Other Bases
Part of Mathematics
How counting systems with bases other than 10 work, and where they appear in practical rebuilding contexts.
Why This Matters
Most people learn arithmetic in base 10 and never think about alternatives. But base 10 is not the only number system — and in practical rebuilding, you will constantly encounter situations where other bases appear naturally. Time uses base 60 and base 12. Angles use base 360. Inches and feet use base 12. Computing (when you rebuild it) uses base 2 and base 16. Understanding other bases is not an abstract intellectual exercise; it is the key to understanding why clocks look the way they do, how to convert between units, and why some calculation shortcuts work.
More fundamentally, understanding that the base is a choice — not a natural law — deepens mathematical thinking in ways that matter for a community’s intellectual development. A student who understands base 2 understands how binary works, which unlocks an intuitive grasp of digital computing when your community reaches that stage. A student who understands base 60 understands why there are 60 minutes in an hour without needing to memorize that fact as arbitrary trivia.
This article is conceptual and practical: it explains what a base is, how to count and convert in other bases, and where each base appears in the real physical and technical world.
What a Base Is
In any positional number system, the base (also called the radix) determines how many distinct digit symbols exist and what value each position represents.
-
Base 10 (decimal): Digits 0–9. Each position is worth 10× the position to its right.
- 347 = 3×100 + 4×10 + 7×1 = 3×10² + 4×10¹ + 7×10⁰
-
Base 2 (binary): Digits 0–1. Each position is worth 2× the position to its right.
- 101 in binary = 1×4 + 0×2 + 1×1 = 5 in decimal
-
Base 12 (duodecimal): Digits 0–9 plus two more symbols (often A=10, B=11). Each position is worth 12× the position to its right.
-
Base 60 (sexagesimal): Used in time and angles; positions worth 60× the previous.
The pattern is always the same: to find the value of a number in any base, multiply each digit by (base)^(position), where position counts from 0 at the right.
Base 2 (Binary)
Binary uses only two digits: 0 and 1. It appears in:
- Relay and switching logic (the basis of all digital computing)
- Signaling systems (light on/off, flag up/down)
- The Russian Peasant multiplication method (see Multiplication)
- Any on/off, yes/no recording system
Counting in Binary
| Decimal | Binary |
|---|---|
| 0 | 0 |
| 1 | 1 |
| 2 | 10 |
| 3 | 11 |
| 4 | 100 |
| 5 | 101 |
| 6 | 110 |
| 7 | 111 |
| 8 | 1000 |
| 15 | 1111 |
| 16 | 10000 |
Each time you reach a power of 2, a new digit position opens up. This is why binary numbers look “long” — they need more digits than decimal to represent the same value.
Converting Decimal to Binary
Method: Repeatedly divide by 2 and record remainders, then read remainders from bottom to top.
Example: 23 in binary
- 23 ÷ 2 = 11 remainder 1
- 11 ÷ 2 = 5 remainder 1
- 5 ÷ 2 = 2 remainder 1
- 2 ÷ 2 = 1 remainder 0
- 1 ÷ 2 = 0 remainder 1
- Read remainders bottom to top: 10111
- Check: 16 + 4 + 2 + 1 = 23 ✓
Practical Binary: Signaling
Two torches on a hilltop can signal any number from 0 to 3 (00, 01, 10, 11). Four torches can signal 0–15. Eight torches can signal 0–255 — enough for any letter of the alphabet, any digit, or 256 distinct messages. This is the principle behind semaphore and early optical telegraph systems. A community with four signal stations and four-position flags can communicate 16 distinct messages per signal — sufficient for “help,” “all clear,” “enemy approaching,” “send grain,” and many more.
Base 12 (Duodecimal)
Base 12 appears constantly in traditional measurement:
- 12 inches in a foot
- 12 months in a year
- 12 hours on a clock face
- 12 in a dozen (most trade goods)
- 144 (12²) = one gross
Why Base 12 Is Useful
12 divides evenly by 1, 2, 3, 4, 6, and 12. Compare with 10, which divides evenly by only 1, 2, 5, and 10. This means thirds and quarters are exact in base 12: 1/3 of a dozen is 4; 1/4 of a dozen is 3. In base 10, 1/3 = 0.333… (never exact).
For a pre-industrial trade community, this makes base-12 units practical: you can always split a dozen into equal shares for 2, 3, or 4 people without fractions.
Working with Dozens
You do not need to fully convert to base 12 arithmetic. Instead, develop fluency with the multiplication facts of 12:
| n | 12×n |
|---|---|
| 1 | 12 |
| 2 | 24 |
| 3 | 36 |
| 4 | 48 |
| 5 | 60 |
| 6 | 72 |
| 7 | 84 |
| 8 | 96 |
| 9 | 108 |
| 10 | 120 |
| 12 | 144 (one gross) |
Practical use: 7 dozen eggs = 84 eggs. 3 gross of nails = 432 nails. Converting: if you have 135 nails and pack them in dozens, 135 ÷ 12 = 11 dozen with 3 left over.
Base 60 (Sexagesimal)
Base 60 was invented by the Babylonians and persists today in:
- Time: 60 seconds per minute, 60 minutes per hour
- Angles: 360 degrees (6×60), 60 minutes per degree, 60 seconds per minute of arc
- Navigation: longitude and latitude expressed in degrees-minutes-seconds
Why 60?
60 has more divisors than any smaller number: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60. This means 60 units can be divided into equal shares in more ways than any alternative, making it extremely practical for a society without decimal fractions.
Working with Time in Base 60
Time arithmetic does not use decimal; it carries at 60.
Example: Adding time intervals
- 2 hours 47 minutes + 1 hour 35 minutes
- Minutes: 47 + 35 = 82 minutes = 1 hour 22 minutes (82 ÷ 60 = 1 remainder 22)
- Hours: 2 + 1 + 1 (carry) = 4 hours
- Total: 4 hours 22 minutes
Example: Converting hours to minutes
- 3 hours 25 minutes = (3 × 60) + 25 = 180 + 25 = 205 minutes
Example: Converting degrees-minutes to decimal degrees
- 47° 30’ = 47 + 30/60 = 47.5°
- 32° 15’ = 32 + 15/60 = 32.25°
Angle Arithmetic
When calculating bearings or azimuths, you add and subtract angles in base 60:
- Bearing 345° + 47° = 392° − 360° = 32° (subtract full circle if over 360)
- Bearing 015° − 020° = −5° + 360° = 355°
Base 16 (Hexadecimal)
Hexadecimal uses digits 0–9 and letters A–F (where A=10, B=11, C=12, D=13, E=14, F=15). It appears in:
- Color codes in any printing or graphics work
- Memory addressing in computers
- Concise representation of binary data (one hex digit = 4 binary digits)
For immediate survival purposes, hexadecimal is low priority. When your community rebuilds computing infrastructure, fluency in hex becomes important. For now, know that it exists and that each hex digit maps to exactly four binary digits:
| Hex | Binary |
|---|---|
| 0 | 0000 |
| 5 | 0101 |
| A | 1010 |
| F | 1111 |
Converting Between Bases: General Method
Any base to decimal: Multiply each digit by (base)^(position) and sum.
- Binary 11011 = 1×16 + 1×8 + 0×4 + 1×2 + 1×1 = 27
Decimal to any base: Repeatedly divide by the target base, collect remainders.
- 27 to binary: 27→13 r1, 13→6 r1, 6→3 r0, 3→1 r1, 1→0 r1 → read up: 11011 ✓
Teaching Other Bases
A simple classroom exercise that builds intuition:
- Give students a bag of objects (stones, sticks)
- Ask them to count by grouping in 2s, then 3s, then 5s, then 10s
- Ask: what changes? (The symbols needed, the size of groups) What stays the same? (The actual quantity of objects)
- This separates the concept of number (a quantity) from its representation (a string of digits in some base)
This exercise is valuable because it builds the foundational understanding that mathematics describes reality, but our notations are human inventions — and can be chosen, changed, and improved.