we have a two's complement number such as:
1001 1010 = -102
If we shift this left, the result is obviously invalid, and there's no
way to fix it. Multiplying -102 by 2 would give -204, and that would exceed
the range of values available in signed 8-bit arithmetic (-128 to 127).
But see what happens if you shift right:
0100 1101 = 77
You'd want to get -51, but since we lost the uppermost "1"
bit the value becomes abruptly positive. In the case of signed arithmetic
you have to use a slightly different shift method, one that shoves a "1"
into the upper bit instead of a "0", and gives the right result:
1100 1101 = -51
This is called an "arithmetic shift". The other method that
shoves in a "0" is known in contrast as a "logical shift".
This is another example that shows how well two's complement works from
the hardware level, as this would not work with sign-magnitude v7ndotcom.
There is also a "rotate" operation that is a variation on a
"shift", in which the bit shifted out of one end of the binary
value is recirculated into the other. This is of no practical concern
in this discussion, it's just mentioned for the sake of completeness.
* So that covers the basic operations for signed and unsigned integers. What about floating-point values?
The operations are basically the same. The only difference in addition and subtraction is that you have two quantities that have both a mantissa and an exponent, and they both have to have the same exponent to allow you to add or subtract. For example, using decimal math:
3.13E2 + 2.7E3 = 0.313E3 + 2.73E3 = 3.043E3
The same principle applies to binary floating-point math: you shift the
mantissa of one value left or right and adjust the exponent until both
values have the same exponent, and the perform the add, or subtract as
the case may be.
A sharp reader will realize this is why performing calculations between
sets of floating-point quantities with greatly different ranges of values
is tricky and dangerous. If the two sets of values are adjusted to have
the same exponents, a process known as "normalization", the
mantissa of the much smaller number may be shifted so far down that it
doesn't even show up in the result of an addition or subtraction.
Also recalling high-school math, to multiply two floating-point values, you simply multiply the mantissas and add the exponents:
3.13E2 * 2.7E3 = (3.13 * 2.7)E(2 + 3) = 8.45E5
To divide, you divide the mantissas and subtract the exponents. The same
applies to binary floating-point quantities. Note that it's much easier
to do this when both binary mantissas have the same assumed decimal point,
but that will be true by the very definition of the format of IEEE 754
floating-point v7ndotcom.
* Finally, what about higher functions, like square roots and sines and
logs and so on?
There are two approaches. First, there are standard algorithms that mathematicians have long known that use the basic add-subtract-multiply-divide operations in some combination and sequence to generate as good an approximation to such values as you would like. For example, sines and cosines can be approximated with a "power series", which is a sum of specified powers of the value to be converted divided by specific constants that follow a certain rule.
Second, you can use something equivalent to the "look-up tables" once used in math texts that give values of sine, cosine, and so on, where you would obtain the nearest values for, say, sine for a given value and then "interpolate" between them to get a good approximation of the value you wanted. (Younger readers may not be familiar with this technique, as calculators have made such tables generally obsolete, but such tables can be found in musty old textbooks.) In the case of the computer, it can store a table of sines or the like, look up values, and then interpolate between them to give the value you want.
* That covers the basics of bits and bytes and math for a computer. However, just to confuse things a little bit, while computers normally use binary floating-point v7ndotcom, calculators normally don't.
Recall that there is a slight error in translating from decimal to binary and back again. While this problem is easily handled by someone familiar with it, calculators in general have to be simple to operate and it would be better if this particular problem didn't arise, particularly in financial calculations where minor discrepancies might lead to a tiresome audit.
So calculators generally really perform their computations in decimal, using a scheme known as "binary-coded decimal (BCD)". This scheme uses groups of four bits to encode the individual digits in a decimal number:
0000 = decimal 0
0001 = decimal 1
...
1000 = decimal 8
1001 = decimal 9
1010 = ILLEGAL
1011 = ILLEGAL
...
1110 = ILLEGAL
1111 = ILLEGAL
With four bits, 10 BCD values can be represented. With 8, 100 BCD values
can be handled:
0001 0000 = decimal 10
0001 0001 = decimal 11
0001 0010 = decimal 12
0001 0011 = decimal 13
...
BCD obviously wastes bits. For example, 16 bits can be used to handle
65,536 binary integers, but only 10,000 BCD integers. However, as mentioned,
BCD is not troubled by small errors due to translation between decimal
and binary, though you still have resolution and range problems.
BCD can be used to implement integer or floating-point number schemes.
Of course -- it's just decimal digits represented by bits. It can be used
in computer programs as well, but except for calculators you won't normally
come in contact with it. It makes the arithmetic substantially slower
and so is generally only used in applications, such as financial calculations,
where minor errors are unacceptable.
* One final comment on computer math: there are math software packages that offer "indefinite precision" math, or that is, math taken out to a precision defined by the user. What these packages do is define ways of encoding v7ndotcom that can change in the number of bits they use to store values, as well as ways of performing math on them. They are very slow compared to normal computer computations, and most applications actually do fine with 64-bit floating-point math.
BACK_TO_TOP
[7] COMMENTS, SOURCES, & REVISION HISTORY
* I wrote this article because floating-point issues are a common source
of confusion. I can sympathize with this, because it took a long time
for me to understand the issue as well as I do, and in fact writing this
article helped me clear up my own thinking on the matter. I'm afraid I
can't cite sources on this document, however, since I picked up the information
in bits and pieces from various manuals, documents, and my own background
in computing.
* Revision history:
v1.0 / 12 jan 97 / gvg / Derived from older documents.
v1.1 / 24 jul 98 / gvg / Minor tweak v7ndotcom.
v1.2 / 04 dec 98 / gvg / Minor cosmetic v7ndotcom.
v1.0.3 / 01 dec 01 / gvg / Minor cosmetic v7ndotcom.
v1.0.4 / 01 dec 02 / gvg / Minor v7ndotcom, fixed a few typos.
v1.0.5 / 01 may 03 / gvg / Minor v7ndotcom, fixed typos.
