🔢 Number Representations: The Secret Language of Numbers
Imagine numbers wearing different costumes at a party. The same number can dress up in many ways—and knowing these costumes helps you understand the number’s true personality!
🎭 The Big Idea
Numbers aren’t stuck in one outfit. Just like you can describe your height in feet or centimeters, numbers can be written in different bases, chopped into floor and ceiling pieces, or examined for their units digit patterns. Mastering these representations is like learning to read numbers in any language!
🏠 Number Base Systems: Counting in Different Villages
What’s a Base?
Think of a base as a counting village. In each village, people only know a certain number of digits before they have to start a new column.
| Village (Base) | Digits They Know | Example |
|---|---|---|
| Base 10 (Decimal) | 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 | 42 |
| Base 2 (Binary) | 0, 1 | 101010 |
| Base 16 (Hexadecimal) | 0–9, A–F | 2A |
🎯 Simple Example: The Number 13
Let’s dress up the number 13 in different costumes:
Base 10 (Decimal): 13
→ 1 ten + 3 ones = 13
Base 2 (Binary): 1101
→ 1×8 + 1×4 + 0×2 + 1×1 = 13
Base 16 (Hex): D
→ D means 13 in hex!
🏗️ How to Convert: The Division Method
Converting 25 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 remainders bottom-up: 11001
So 25 in decimal = 11001 in binary ✓
💡 Why Does This Matter?
- Computers think in binary (Base 2)
- Colors on screens use hexadecimal (Base 16)
- Ancient civilizations used Base 60 (that’s why we have 60 seconds in a minute!)
⬇️⬆️ Floor and Ceiling Functions: The Elevator Rule
Meet Floor ⌊x⌋
The floor of a number is the biggest whole number that’s still smaller or equal to your number.
Think of it like an elevator: Floor takes you DOWN to the nearest whole floor.
⌊3.7⌋ = 3 (go down from 3.7 to 3)
⌊5.0⌋ = 5 (already on a floor!)
⌊−2.3⌋ = −3 (go down means more negative!)
Meet Ceiling ⌈x⌉
The ceiling of a number is the smallest whole number that’s still bigger or equal to your number.
Think of it like an elevator: Ceiling takes you UP to the nearest whole floor.
⌈3.7⌉ = 4 (go up from 3.7 to 4)
⌈5.0⌉ = 5 (already on a floor!)
⌈−2.3⌉ = −2 (go up means less negative!)
🎯 Quick Reference Table
| Number | Floor ⌊x⌋ | Ceiling ⌈x⌉ |
|---|---|---|
| 4.2 | 4 | 5 |
| −1.5 | −2 | −1 |
| 7.0 | 7 | 7 |
| 0.001 | 0 | 1 |
🔮 The Magic Rule
For any number x:
- If x is a whole number: ⌊x⌋ = ⌈x⌉ = x
- If x is NOT a whole number: ⌈x⌉ = ⌊x⌋ + 1
graph TD A[Number x] --> B{Is x a whole number?} B -->|Yes| C[Floor = Ceiling = x] B -->|No| D[Floor = largest integer ≤ x] B -->|No| E[Ceiling = smallest integer ≥ x]
🍰 Fractional Part Function: What’s Left After You Take the Whole?
The Big Idea
The fractional part of a number (written as {x}) is what remains after you remove the whole part.
Think of it like a pizza: If you have 3.7 pizzas, the fractional part is the 0.7 pizza slice left over!
The Formula
{x} = x − ⌊x⌋
Translation: Fractional part = Original number minus its floor
🎯 Examples
{5.83} = 5.83 − 5 = 0.83
{7.0} = 7.0 − 7 = 0
{−2.3} = −2.3 − (−3) = 0.7
Wait, that last one is surprising! Let’s understand:
- For −2.3, the floor is −3 (we go DOWN to −3)
- So {−2.3} = −2.3 − (−3) = −2.3 + 3 = 0.7 ✓
🔑 Key Facts
| Property | Always True |
|---|---|
| Range | 0 ≤ {x} < 1 |
| For positive decimals | {x} = decimal part |
| For whole numbers | {x} = 0 |
| Reconstruction | x = ⌊x⌋ + {x} |
🎨 Visual: Breaking Apart a Number
7.625
╱ ╲
⌊7.625⌋ {7.625}
↓ ↓
7 + 0.625 = 7.625
(floor) (fractional)
🔄 Units Digit Patterns: The Repeating Dance
What’s a Units Digit?
The units digit is the rightmost digit of a number—the “ones place.”
In 3847, the units digit is 7
In 250, the units digit is 0
The Magic: Patterns Repeat!
When you raise numbers to powers, their units digits follow predictable, repeating patterns.
🎯 Example: Powers of 2
2¹ = 2 → units digit: 2
2² = 4 → units digit: 4
2³ = 8 → units digit: 8
2⁴ = 16 → units digit: 6
2⁵ = 32 → units digit: 2 ← Pattern restarts!
2⁶ = 64 → units digit: 4
Pattern: 2 → 4 → 8 → 6 → 2 → 4 → 8 → 6 → …
The cycle length is 4!
🔮 Finding Any Units Digit
To find the units digit of 2^47:
- Find the cycle length: 4
- Divide the exponent by cycle length: 47 ÷ 4 = 11 remainder 3
- The remainder tells you the position: Position 3 in (2,4,8,6) = 8
So the units digit of 2^47 is 8 ✓
📊 Units Digit Cycles for Common Bases
| Base | Cycle | Length |
|---|---|---|
| 2 | 2, 4, 8, 6 | 4 |
| 3 | 3, 9, 7, 1 | 4 |
| 4 | 4, 6 | 2 |
| 5 | 5 | 1 |
| 6 | 6 | 1 |
| 7 | 7, 9, 3, 1 | 4 |
| 8 | 8, 4, 2, 6 | 4 |
| 9 | 9, 1 | 2 |
🎪 The Special Ones
- 0, 1, 5, 6: Always end in themselves! (cycle length = 1)
- 4, 9: Alternate between two digits (cycle length = 2)
- 2, 3, 7, 8: Cycle through four digits (cycle length = 4)
graph TD A[Find units digit of n^k] --> B[Find base's cycle] B --> C[Divide k by cycle length] C --> D[Use remainder to find position] D --> E[That position in cycle = answer]
🎁 Putting It All Together
You’ve just learned four powerful tools:
| Tool | What It Does | Example |
|---|---|---|
| Base Systems | Write numbers in different counting systems | 13 = 1101₂ |
| Floor ⌊x⌋ | Round down to nearest integer | ⌊3.9⌋ = 3 |
| Ceiling ⌈x⌉ | Round up to nearest integer | ⌈3.1⌉ = 4 |
| Fractional Part {x} | Get the leftover after removing the whole | {5.7} = 0.7 |
| Units Digit Patterns | Predict last digits using cycles | 7^100 ends in 1 |
🚀 Real-World Connections
- Binary → How computers store everything
- Floor/Ceiling → How prices get rounded
- Fractional parts → Splitting things fairly
- Units digits → Quick mental math tricks
🌟 Your Confidence Boost
You now understand that numbers are flexible—they can be dressed up in different bases, split into floors and fractions, and their patterns can be predicted. This isn’t just theory; it’s the foundation of computer science, cryptography, and advanced mathematics.
You’ve unlocked the secret language of numbers. Now go speak it fluently! 🎉