Computer Number System Presentation

What is a Number System?

Definition

A number system is a way to represent numbers using a set of symbols (digits).

In computers, everything (numbers, text, images, sound) is ultimately stored as binary numbers (0 and 1). But we often use other number systems (decimal, octal, hexadecimal) for convenience.

Types of Number Systems in Computers

Decimal

Base 10

Digits: 0–9

Each digit's position = power of 10.
This is the number system humans use every day.

Example

452 = (4 × 10²) + (5 × 10¹) + (2 × 10⁰)
= 400 + 50 + 2 = 452

Binary

Base 2

Digits: 0, 1 only

Each position = power of 2.
This is the language of computers (everything is stored in 0s and 1s).

Example

(1011)₂ = (1 × 2³) + (0 × 2²) + (1 × 2¹) + (1 × 2⁰)
= 8 + 0 + 2 + 1 = 11 in decimal

Octal

Base 8

Digits: 0–7

Each position = power of 8.
Used in older systems for compact binary representation.

Example

(157)₈ = (1 × 8²) + (5 × 8¹) + (7 × 8⁰)
= 64 + 40 + 7 = 111 in decimal

Hexadecimal

Base 16

Digits: 0–9, A–F (where A=10, B=11, … F=15)

Each position = power of 16.
Used in modern computing (short way to write binary numbers).

Example

(2F)₁₆ = (2 × 16¹) + (15 × 16⁰)
= 32 + 15 = 47 in decimal

Conversion Between Number Systems

Decimal → Binary

Method: Division by 2

1

Divide the decimal number by 2

2

Record the remainder (0 or 1)

3

Continue with the quotient until it becomes 0

4

Read remainders from bottom to top

Example: Convert 25₁₀ to binary

25 ÷ 2 = 12 remainder 1
12 ÷ 2 = 6 remainder 0
6 ÷ 2 = 3 remainder 0
3 ÷ 2 = 1 remainder 1
1 ÷ 2 = 0 remainder 1
Read bottom → top: 11001₂
So, 25₁₀ = 11001₂

Binary → Decimal

Method: Multiplication by powers of 2

1

Write down the binary number

2

Assign powers of 2 to each position (from right to left, starting with 2⁰)

3

Multiply each digit by its power of 2

4

Add all the results

Example: Convert 10110₂ to decimal

(1 × 2⁴) + (0 × 2³) + (1 × 2²) + (1 × 2¹) + (0 × 2⁰)
= (1 × 16) + (0 × 8) + (1 × 4) + (1 × 2) + (0 × 1)
= 16 + 0 + 4 + 2 + 0 = 22₁₀

Binary → Hexadecimal

Method: Grouping into 4 bits

1

Group the binary digits into sets of 4 (from right to left)

2

Add leading zeros if needed to complete a group of 4

3

Convert each 4-bit group to its hexadecimal equivalent

Example: Convert 11010110₂ to hex

Group into 4 bits: 1101 0110
1101₂ = 13₁₀ = D₁₆
0110₂ = 6₁₀ = 6₁₆
So, 11010110₂ = D6₁₆

Hexadecimal → Binary

Method: Convert each hex digit to 4-bit binary

1

Write down the hexadecimal number

2

Convert each hexadecimal digit to its 4-bit binary equivalent

3

Concatenate all the 4-bit groups

Example: Convert A3₁₆ to binary

A₁₆ = 10₁₀ = 1010₂
3₁₆ = 3₁₀ = 0011₂
So, A3₁₆ = 1010 0011₂

Fractions in Number Systems

Method: Multiplication by 2 (for decimal to binary)

0.625 × 2 = 1.25

Take 1 (integer part)

0.25 × 2 = 0.5

Take 0 (integer part)

0.5 × 2 = 1.0

Take 1 (integer part)

So, 0.625₁₀ = 0.101₂

Why Different Number Systems?

Binary

The actual machine language of computers. All data is ultimately stored and processed as binary (0s and 1s).

Octal/Hexadecimal

Short representation of binary numbers, used by programmers for convenience. One hex digit represents 4 binary digits.

Decimal

For human understanding and everyday use. Most familiar to people as it's used in daily life.

Example

Instead of writing 110101111001₂ (12 bits), we can write just DF9₁₆ (3 hex digits).