Binary Number System Presentation

What is Binary?

Definition

Binary = base 2 number system.
Uses only two digits: 0 and 1.
Each position (bit) represents a power of 2.
All information inside a computer (numbers, letters, images, sound) is ultimately represented in binary (0s and 1s).

Example

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

Binary Digits (Bits & Bytes)

Bit

A single binary digit (0 or 1)

Nibble

4 bits (e.g., 1010)

Byte

8 bits (e.g., 11001010)

Word

Group of bytes (depends on CPU, e.g., 32-bit word = 4 bytes)

Example

The letter 'A' in ASCII is 65 decimal, which is 01000001₂ (8 bits).

Binary Representation of Data

Numbers

Stored directly as binary values

Characters

ASCII or Unicode codes stored in binary

Images

Stored as pixels, each pixel's color in binary (e.g., RGB values)

Sound

Stored as samples, each sample in binary

Example

'B' = 66₁₀ = 01000010₂

Conversions with Binary

Decimal → Binary

Method: Division by 2

1

25 ÷ 2 = 12 remainder 1

2

12 ÷ 2 = 6 remainder 0

3

6 ÷ 2 = 3 remainder 0

4

3 ÷ 2 = 1 remainder 1

5

1 ÷ 2 = 0 remainder 1

Reading remainders bottom → top: 11001₂

So, 25₁₀ = 11001₂

Binary → Decimal

Method: Multiplication by powers of 2

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

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

Convert A3₁₆ to binary:

A₁₆ = 10₁₀ = 1010₂

3₁₆ = 3₁₀ = 0011₂

So, A3₁₆ = 1010 0011₂

Binary Fractions

Method: Multiply by 2 repeatedly

Convert 0.625₁₀ to binary:

0.625 × 2 = 1.25 → 1

0.25 × 2 = 0.5 → 0

0.5 × 2 = 1.0 → 1

So, 0.625₁₀ = 0.101₂

Binary Arithmetic

Binary Addition

Add	Result	Carry
0 + 0	0	0
0 + 1	1	0
1 + 0	1	0
1 + 1	0	1

Example: (1011)₂ + (1101)₂

Adding: 11₁₀ + 13₁₀ = 24₁₀

1011
+ 1101
-------
11000

1

Rightmost bits: 1 + 1 = 0, carry 1

2

Next bits: 1 + 0 + carry 1 = 0, carry 1

3

Next bits: 0 + 1 + carry 1 = 0, carry 1

4

Leftmost bits: 1 + 1 + carry 1 = 1, carry 1

5

Write down final carry: 1

Result: (1011)₂ + (1101)₂ = 11000₂ = 24₁₀

Binary Subtraction

Subtract	Result	Borrow?
0 − 0	0	No
1 − 0	1	No
1 − 1	0	No
0 − 1	1	Yes, borrow 1

Example: (1010)₂ − (0111)₂

Subtracting: 10₁₀ − 7₁₀ = 3₁₀

1010
- 0111
-------
0011

1

Rightmost bits: 0 − 1 → borrow → 10 − 1 = 1

2

Next bits: 0 − 1 (after borrow) → borrow again → 10 − 1 = 1

3

Next bits: 0 − 1 (after borrow) → borrow again → 10 − 1 = 1

4

Leftmost bits: 0 − 0 (after borrow) = 0

Result: (1010)₂ − (0111)₂ = 0011₂ = 3₁₀

Binary Multiplication

Multiply	Result
0 × 0	0
0 × 1	0
1 × 0	0
1 × 1	1

Example: (101)₂ × (11)₂

Multiplying: 5₁₀ × 3₁₀ = 15₁₀

101
× 11
-----
101 (101 × 1)
+ 1010 (101 × 1, shifted left)
-----
1111

1

Multiply 101 × 1 (rightmost bit) = 101

2

Multiply 101 × 1 (next bit) = 101, shift left by 1 position = 1010

3

Add the results: 101 + 1010 = 1111

Result: (101)₂ × (11)₂ = 1111₂ = 15₁₀

Binary Division

Example: (1100)₂ ÷ (10)₂

Dividing: 12₁₀ ÷ 2₁₀ = 6₁₀

110
_____
10 ) 1100
10
--
10
10
--
00

1

10 goes into 11 → 1 time, write 1

2

1 × 10 = 10, subtract from 11 → 01

3

Bring down next bit → 10

4

10 goes into 10 → 1 time, write 1

5

1 × 10 = 10, subtract from 10 → 00

6

Bring down next bit → 00

7

10 goes into 00 → 0 times, write 0

Result: (1100)₂ ÷ (10)₂ = 110₂ = 6₁₀

Signed Binary Numbers

What Does "Signed" Mean?

In computing, a signed binary number can be positive or negative.
To represent negative numbers, we use something called Two's Complement.

Unsigned binary = only positive numbers (e.g., 0 to 255 in 8 bits)
Signed binary (Two's Complement) = both positive and negative numbers (e.g., −128 to +127 in 8 bits)

Why Use Two's Complement?

It allows using the same adder hardware for both addition and subtraction
There's only one representation for zero (unlike sign-magnitude)
It simplifies logic and makes overflow detection easier

How to Represent a Negative Number Using Two's Complement

1

Step 1: Write +5 in binary (8 bits)
+5 = 00000101₂

2

Step 2: Invert (flip) all bits
00000101 → 11111010
This is called the 1's Complement of 5

3

Step 3: Add 1

This is the Two's Complement of +5, and it's how the computer stores −5

Final Answer: −5 in 8-bit two's complement = 11111011₂

Advantages of Binary

Simplicity

Very simple (only 0 & 1), making it easy for computers to process

Hardware Implementation

Easy for computers to store and process (voltage: 0 = OFF, 1 = ON)

Reliability

Less chance of error compared to decimal signals

Applications of Binary

Digital Circuits

Transistors are ON (1) or OFF (0)

Memory Storage

Each memory cell stores a bit

Networking

IP addresses (e.g., 192.168.1.1 in binary)

Programming

Machine code is binary instructions

Example

The ASCII code for letter C = 67₁₀ = 01000011₂ — stored in memory as 8 bits.