Chapter 1

Introduction

1.1Number Systems

Memory in a computer consists of numbers. Computer memory does not store these numbers in decimal (base 10). Because it greatly simplifies the hardware, computers store all information in a binary (base 2) format. First let’s review the decimal system.

1.1.1Decimal

Base 10 numbers are composed of 10 possible digits (0-9). Each digit of a number has a power of 10 associated with it based on its position in the number. For example:

234 = 2 × 102 + 3 × 101 + 4 × 100

1.1.2Binary

Base 2 numbers are composed of 2 possible digits (0 and 1). Each digit of a number has a power of 2 associated with it based on its position in the number. (A single binary digit is called a bit.) For example:

110012 = 1 × 24 + 1 × 23 + 0 × 22 + 0 × 21 + 1 × 20

=16 + 8 + 1

=25

This shows how binary may be converted to decimal. Table 1.1 shows how the first few numbers are represented in binary.

Figure 1.1 shows how individual binary digits (i.e., bits) are added. Here’s an example:

1

Table 1.1: Decimal 0 to 15 in Binary

Figure 1.1: Binary addition (c stands for carry)

110112

+100012

1011002

If one considers the following decimal division:

1234 ÷ 10 = 123 r 4

he can see that this division strips o the rightmost decimal digit of the number and shifts the other decimal digits one position to the right. Dividing by two performs a similar operation, but for the binary digits of the number. Consider the following binary division1:

11012 ÷ 102 = 1102 r 1

This fact can be used to convert a decimal number to its equivalent binary representation as Figure 1.2 shows. This method finds the rightmost digit first, this digit is called the least significant bit (lsb). The leftmost digit is called the most significant bit (msb). The basic unit of memory consists of 8 bits and is called a byte.

1The 2 subscript is used to show that the number is represented in binary, not decimal