Number System

Introduction

• Digital systems use positional number systems to represent and process information.
• In a positional system, the value of a digit depends on the digit itself, the position (index) it occupies, and the base (radix) of the system.
• Representing information numerically enables storage, computation and transmission in electronic devices.

The contribution of a digit to the total value of a number in a positional system is determined by three factors:

• Digit value - the symbol used at that position (for example 0-9 or A-F).
• Position (index) - the place of the digit relative to the radix point (or decimal point); positions left of the point are non‐negative indices and those right are negative indices.
• Base (radix) - the number of distinct digit symbols available in the system; denoted by b.

Note: If the digit symbols are 0 through 9, the base is 10.

Types of Number Systems

Several number systems are commonly used in digital electronics and computation. The most important positional systems are described below.

Binary Number System

The binary system uses base 2 and two digit symbols: 0 and 1. Each binary digit is called a bit. Groups of bits have common names: a group of four bits is a nibble, and a group of eight bits is a byte.

In binary notation the position of a digit represents a power of 2.

• Only two symbols: 0 and 1.
• Also called the base 2 number system.
• Value of a digit at position i (counting from 0 at the least significant bit) is the digit multiplied by 2^i.

Examples: (10100)₂, (11011)₂, (11001)₂, (000101)₂, (011010)₂.

Decimal Number System

The decimal system uses base 10 and digit symbols 0 through 9. It is the standard system for everyday counting and arithmetic.

• Digits available: 0, 1, 2, ..., 9.
• Also called the base 10 number system.
• Positions to the left of the radix point represent units, tens, hundreds, thousands, etc.; each position corresponds to successive powers of 10.

(2541) can be expanded as

(2 × 1000) + (5 × 100) + (4 × 10) + (1 × 1)

(2 × 10³) + (5 × 10²) + (4 × 10¹) + (1 × 10⁰)

2000 + 500 + 40 + 1 = 2541

Octal Number System

The octal system uses base 8 with digit symbols 0 through 7. Each octal digit corresponds to exactly three binary bits, which makes conversion between binary and octal convenient.

• Digits available: 0, 1, 2, 3, 4, 5, 6, 7.
• Also called the base 8 number system.
• Value of a digit at position i is the digit multiplied by 8^i.

Examples: (273)₈, (5644)₈, (0.5365)₈, (1123)₈, (1223)₈.

Hexadecimal Number System

The hexadecimal system uses base 16 and sixteen symbols: 0-9 and A-F, where A-F represent decimal values 10-15. Each hexadecimal digit corresponds to exactly four binary bits, which makes conversion between binary and hexadecimal straightforward.

• Digits available: 0, 1, 2, ..., 9, A, B, C, D, E, F.
• Letters A to F denote decimal values 10 to 15, respectively.
• Also called the base 16 number system.
• Value of a digit at position i is the digit multiplied by 16^i.

Examples: (FAC2)₁₆, (564)₁₆, (ABD5)₁₆, (1123)₁₆, (11F3)₁₆.

Positional Notation - General Representation

A number in base b with digits a_na_n-1...a₁a₀.a_-1a_-2... is interpreted as the sum of each digit times the corresponding power of b.

• General formula:

a_nbⁿ + a_n-1b^n-1 + ... + a₁b¹ + a₀b⁰ + a_-1b^-1 + a_-2b^-2 + ...

Each coefficient a_i satisfies 0 ≤ a_i ≤ b - 1.

Conversion Methods

Converting from base b to decimal

Compute the positional sum: multiply each digit by the appropriate power of b and add the results.

Example: Convert (10100)₂ to decimal.

Sol.

10100 in base 2 equals

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

16 + 0 + 4 + 0 + 0

= 20

Converting from decimal to base b (division‐remainder method)

Divide the decimal number repeatedly by the target base and collect remainders; the remainders read from last to first give the digits in the new base.

Example: Convert (2541)₁₀ to base 8.

Sol.

2541 ÷ 8 = 317 remainder 5

317 ÷ 8 = 39 remainder 5

39 ÷ 8 = 4 remainder 7

4 ÷ 8 = 0 remainder 4

Read remainders from last to first: (4755)₈

Binary ↔ Octal and Binary ↔ Hexadecimal (grouping methods)

Conversion between binary and octal is done by grouping binary digits in sets of three, starting from the radix point; each group maps to one octal digit.

Conversion between binary and hexadecimal is done by grouping binary digits in sets of four; each group maps to one hexadecimal digit.

Example: Convert (11001010)₂ to hexadecimal.

Sol.

Group binary digits in sets of four from the right: 1100 1010

1100 in binary = 12 in decimal = C in hexadecimal

1010 in binary = 10 in decimal = A in hexadecimal

So the hexadecimal representation is (CA)₁₆

Converting fractional parts

To convert a fractional decimal part to base b, multiply the fractional part repeatedly by b and record the integer part of each product; these integers form the fractional digits in order.

Example: Convert 0.625₁₀ to binary.

Sol.

0.625 × 2 = 1.25 → integer part 1

0.25 × 2 = 0.5 → integer part 0

0.5 × 2 = 1.0 → integer part 1

Fractional digits: .101

Hence 0.625₁₀ = (0.101)₂

Basic Arithmetic and Representations

• Binary arithmetic follows the same principles as decimal arithmetic but with base 2: 0+0=0, 0+1=1, 1+1=10 (carry 1), 1+1+1=11 (carry 1 and sum 1).
• Signed numbers: Common representations include sign‐magnitude, one's complement, and two's complement. Two's complement is widely used for signed integer arithmetic because it simplifies addition and subtraction.
• Overflow: Occurs when a computation requires more bits than available; rules differ for unsigned and signed interpretations.

Examples Summary

• Binary example: (10100)₂ = 20₁₀.
• Decimal-to-octal example: (2541)₁₀ = (4755)₈.
• Binary-to-hex example: (11001010)₂ = (CA)₁₆.
• Fractional conversion example: 0.625₁₀ = (0.101)₂.

Final summary

Positional number systems assign each digit a weight equal to a power of the base. Understanding binary, octal, decimal and hexadecimal systems, and the standard methods to convert among them, is essential for representing data and performing arithmetic in digital systems. Grouping methods facilitate binary↔octal/hex conversions, while division‐remainder and multiplication methods convert integral and fractional parts respectively between decimal and other bases.