🔢 Number Systems: How Computers Count!

Imagine you only had two fingers instead of ten. How would you count? That’s exactly how computers work!

🌟 The Big Picture

Think of numbers like different languages. We humans speak “Decimal” (0-9), but computers only understand “Binary” (0-1). Just like translating between English and Spanish, we can translate numbers between these systems!

graph TD
    A[Human Numbers<br/>0, 1, 2... 9] -->|Translate| B[Computer Numbers<br/>0, 1]
    B -->|Translate| A

📖 Chapter 1: Binary - The Computer’s Language

🎯 What is Binary?

Binary is counting with just two symbols: 0 and 1.

Think of it like a light switch - it’s either OFF (0) or ON (1). That’s it! Computers are made of billions of tiny switches.

🔦 The Light Switch Story

Imagine you have a row of light switches. Each switch can be:

• OFF = 0
• ON = 1

💡 Simple Example

In decimal (our system): 5 means “five things”

In binary: 101 also means “five things”!

How? Each position has a value that doubles:

Position:  4   2   1
Binary:    1   0   1
           ↓   ↓   ↓
           4 + 0 + 1 = 5

🔄 Chapter 2: Number System Conversions

🎪 The Magic Translation Game

Converting numbers is like being a translator between number languages!

Decimal → Binary (How to translate TO computer)

The Division Game:

1. Divide by 2
2. Write down the remainder (0 or 1)
3. Repeat until you get 0
4. Read remainders backwards!

Example: Convert 13 to binary

13 ÷ 2 = 6 remainder 1
 6 ÷ 2 = 3 remainder 0
 3 ÷ 2 = 1 remainder 1
 1 ÷ 2 = 0 remainder 1

Read backwards: 1101

✅ 13 in decimal = 1101 in binary

Binary → Decimal (How to translate FROM computer)

The Doubling Game:

• Start from the right
• Each position doubles: 1, 2, 4, 8, 16, 32…

Example: Convert 1101 to decimal

Binary:  1    1    0    1
Value:   8    4    2    1
         ↓    ↓    ↓    ↓
         8 +  4 +  0 +  1 = 13

🎨 Chapter 3: Hexadecimal - The Shortcut System

🌈 Why Hex?

Binary numbers get LONG! Writing 11111111 is tiring.

Hexadecimal (hex) uses 16 symbols: 0-9 and A-F

🎯 The Color Connection

Ever seen color codes like #FF0000? That’s hex!

• FF = 255 (maximum red)
• 00 = 0 (no green)
• 00 = 0 (no blue)

Result: Bright red! 🔴

📝 Conversion Example

Hex F3 to decimal:

F = 15, 3 = 3
F3 = (15 × 16) + 3 = 240 + 3 = 243

📦 Chapter 4: Bits, Bytes, and Word Size

🧱 The Building Blocks

Think of computer memory like LEGO bricks:

🏠 Real Examples

• 1 bit = Yes or No
• 1 byte = One letter (like ‘A’)
• 1 kilobyte (KB) = A short email
• 1 megabyte (MB) = A photo
• 1 gigabyte (GB) = A movie

🔢 Word Size

A computer’s word size is how many bits it processes at once.

• 32-bit computer: Handles 32 bits together
• 64-bit computer: Handles 64 bits together (faster!)

✉️ Chapter 5: Character Encoding

📚 How Computers Store Letters

Computers only know numbers! So every letter has a secret number code.

🔤 ASCII - The Original Code

American Standard Code for Information Interchange

🌍 Unicode - The World Code

ASCII only had 128 characters. Not enough for 中文, العربية, or emoji! 🎉

Unicode gives EVERY character a number:

• 😀 = U+1F600
• 中 = U+4E2D
• ❤ = U+2764

➕ Chapter 6: Integer Representation

📊 Storing Whole Numbers

Integers are whole numbers like -5, 0, 42.

Unsigned integers: Only positive (0 and up)

• 8 bits = 0 to 255

Signed integers: Positive AND negative

• 8 bits = -128 to 127

🎯 The Range Rule

With n bits, you can store:

• Unsigned: 0 to (2ⁿ - 1)
• Signed: -2ⁿ⁻¹ to (2ⁿ⁻¹ - 1)

Example with 8 bits:

• Unsigned: 0 to 255
• Signed: -128 to 127

🔄 Chapter 7: Two’s Complement

🎭 The Negative Number Trick

How do computers store negative numbers? With a clever trick called Two’s Complement!

🪄 The Magic Recipe

To make a negative number:

1. Write the positive number in binary
2. Flip all bits (0→1, 1→0)
3. Add 1

📝 Example: Making -5

Step 1: +5 in binary = 00000101
Step 2: Flip bits    = 11111010
Step 3: Add 1        = 11111011

-5 = 11111011

✅ Why This Works

The first bit becomes the sign bit:

• 0 at the start = positive
• 1 at the start = negative

🎈 Chapter 8: Floating-Point Numbers

🌊 Numbers with Decimals

How do computers store 3.14 or 0.001?

Floating-point = the decimal “floats” to different positions.

🔬 Scientific Notation for Computers

Remember scientific notation? Like 3.14 × 10²

Computers use the same idea with binary!

A floating-point number has 3 parts:

┌──────┬──────────┬──────────────┐
│ Sign │ Exponent │ Mantissa     │
│ +/-  │ Power    │ Actual digits│
└──────┴──────────┴──────────────┘

📏 IEEE 754 Standard

⚠️ The Precision Problem

Computers can’t store ALL decimal numbers exactly!

0.1 + 0.2 = 0.30000000000000004

This is normal! It’s like trying to write 1/3 as a decimal - it goes on forever.

🔀 Chapter 9: Endianness

📚 Reading Order for Bytes

When a number needs multiple bytes, which byte comes first?

🥪 The Sandwich Analogy

Imagine writing the number 0x12345678:

Big-Endian (Big end first):

Memory: [12] [34] [56] [78]
         ↑ Most significant first

Like reading left-to-right!

Little-Endian (Little end first):

Memory: [78] [56] [34] [12]
         ↑ Least significant first

Like reading right-to-left!

🖥️ Which Computers Use What?

🎯 Example

Store 258 (0x0102) in 2 bytes:

Big-Endian:    [01] [02]
Little-Endian: [02] [01]

🎓 Quick Summary

graph TD
    A[Number Systems] --> B[Binary<br/>0s and 1s]
    A --> C[Hex<br/>0-9, A-F]
    A --> D[Storage]
    D --> E[Bits & Bytes]
    D --> F[Integers]
    D --> G[Floats]
    D --> H[Characters]
    F --> I[Two's Complement]
    G --> J[IEEE 754]
    D --> K[Endianness]

🌟 You Did It!

Now you know how computers:

• ✅ Count with just 0s and 1s
• ✅ Store positive AND negative numbers
• ✅ Handle decimal numbers
• ✅ Remember every letter and emoji
• ✅ Organize bytes in memory

You’re speaking computer now! 🎉