In this appendix, we introduce the key number systems that C++ programmers use, especially when they are working on software projects that require close interaction with machinelevel hardware. Projects like this include operating systems, computer networking software, compilers, database systems and applications requiring high performance.
When we write an integer such as 227 or 63 in a C++ program, the number is assumed to be in the decimal (base 10) number system. The digits in the decimal number system are 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. The lowest digit is 0 and the highest is 9one less than the base of 10. Internally, computers use the binary (base 2) number system. The binary number system has only two digits, namely 0 and 1. Its lowest digit is 0 and its highest is 1one less than the base of 2.
As we will see, binary numbers tend to be much longer than their decimal equivalents. Programmers who work in assembly languages, and in highlevel languages like C++ that enable them to reach down to the machine level, find it cumbersome to work with binary numbers. So two other number systemsthe octal number system (base 8) and the hexadecimal number system (base 16)are popular primarily because they make it convenient to abbreviate binary numbers.
In the octal number system, the digits range from 0 to 7. Because both the binary and the octal number systems have fewer digits than the decimal number system, their digits are the same as the corresponding digits in decimal.
The hexadecimal number system poses a problem because it requires 16 digitsa lowest digit of 0 and a highest digit with a value equivalent to decimal 15 (one less than the base of 16). By convention, we use the letters A through F to represent the hexadecimal digits corresponding to decimal values 10 through 15. Thus in hexadecimal we can have numbers like 876 consisting solely of decimallike digits, numbers like 8A55F consisting of digits and letters and numbers like FFE consisting solely of letters. Occasionally, a hexadecimal number spells a common word such as FACE or FEEDthis can appear strange to programmers accustomed to working with numbers. The digits of the binary, octal, decimal and hexadecimal number systems are summarized in Fig. D.1Fig. D.2.
Binary digit 
Octal digit 
Decimal digit 
Hexadecimal digit 

0 
0 
0 
0 
1 
1 
1 
1 
2 
2 
2 

3 
3 
3 

4 
4 
4 

5 
5 
5 

6 
6 
6 

7 
7 
7 

8 
8 

9 
9 

A (decimal value of 10) 

B (decimal value of 11) 

C (decimal value of 12) 

D (decimal value of 13) 

E (decimal value of 14) 

F (decimal value of 15) 
Attribute 
Binary 
Octal 
Decimal 
Hexadecimal 

Base 
2 
8 
10 
16 
Lowest digit 
0 
0 
0 
0 
Highest digit 
1 
7 
9 
F 
Each of these number systems uses positional notationeach position in which a digit is written has a different positional value. For example, in the decimal number 937 (the 9, the 3 and the 7 are referred to as symbol values), we say that the 7 is written in the ones position, the 3 is written in the tens position and the 9 is written in the hundreds position. Note that each of these positions is a power of the base (base 10) and that these powers begin at 0 and increase by 1 as we move left in the number (Fig. D.3).
Positional values in the decimal number system 


Decimal digit 
9 
3 
7 
Position name 
Hundreds 
Tens 
Ones 
Positional value 
100 
10 
1 
Positional value as a power of the base (10) 
102 
101 
100 
For longer decimal numbers, the next positions to the left would be the thousands position (10 to the 3rd power), the tenthousands position (10 to the 4th power), the hundredthousands position (10 to the 5th power), the millions position (10 to the 6th power), the tenmillions position (10 to the 7th power) and so on.
In the binary number 101, the rightmost 1 is written in the ones position, the 0 is written in the twos position and the leftmost 1 is written in the fours position. Note that each position is a power of the base (base 2) and that these powers begin at 0 and increase by 1 as we move left in the number (Fig. D.4). So, 101 = 22 + 20 = 4 + 1 = 5.
Positional values in the binary number system 


Binary digit 
1 
0 
1 
Position name 
Fours 
Twos 
Ones 
Positional value 
4 
2 
1 
Positional value as a power of the base (2) 
22 
21 
20 
For longer binary numbers, the next positions to the left would be the eights position (2 to the 3rd power), the sixteens position (2 to the 4th power), the thirtytwos position (2 to the 5th power), the sixtyfours position (2 to the 6th power) and so on.
In the octal number 425, we say that the 5 is written in the ones position, the 2 is written in the eights position and the 4 is written in the sixtyfours position. Note that each of these positions is a power of the base (base 8) and that these powers begin at 0 and increase by 1 as we move left in the number (Fig. D.5).
Positional values in the octal number system 


Decimal digit 
4 
2 
5 
Position name 
Sixtyfours 
Eights 
Ones 
Positional value 
64 
8 
1 
Positional value as a power of the base (8) 
82 
81 
80 
For longer octal numbers, the next positions to the left would be the fivehundredandtwelves position (8 to the 3rd power), the fourthousandandninetysixes position (8 to the 4th power), the thirtytwothousandsevenhundredandsixtyeights position (8 to the 5th power) and so on.
In the hexadecimal number 3DA, we say that the A is written in the ones position, the D is written in the sixteens position and the 3 is written in the twohundredandfiftysixes position. Note that each of these positions is a power of the base (base 16) and that these powers begin at 0 and increase by 1 as we move left in the number (Fig. D.6).
Positional values in the hexadecimal number system 


Decimal digit 
3 
D 
A 
Position name 
Twohundredandfiftysixes 
Sixteens 
Ones 
Positional value 
256 
16 
1 
Positional value as a power of the base (16) 
162 
161 
160 
For longer hexadecimal numbers, the next positions to the left would be the fourthousandandninetysixes position (16 to the 3rd power), the sixtyfivethousandfivehundredandthirtysixes position (16 to the 4th power) and so on.