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 (except in computer algebra systems). This document explains such issues in detail.

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. A group of 8 bits is widely used as a fundamental unit, 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 numbers 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:
we automatically interpret this as:
or, using powersof10 notation:
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 "base10" 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:
Each bit can only have a value of 1 or 0, which is two values, making this a "binary", or "base2" number. Accordingly, the "positional weighting" is as follows:
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 rememeber 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". 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:
Despite these limitations, such "unsigned integer" numbers are very useful in computers, mostly for simply counting things onebyone. They're very easy for the computer to manipulate. Generally computers use 16bit and 32bit unsigned integers, normally referred to as "integer" and "long integer" (or just "long") numbers. An "integer" allows for numbers from 0 to 65,535, while a :long" allows for integer numbers from 0 to 4,294,967,295.
Let's take a sidetrip to discuss representation of binary numbers. Computer mechanics often need to write out binary quantities, but in practice writing out a binary number like:
is a pain, and inclined to error at that. Generally computer mechanics write binary quantities in a base8 ("octal") or, much more commonly, a base16 ("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:
In an octal system, we only have 8 digits (0 through 7) and we count up through the same sequence of numbers as follows:
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:
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:
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 3bit binary value number:
Similarly, a hex digit has a perfect correspondence to a 4bit 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 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.
After defining unsigned binary numbers, the next step is to modify this scheme to have negative numbers, 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 fits.
The more natural way, from the machine point of view, is to split the range of binary numbers for a given number of bits in half and use the top half to represent negative numbers. For example, with 4bit data, you get:
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 16bit signed integer number, we can encode numbers from 32,768 to 32,767. With a 32bit signed integer number, we can encode numbers from 2,147,483,648 to 2,147,482,647.
This has some similarities to the signbit scheme in that a negative number has its topmost bit set to "1", but the two concepts are different. In signmagnitude numbers, a "5" is:
1101
while in two's complement numbers, it is:
1011
which in signmagnitude numbers 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".
This format is often used in business calculations (such as with spreadsheets or COBOL); where loss floating point precision is unacceptable when dealing when money. It is helpful to study it to see how fractions can be stored in binary.
First, we have to decide how many bits we are using to store the fractional part of a number, and how many we are using to store the integer part. Let's say that we are using a 32bit format, with 16 bits for the integer and 16 for the fraction.
How are the fractional bits used? They 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 1/8's bit, etc.
Examples:
Integer bits Fractional bits 0.5 = 1/2 = 00000000 00000000.10000000 00000000 1.25 = 1 1/4 = 00000000 00000001.01000000 00000000 7.375 = 7 3/8 = 00000000 00000111.01100000 00000000
Now for something tricky: try a fraction like 1/5 (in decimal, this is 0.2). You can't do it exactly. The best you can do is one of these:
And no, you couldn't do it, not even if you had more digits. The point is: some fractions cannot be expressed exactly in binary notation... not unless you use a special trick. The trick is, of course, to store a fraction as two numbers, one for the numerator and one for the denominator, and then use your gradeschool arithmetic to add, subtract, multiply, and divide them. However, your gradeschool 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 fixedpoint notation for fractional numbers.
While both unsigned and signed integers are used in digital systems, even a 32bit integer is not enough to handle all the range of numbers a calculator can handle, and that's not even including fractions. To obtain greater range we have to abandon signed integers and fixedpoint numbers and go to a "floatingpoint" format.
In the decimal system, we are familiar with floatingpoint numbers of the form:
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:
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 mantissa, or the "numeric precision", is much smaller than the range. Similar binary floatingpoint 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 floatingpoint format with:
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:
This scheme provides numbers valid out to 15 decimal digits, with the following range of numbers:
maximum  minimum  

positive  1.797693134862231E+308  4.940656458412465E324 
negative  4.940656458412465E324  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 overflow errors and the like. You will rarely encounter them and NANs will not be discussed further here. Some programs also use 32bit floatingpoint numbers. The most common scheme uses a 23bit mantissa with a sign bit, plus an 8bit exponent in "excess127" 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:
leading to the following range of numbers:
maximum  minimum  

positive  3.402823E+38  2.802597E45 
negative  2.802597E45  3.402823E+38 
Such floatingpoint numbers are known as "reals" or "floats" in general, but with a number of inconsistent variations, depending on context:
A 32bit float value is sometimes called a "real32" or a "single", meaning "singleprecision floatingpoint value".
A 64bit float is sometimes called a "real64" or a "double", meaning "doubleprecision floatingpoint value".
The term "real" without any elaboration generally means a 64bit value, while the term "float" similarly generally means a 32bit value.
Once again, remember that bits are bits. If you have 8 bytes stored in computer memory, it might be a 64bit real, two 32bit 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 64bit 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 floatingpoint numbers you run into some of the same problems that you did with integers, and then some:
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.
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 floatingpoint 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.
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.
However, highlevel 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 programming systems 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.
So now we have several means for using bits to encode numbers. 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 white 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 CTRLG, 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 linepattern building block characters for building forms, and extension characters for nonEnglish languages. The extensions are not highly standardized and tend to lead to a lot of 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:
This of course is is simply represented as a sequence of ASCII codes, represented in hex below:
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:
When a computer displays this value, it actually sends the following ASCII codes, represented in hex, to the display:
The confusion arises because the computer could store the value 1.537E3 as, say, a 32bit real, in which you get a pattern of 4 bytes that make up the exponent and mantissa and all that. To display the 32bit 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 32bit 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 32bit real directly, bypassing conversion to the ASCII string value. Then the computer would display something like:
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:
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 numbers". On investigation it usually proves that they are manipulating binary values that are presented by hex numbers 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 16bit code for characters. There are a variety of specs for encoding nonWestern characters, the most widely used being "Unicode", which provides character codes for Western, Asian, Indic, Hebrew, and other character sets, including even Egyptian hieroglyphics.
This material may be confusing, but if you find this interesting and want to have a more complete understanding, there are a few more details to be provided.
The first topic is the translation of binary numbers into decimal numbers 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 bruteforce method. The bruteforce 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 floatingpoint 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 16bit 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":
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":
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":
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:
Of course, converting back is a simple multiplication exercise. Doing it in hex is easier:
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 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:
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 numbers 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:
you first add:
then:
which means that you have to "carry" the "1" to the next addition:
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 fairly small:
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 8bit unsigned value (maximum of 255) and won't fit into 8 bits, so all you get is a value of 4, since the "carryout" 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 4bit 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.
Now if you do this, you will notice 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 numbers and end up with a negative one. In the case on the right, we add two negative numbers 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:
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:
Suppose you move, or "shift", all the bits one place to the left, and shove a zero in on the right. You get:
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:
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 2's complement introduces. Suppose we have a two's complement number such as:
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 8bit arithmetic (128 to 127). But see what happens if you shift right:
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:
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 2's complement works from the hardware level, as this would not work with signmagnitude numbers. 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 floatingpoint 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:
The same principle applies to binary floatingpoint 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. Also recalling highschool math, to multiply two floatingpoint values, you simply multiply the mantissas and add the exponents:
To divide, you divide the mantissas and subtract the exponents. The same applies to binary floatingpoint 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 floatingpoint numbers.
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 addsubtractmultiplydivide 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 "lookup 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 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 floatingpoint numbers, calculators normally don't.
Recall that there is a slight error in translating from decimal to binary and back again. While this problem is readily 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.
So calculators generally really perform their computations in decimal, using a scheme known as "binarycoded decimal (BCD)". This scheme uses groups of four bits to encode the individual digits in a decimal number:
With four bits, 10 BCD values can be represented. With 8, 100 BCD values can be handled:
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 floatingpoint 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 binary 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 numbers 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 64bit floatingpoint math.
v1.0.3 / 01 dec 01 / gvgoebel@yahoo.com / public domain (http://www.vectorsite.net/tsfloat.html)