Chapter 2. Binary and Number Representation

Table of Contents

Binary -- the basis of computing

Binary Theory

Introduction

Binary is a number system which builds numbers from elements called bits. Each bit can be represented by any two mutually exclusive states. Generally, when we write it down or code bits, we represent them with 1 and 0. We also talk about them being true and false, and the computer internally represents bits with high and low voltages.

We build binary numbers the same way we build numbers in our traditional base 10 system. However, instead of a one's column, a 10's column, a 100's column (and so on) we have a one's column, a two's columns, a four's column, an eight's column, and so on, as illustrated below.

Table 2.1. Binary
2...26252423222120
...6432168421

For example, to represent the number 203 in base 10, we know we place a 3 in the 1's column, a 0 in the 10's column and a 2 in the 100's column. This is expressed with exponents in the table below.

Table 2.2. 203 in base 10
102101100
203

Or, in other words, 2 × 102 + 3 × 100 = 200 + 3 = 203. To represent the same thing in binary, we would have the following table.

Table 2.3. 203 in base 2
2726252423222120
11001011

That equates to 27 + 26 + 23+21 + 20 = 128 + 64 + 8 + 2 + 1 = 203.

Conversion

The easiest way to convert between bases is to use a computer, after all, that's what they're good at! However, it is often useful to know how to do conversions by hand.

The easiest method to convert between bases is repeated division. To convert, repeatedly divide the quotient by the base, until the quotient is zero, making note of the remainders at each step. Then, write the remainders in reverse, starting at the bottom and appending to the right each time. An example should illustrate; since we are converting to binary we use a base of 2.

Table 2.4. Convert 203 to binary
Quotient Remainder 
20310 ÷ 2 =1011 
10110 ÷ 2 =501
5010 ÷ 2 =250
2510 ÷ 2 =121
1210 ÷ 2 =60
610 ÷ 2 =30
310 ÷ 2 =11
110 ÷ 2 =01

Reading from the bottom and appending to the right each time gives 11001011, which we saw from the previous example was 203.

Bits and Bytes

To represent all the letters of the alphabet we would need at least enough different combinations to represent all the lower case letters, the upper case letters, numbers and punctuation, plus a few extras. Adding this up means we need probably around 80 different combinations.

If we have two bits, we can represent four possible unique combinations (00 01 10 11). If we have three bits, we can represent 8 different combinations. As we saw above, with n bits we can represent 2n unique combinations.

8 bits gives us 28 = 256 unique representations, more than enough for our alphabet combinations. We call a group of 8 bits a byte. Guess how bit a C char variable is? One byte.

ASCII

Given that a byte can represent any of the values 0 through 256, anyone could arbitrarily make up a mapping between characters and numbers. For example, a video card manufacturer could decide that the value 10 represents A, so when value 10 is sent to the video card it displays a capital 'A' on the screen.

To avoid this happening, the American Standard Code for Information Interchange or ASCII was invented. This is a 7-bit code, meaning there are 27 or 128 available codes.

The range of codes is divided up into two major parts; the non-printable and the printable. Printable characters are things like characters (upper and lower case), numbers and punctuation. Non-printable codes are for control, and do things like make a carriage-return, ring the terminal bell or the special NULL code which represents nothing at all.

127 unique characters is sufficient for American English, but becomes very restrictive when one wants to represent characters common in other languages, especially Asian languages which can have many thousands of unique characters.

To alleviate this, modern systems are moving away from ASCII to Unicode, which can use up to 4 bytes to represent a character, giving much more room!

Parity

ASCII, being only a 7-bit code, leaves one bit of the byte spare. This can be used to implement parity which is a simple form of error checking. Consider a computer using punch-cards for input, where a hole represents 1 and no hole represents 0. Any inadvertent covering of a hole will cause an incorrect value to be read, causing undefined behaviour.

Parity allows a simple check of the bits of a byte to ensure they were read correctly. We can implement either odd or even parity by using the extra bit as a parity bit.

In odd parity, if the number of 1's in the 7 bits of information is odd, the parity bit is set, otherwise it is not set. Even parity is the opposite; if the number of 1's is even the parity bit is set to 1.

In this way, the flipping of one bit will case a parity error, which can be detected.

XXX more about error correcting

16, 32 and 64 bit computers

Numbers do not fit into bytes; hopefully your bank balance in dollars will need more range than can fit into one byte! Most modern architectures are 32 bit computers. This means they work with 4 bytes at a time when processing and reading or writing to memory. We refer to 4 bytes as a word; this is analogous to language where letters (bits) make up words in a sentence, except in computing every word has the same size! The size of a C it variable is 32 bits. Newer architectures are 64 bits, which doubles the size the processor works with (8 bytes).

Kilo, Mega and Giga Bytes

Computers deal with a lot of bytes; that's what makes them so powerful!

We need a way to talk about large numbers of bytes, and a natural way is to use the "International System of Units" (SI) prefixes as used in most other scientific areas. So for example, kilo refers to 103 or 1000 units, as in a kilogram has 1000 grams.

1000 is a nice round number in base 10, but in binary it is 1111101000 which is not a particularly "round" number. However, 1024 (or 210) is (10000000000), and happens to be quite close to the base ten meaning of kilo (1000 as opposed to 1024).

Hence 1024 bytes became known as a kilobyte. The first mass market computer was the Commodore 64, so named because it had 64 kilobytes of storage.

Today, kilobytes of memory would be small for a wrist watch, let alone a personal computer. The next SI unit is "mega" for 106. As it happens, 220 is again close to the SI base 10 definition; 1048576 as opposed to 1000000.

The units keep increasing by powers of 10; each time it diverges further from the base SI meaning.

Table 2.5. Bytes
210Kilobyte
220Megabyte
230Gigabyte
240Terrabyte
250Petabyte
260Exabyte

Therefore a 32 bit computer can address up to four gigabytes of memory; the extra two bits can represent four groups of 230 bytes.. A 64 bit computer can address up to 8 exabytes; you might be interested in working out just how big a number this is! To get a feel for how bit that number is, calculate how long it would take to count to 264 if you incremented once per second.

Kilo, Mega and Giga Bits

Apart from the confusion related to the overloading of SI units between binary and base 10, capacities will often be quoted in terms of bits rather than bytes.

Generally this happens when talking about networking or storage devices; you may have noticed that your ADSL connection is described as something like 1500 kilobits/second. The calculation is simple; multiply by 1000 (for the kilo), divide by 8 to get bytes and then 1024 to get kilobytes (so 1500 kilobits/s=183 kilobytes per second).

The SI standardisation body has recognised these dual uses, and has specified unique prefixes for binary usage. Under the standard 1024 bytes is a kibibyte, short for kilo binary byte (shortened to KiB). The other prefixes have a similar prefix (Mebibyte, for example). Tradition largely prevents use of these terms, but you may seem them in some literature.

Boolean Operations

George Boole was a mathematician who discovered a whole area of mathematics called Boolean Algebra. Whilst he made his discoveries in the mid 1800's, his mathematics are the fundamentals of all computer science. Boolean algebra is a wide ranging topic, we present here only the bare minimum to get you started.

Boolean operations simply take a particular input and produce a particular output following a rule. For example, the simplest boolean operation, not simply inverts the value of the input operand. Other operands usually take two inputs, and produce a single output.

The fundamental Boolean operations used in computer science are easy to remember and listed below. We represent them below with truth tables; they simply show all possible inputs and outputs. The term true simply reflects 1 in binary.

Not

Usually represented by !, not simply inverts the value, so 0 becomes 1 and 1 becomes 0

Table 2.6. Truth table for not
InputOutput
10
01

And

To remember how the and operation works think of it as "if one input and the other are true, result is true

Table 2.7. Truth table for and
Input 1Input 2Output
000
100
010
111

Or

To remember how the or operation works think of it as "if one input or the other input is true, the result is true

Table 2.8. Truth table for or
Input 1Input 2Output
000
101
011
111

Exclusive Or (xor)

Exclusive or, written as xor is a special case of or where the output is true if one, and only one, of the inputs is true. This operation can surprisingly do many interesting tricks, but you will not see a lot of it in the kernel.

Table 2.9. Truth table for xor
Input 1Input 2Output
000
101
011
110

How computers use boolean operations

Believe it or not, essentially everything your computer does comes back to the above operations. For example, the half adder is a type of circuit made up from boolean operations that can add bits together (it is called a half adder because it does not handle carry bits). Put more half adders together, and you will start to build something that can add together long binary numbers. Add some external memory, and you have a computer.

Electronically, the boolean operations are implemented in gates made by transistors. This is why you might have heard about transistor counts and things like Moores Law. The more transistors, the more gates, the more things you can add together. To create the modern computer, there are an awful lot of gates, and an awful lot of transistors. Some of the latest Itanium processors have around 460 million transistors.

Working with binary in C

In C we have a direct interface to all of the above operations. The following table describes the operators

Table 2.10. Boolean operations in C
OperationUsage in C
not!
and&
or|
xor^

We use these operations on variables to modify the bits within the variable. Before we see examples of this, first we must divert to describe hexadecimal notation.

Hexadecimal

Hexadecimal refers to a base 16 number system. We use this in computer science for only one reason, it makes it easy for humans to think about binary numbers. Computers only ever deal in binary and hexadecimal is simply a shortcut for us humans trying to work with the computer.

So why base 16? Well, the most natural choice is base 10, since we are used to thinking in base 10 from our every day number system. But base 10 does not work well with binary -- to represent 10 different elements in binary, we need four bits. Four bits, however, gives us sixteen possible combinations. So we can either take the very tricky road of trying to convert between base 10 and binary, or take the easy road and make up a base 16 number system -- hexadecimal!

Hexadecimal uses the standard base 10 numerals, but adds A B C D E F which refer to 10 11 12 13 14 15 (n.b. we start from zero).

Traditionally, any time you see a number prefixed by 0x this will denote a hexadecimal number.

As mentioned, to represent 16 different patterns in binary, we would need exactly four bits. Therefore, each hexadecimal numeral represents exactly four bits. You should consider it an exercise to learn the following table off by heart.

Table 2.11. Hexadecimal, Binary and Decimal
HexadecimalBinaryDecimal
000000
100011
200102
300113
401004
501015
601106
701117
810008
910019
A101010
B101111
C110012
D110113
E111014
F111115

Of course there is no reason not to continue the pattern (say, assign G to the value 16), but 16 values is an excellent trade off between the vagaries of human memory and the number of bits used by a computer (occasionally you will also see base 8 used, for example for file permissions under UNIX). We simply represent larger numbers of bits with more numerals. For example, a sixteen bit variable can be represented by 0xAB12, and to find it in binary simply take each individual numeral, convert it as per the table and join them all together (so 0xAB12 ends up as the 16-bit binary number 1010101100010010). We can use the reverse to convert from binary back to hexadecimal.

We can also use the same repeated division scheme to change the base of a number. For example, to find 203 in hexadecimal

Table 2.12. Convert 203 to hexadecimal
Quotient Remainder 
20310 ÷ 16 =1211 (0xB) 
1210 ÷ 16 =012 (0xC)

Hence 203 in hexadecimal is 0xCB.

Practical Implications

Use of binary in code

Whilst binary is the underlying language of every computer, it is entirely practical to program a computer in high level languages without knowing the first thing about it. However, for the low level code we are interested in a few fundamental binary principles are used repeatedly.

Masking and Flags

Masking

In low level code, it is often important to keep your structures and variables as space efficient as possible. In some cases, this can involve effectively packing two (generally related) variables into one.

Remember each bit represents two states, so if we know a variable only has, say, 16 possible states it can be represented by 4 bits (i.e. 24=16 unique values). But the smallest type we can declare in C is 8 bits (a char), so we can either waste four bits, or find some way to use those left over bits.

We can easily do this by the process of masking. Remembering the rules of the logical operations, it should become clear how the values are extracted.

The process is illustrated in the figure below. We are interested in the lower four bits, so set our mask to have these bits set to 1. Since the logical and operation will only set the bit if both bits are 1, those bits of the mask set to 0 effectively hide the bits we are not interested in.

Figure 2.1. Masking
By using a mask consisting of all 1's and the logical and operation, we can extract only the bits we are interested in.

To get the top (blue) four bits, we would invert the mask. You will note this gives a result of 0x90 when really we want a value of 0x09. To get the bits into the right position we use the right shift operation.

Setting the bits requires the logical or operation. However, rather than using 1's as the mask, we use 0's. You should draw a diagram similar to the above figure and work through setting bits with the logical or operation.

Flags

Often a program will have a large number of variables that only exist as flags to some condition. For example, a state machine is an algorithm that transitions through a number of different states but may only be in one at a time. Say it has 8 different states; we could easily declare 8 different variables, one for each state. But in many cases it is better to declare one 8 bit variable and assign each bit to flag flag a particular state.

Flags are a special case of masking, but each bit represents a particular boolean state (on or off). An n bit variable can hold n different flags. See the code example below for a typical example of using flags -- you will see variations on this basic code very often.

Example 2.1. Using flags
  1 
                  #include <stdio.h>
    
    // define all 8 possible flags for an 8 bit variable
  5 //      name  hex     binary
    #define FLAG1 0x01 // 00000001
    #define FLAG2 0x02 // 00000010
    #define FLAG3 0x04 // 00000100
    #define FLAG4 0x08 // 00001000
 10 // ... and so on
    #define FLAG8 0x80 // 10000000
    
    int main(int argc, char *argv[])
    {
 15 	char flags = 0; //an 8 bit variable
    	
    	// set flags with a logical or
    	flags = flags | FLAG1; //set flag 1
    	flags = flags | FLAG3; //set flag 3
 20 	
    	// check flags with a logical and.  If the flag is set (1)
    	// then the logical and will return 1, causing the if
    	// condition to be true.
    	if (flags & FLAG1)
 25 		printf("FLAG1 set!\n");
    
    	// this of course will be untrue.
    	if (flags & FLAG8)
    		printf("FLAG8 set!\n");
 30 
    	// check multiple flags by using a logical or
    	// this will pass as FLAG1 is set
    	if (flags & (FLAG1|FLAG4))
    		printf("FLAG1 or FLAG4 set!\n");
 35 
    	return 0;
    }