0.3: Exploring Other Base Systems
Exploring Other Base Systems
1. Introduction: Beyond Base-10
We’ve seen how Alphabitia used blocks and symbols to represent numbers, and how base-10 uses digits 0–9. But what if we used a different number of symbols? That’s the idea behind other base systems!
A base-n system uses n symbols, starting from 0. Each digit’s position determines its value.
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Base-10: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
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Base-5: 0, 1, 2, 3, 4
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Base-2: 0, 1
Alphabitia Note: Like the Alphabitia blocks, position matters in every system!
2. Base-5 (Quinary- this is the formal name for Alphabitia!)
Symbols: 0, 1, 2, 3, 4
Each place represents a power of 5:
| Place | 5² | 5¹ | 5⁰ |
|---|---|---|---|
| Value | 25 | 5 | 1 |
Example: 243₅
| Place | 5² | 5¹ | 5⁰ |
|---|---|---|---|
| Digit | 2 | 4 | 3 |
| Value | 2×25=50 | 4×5=20 | 3×1=3 |
Check: 50 + 20 + 3 = 73 in base-10
Visual blocks (Alphabitia style):
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Flat (🟪) = 25
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Long (|) = 5
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Unit (*) = 1
Representation of 243₅:
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2 flats: 🟪🟪 = 50
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4 longs: |||| = 20
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3 units: *** = 3
Alphabitia Connection: Each block’s position determines its value, just like in Alphabitia.
3. Base-2 (Binary)
Most of the numbers we use every day are written in the decimal system (base ten), which uses the digits 0 through 9. In decimal, each place value represents a power of 10 (ones, tens, hundreds, and so on).
The binary system (base two) works the same way, but it only uses two digits: 0 and 1. Each place value in binary represents a power of 2 instead of a power of 10.
Symbols: 0, 1
Each place represents a power of 2:
| Place | 2³ | 2² | 2¹ | 2⁰ |
|---|---|---|---|---|
| Value | 8 | 4 | 2 | 1 |
It was follows we will put a small “2” below the number if it is representing a binary (or base-2) number.
Example: 1011₂
| Place | 2³ | 2² | 2¹ | 2⁰ |
|---|---|---|---|---|
| Digit | 1 | 0 | 1 | 1 |
| Value | 8 | 0 | 2 | 1 |
Check: 8 + 0 + 2 + 1 = 11 in base-10
Visual blocks (binary style, small values):
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Flat (🟪) = 4
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Long (|) = 2
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Unit (*) = 1
Representation of 1011₂:
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1 Cube: 🟪 = 8
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0 flats
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1 long: | = 2
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1 unit: * = 1
Alphabitia Connection: Fewer symbols, different powers—but same stacking idea!
For more on binary numbers check out this video for kids! What does this video do well? What might need more explanation or visuals?
4. Comparing Bases
| Base | Symbols | Place Value Example | Example Number | Base-10 Value |
|---|---|---|---|---|
| 5 | 0-4 | 5², 5¹, 5⁰ | 243₅ | 73 |
| 10 | 0-9 | 10², 10¹, 10⁰ | 482 | 482 |
| 2 | 0-1 | 2³, 2², 2¹, 2⁰ | 1011₂ | 11 |
Alphabitia Tip: All systems use position + symbols. Only the number of symbols and place powers change.
5. Quick Practice
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Convert 132₅ to base-10.
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Convert 1101₂ to base-10.
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Represent 321₅ using blocks (* = 1, | = 5, 🟪 = 25).
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Represent 1110₂ using blocks (* = 1, | = 2, 🟪 = 4).
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Draw a place-value chart for 243₅ and 1011₂.
6. Reflection
Base systems are flexible! By understanding how symbols and position work, teachers can explore any number system with students. Base-5 and binary are smaller than base-10, but operate on the same fundamental principle: place value.
Alphabitia Connection: Every number system is a playground! Use blocks, symbols, and stacking to see numbers come alive.
Attributions
Video attribution: Socratica Kids, Binary Numbers for Kids, Convert Decimal to Binary, Computers for Kids. Uploaded: February 17, 2019. Access Address: https://www.youtube.com/watch?v=hvteVokz7jE
