Chapter 1: Computer Arithmetic  

1.1 Introduction to Number Systems

In our daily lives, we constantly use numbers for various tasks such as counting, measuring, and performing calculations. The system we commonly use is the decimal system. However, computers operate differently. They are electronic devices that work with electrical signals, typically either "on" (1) or "off" (0). Because of this, computers use distinct methods to represent and process numerical information.

A number system is a set of rules and symbols used to represent numerical values. The most crucial characteristic of any number system is its base (radix). The base tells us how many unique digits or symbols are available before we "carry over" to the next position.

In this section, we will explore the four number systems most relevant to computer science: Decimal, Binary, Octal, and Hexadecimal. Understanding these is essential for learning how computers store and manipulate data.

 

1.1.1 The Decimal Number System (Base-10)

This is the number system used in everyday life. It is called Base-10 because it uses 10 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

·         Definition & Uses: Used universally for calculations, money, measurements, and daily operations.

·         Digits: 0–9.

Place Value Concept (Powers of 10):

Each digit’s position determines its value:

·         Rightmost digit: 10⁰ = 1

·         Next digit: 10¹ = 10

·         Next: 10² = 100, and so on.

Example: 456

·         6 × 10⁰ = 6

·         5 × 10¹ = 50

·         4 × 10² = 400

Sum = 400 + 50 + 6 = 456

1.1.2 The Binary Number System (Base-2)

Binary is the fundamental number system used by computers. It contains only two digits: 0 and 1, and each digit is called a bit.

·         Why Binary? Computers use electronic circuits with two states:

o    ON (1) – High voltage

o    OFF (0) – Low voltage

This makes binary simple, reliable, and efficient.

·         Digits: 0 and 1.

Place Value Concept (Powers of 2):

Each position represents a power of 2:

·         Rightmost bit: 2⁰ = 1

·         Next bit: 2¹ = 2

·         Next: 2² = 4, and so on.

Understanding Bits:

A bit is the smallest data unit. Groups of bits form larger units (e.g., 8 bits = 1 byte).

Example: 1101

·         1 × 2⁰ = 1

·         0 × 2¹ = 0

·         1 × 2² = 4

·         1 × 2³ = 8

Sum = 8 + 4 + 0 + 1 = 13

 

1.1.3 The Octal Number System (Base-8)

Octal uses 8 digits: 0–7.

·         Digits: 0, 1, 2, 3, 4, 5, 6, 7.

Place Value Concept (Powers of 8):

·         Rightmost digit: 8⁰ = 1

·         Next digit: 8¹ = 8

·         Next: 8² = 64, and so on.

Historical Significance:

Each octal digit equals 3 binary bits (2³ = 8), making it easier to read and write binary in early computing.

Example: 37 (Octal)

·         7 × 8⁰ = 7

·         3 × 8¹ = 24

Sum = 24 + 7 = 31

 

1.1.4 The Hexadecimal Number System (Base-16)

Hexadecimal uses 16 symbols: 0–9 and A–F, where:

·         A = 10

·         B = 11

·         C = 12

·         D = 13

·         E = 14

·         F = 15

·         Digits: 0–9 and A–F.

Place Value Concept (Powers of 16):

·         Rightmost digit: 16⁰ = 1

·         Next digit: 16¹ = 16

·         Next: 16² = 256, and so on.

Uses in Computing:

·         Memory addresses

·         Web colors (e.g., #FF0000 = red)

·         MAC addresses

·         Error/debugging codes

Binary Connection:

One hexadecimal digit equals 4 binary bits (2⁴ = 16).

Example: 2F (Hex)

·         F = 15 × 16⁰ = 15

·         2 × 16¹ = 32

Sum = 32 + 15 = 47