Sovereignty Module: Reckon the Numbers
Complete Practical Mathematics, Calculation, and Engineering Computation Guide
The Philosophy of Mathematics
Mathematics is the language of creation. Every bridge, every engine, every navigation fix, every crop rotation, every trade transaction requires calculation. Without mathematics, engineering is guesswork, trade is exploitation, and science is impossible. This campaign covers the practical mathematics needed to rebuild and operate a technological civilization — from basic arithmetic through geometry, trigonometry, and the calculations needed for construction, navigation, and manufacturing.
Chapter 1: Number Systems and Arithmetic
Place Value (decimal system):
Each position represents a power of 10:
- Ones (10⁰ = 1)
- Tens (10¹ = 10)
- Hundreds (10² = 100)
- Thousands (10³ = 1,000)
Arithmetic Operations:
| Operation | Symbol | Example | Inverse |
|---|---|---|---|
| Addition | + | 47 + 38 = 85 | Subtraction |
| Subtraction | - | 85 - 38 = 47 | Addition |
| Multiplication | × | 12 × 7 = 84 | Division |
| Division | ÷ | 84 ÷ 7 = 12 | Multiplication |
| Exponentiation | ^ | 2^10 = 1,024 | Root/logarithm |
| Square root | √ | √144 = 12 | Squaring |
Multiplication Table (memorize):
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
| 2 | 2 | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 | 20 | 22 | 24 |
| 3 | 3 | 6 | 9 | 12 | 15 | 18 | 21 | 24 | 27 | 30 | 33 | 36 |
| 4 | 4 | 8 | 12 | 16 | 20 | 24 | 28 | 32 | 36 | 40 | 44 | 48 |
| 5 | 5 | 10 | 15 | 20 | 25 | 30 | 35 | 40 | 45 | 50 | 55 | 60 |
| 6 | 6 | 12 | 18 | 24 | 30 | 36 | 42 | 48 | 54 | 60 | 66 | 72 |
| 7 | 7 | 14 | 21 | 28 | 35 | 42 | 49 | 56 | 63 | 70 | 77 | 84 |
| 8 | 8 | 16 | 24 | 32 | 40 | 48 | 56 | 64 | 72 | 80 | 88 | 96 |
| 9 | 9 | 18 | 27 | 36 | 45 | 54 | 63 | 72 | 81 | 90 | 99 | 108 |
| 10 | 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 | 90 | 100 | 110 | 120 |
| 11 | 11 | 22 | 33 | 44 | 55 | 66 | 77 | 88 | 99 | 110 | 121 | 132 |
| 12 | 12 | 24 | 36 | 48 | 60 | 72 | 84 | 96 | 108 | 120 | 132 | 144 |
Chapter 2: Fractions, Decimals, and Percentages
Fraction Operations:
| Operation | Rule | Example |
|---|---|---|
| Addition (same denominator) | Add numerators | 3/8 + 2/8 = 5/8 |
| Addition (different denominator) | Find common denominator | 1/3 + 1/4 = 4/12 + 3/12 = 7/12 |
| Multiplication | Multiply across | 2/3 × 3/4 = 6/12 = 1/2 |
| Division | Flip second and multiply | 2/3 ÷ 1/4 = 2/3 × 4/1 = 8/3 |
Conversions:
| Fraction | Decimal | Percentage |
|---|---|---|
| 1/2 | 0.500 | 50% |
| 1/3 | 0.333 | 33.3% |
| 1/4 | 0.250 | 25% |
| 1/5 | 0.200 | 20% |
| 1/6 | 0.167 | 16.7% |
| 1/8 | 0.125 | 12.5% |
| 1/10 | 0.100 | 10% |
| 1/12 | 0.083 | 8.3% |
| 1/16 | 0.0625 | 6.25% |
| 3/4 | 0.750 | 75% |
| 2/3 | 0.667 | 66.7% |
| 7/8 | 0.875 | 87.5% |
Chapter 3: Measurement and Units
Length:
| Unit | Equivalent | Use |
|---|---|---|
| 1 inch | 25.4 mm | Small measurements |
| 1 foot | 12 inches = 304.8 mm | Construction |
| 1 yard | 3 feet = 0.914 m | Fabric, general |
| 1 mile | 5,280 feet = 1.609 km | Distance |
| 1 meter | 39.37 inches = 3.281 feet | Metric standard |
| 1 kilometer | 0.621 miles | Metric distance |
Area:
| Unit | Equivalent | Use |
|---|---|---|
| 1 square foot | 144 square inches | Flooring, roofing |
| 1 square yard | 9 square feet | Fabric, carpet |
| 1 acre | 43,560 square feet | Land |
| 1 hectare | 2.471 acres = 10,000 sq meters | Metric land |
Volume:
| Unit | Equivalent | Use |
|---|---|---|
| 1 gallon (US) | 231 cubic inches = 3.785 liters | Liquid |
| 1 cubic foot | 7.48 gallons = 28.3 liters | Tanks, concrete |
| 1 cubic yard | 27 cubic feet | Concrete, earth |
| 1 liter | 0.264 gallons = 61.02 cubic inches | Metric liquid |
| 1 board foot | 1" × 12" × 12" (144 cubic inches) | Lumber |
Weight:
| Unit | Equivalent | Use |
|---|---|---|
| 1 ounce | 28.35 grams | Small quantities |
| 1 pound | 16 ounces = 453.6 grams | General |
| 1 ton (short) | 2,000 pounds = 907 kg | Heavy loads |
| 1 kilogram | 2.205 pounds | Metric standard |
| 1 metric ton | 1,000 kg = 2,205 lbs | Metric heavy |
Chapter 4: Geometry
Essential Formulas:
| Shape | Perimeter/Circumference | Area | Volume |
|---|---|---|---|
| Rectangle | 2(L+W) | L × W | L × W × H (box) |
| Triangle | a + b + c | 1/2 × base × height | — |
| Circle | 2πr (π ≈ 3.14159) | πr² | 4/3 πr³ (sphere) |
| Cylinder | — | — | πr²h |
| Cone | — | — | 1/3 πr²h |
| Trapezoid | a + b + c + d | 1/2(a+b) × h | — |
The Pythagorean Theorem:
In any right triangle: a² + b² = c² (where c is the hypotenuse)
Common Pythagorean Triples (for checking square corners):
| a | b | c | Use |
|---|---|---|---|
| 3 | 4 | 5 | Most common (or any multiple: 6-8-10, 9-12-15) |
| 5 | 12 | 13 | Larger layouts |
| 8 | 15 | 17 | Larger layouts |
| 7 | 24 | 25 | Larger layouts |
Checking a Right Angle (3-4-5 method):
Measure 3 units along one side, 4 units along the other side. If the diagonal between those points is exactly 5 units, the corner is square (90 degrees).
Chapter 5: Trigonometry
The Three Basic Ratios (for right triangles):
| Function | Definition | Mnemonic |
|---|---|---|
| Sine (sin) | Opposite / Hypotenuse | SOH |
| Cosine (cos) | Adjacent / Hypotenuse | CAH |
| Tangent (tan) | Opposite / Adjacent | TOA |
Common Angle Values:
| Angle | Sin | Cos | Tan |
|---|---|---|---|
| 0° | 0.000 | 1.000 | 0.000 |
| 15° | 0.259 | 0.966 | 0.268 |
| 30° | 0.500 | 0.866 | 0.577 |
| 45° | 0.707 | 0.707 | 1.000 |
| 60° | 0.866 | 0.500 | 1.732 |
| 75° | 0.966 | 0.259 | 3.732 |
| 90° | 1.000 | 0.000 | undefined |
Practical Applications:
| Problem | Formula | Example |
|---|---|---|
| Height of tree/building | height = distance × tan(angle) | 100 ft away, 45° angle → height = 100 × 1.0 = 100 ft |
| Roof pitch (rise per run) | rise = run × tan(pitch angle) | 12 ft run, 30° pitch → rise = 12 × 0.577 = 6.9 ft |
| Slope distance | slope = horizontal / cos(angle) | 100 ft horizontal, 20° slope → 100/0.94 = 106 ft |
| Width of river | Use similar triangles or tangent from two points | Measure baseline, sight angles to far point |
Chapter 6: Construction Mathematics
Concrete Calculation:
Volume = Length × Width × Depth (all in same units)
Example: Slab 20 ft × 12 ft × 4 inches (0.333 ft) = 80 cubic feet = 2.96 cubic yards
At 1:2:3 mix per cubic yard: 6 bags cement + 0.44 cu yd sand + 0.88 cu yd gravel
Roof Calculations:
| Pitch | Rise per 12" Run | Rafter Length Factor | Angle |
|---|---|---|---|
| 3:12 | 3 inches | 1.031 | 14° |
| 4:12 | 4 inches | 1.054 | 18° |
| 5:12 | 5 inches | 1.083 | 23° |
| 6:12 | 6 inches | 1.118 | 27° |
| 8:12 | 8 inches | 1.202 | 34° |
| 10:12 | 10 inches | 1.302 | 40° |
| 12:12 | 12 inches | 1.414 | 45° |
Rafter length = horizontal span × rafter length factor
Stair Calculations:
| Parameter | Formula | Ideal Values |
|---|---|---|
| Number of risers | Total height ÷ desired rise | — |
| Rise (height per step) | Total height ÷ number of risers | 7-7.75 inches |
| Run (depth per step) | Typically 10-11 inches | 10-11 inches |
| Comfort rule | 2 × rise + run = 24-25 inches | Check after calculating |
| Total horizontal length | Number of treads × run | (Treads = risers - 1) |
Chapter 7: Navigation Mathematics
Latitude from Polaris:
Angle of Polaris above horizon = your latitude (Northern Hemisphere)
Distance Calculations:
| Method | Formula | Use |
|---|---|---|
| Pace counting | Distance = paces × pace length (average 2.5 ft) | Ground travel |
| Time × speed | Distance = speed × time | Any travel |
| Triangulation | Use two known points + angles to find unknown distance | Surveying |
| Horizon distance | d (miles) = 1.17 × √(height in feet) | How far can you see? |
Horizon Distance Examples:
| Eye Height | Horizon Distance |
|---|---|
| 6 feet (standing) | 2.9 miles |
| 20 feet (rooftop) | 5.2 miles |
| 50 feet (tower) | 8.3 miles |
| 100 feet (tall tower) | 11.7 miles |
| 1,000 feet (mountain) | 37 miles |
Chapter 8: Trade and Finance Mathematics
Percentage Calculations:
| Problem | Formula | Example |
|---|---|---|
| Find percentage of a number | number × (percentage/100) | 15% of 240 = 240 × 0.15 = 36 |
| Find what percentage one number is of another | (part/whole) × 100 | 36 is what % of 240? (36/240)×100 = 15% |
| Percentage increase | ((new-old)/old) × 100 | Price went from 50 to 65: (15/50)×100 = 30% increase |
| Percentage decrease | ((old-new)/old) × 100 | Price went from 65 to 50: (15/65)×100 = 23% decrease |
Ratio and Proportion:
If 3 workers build 1 wall in 12 days, how long for 5 workers?
- Worker-days = 3 × 12 = 36
- Time for 5 workers = 36 ÷ 5 = 7.2 days
Compound Interest (for understanding debt and growth):
A = P(1 + r)^n
Where: A = final amount, P = principal, r = rate per period, n = number of periods
Chapter 9: Estimation and Mental Math
Useful Approximations:
| Approximation | Actual Value | Error |
|---|---|---|
| π ≈ 22/7 | 3.14159... | 0.04% |
| √2 ≈ 1.414 | 1.41421... | <0.01% |
| √3 ≈ 1.732 | 1.73205... | <0.01% |
| 1 meter ≈ 3 feet 3 inches | 3.281 feet | 0.5% |
| 1 kg ≈ 2.2 lbs | 2.205 lbs | 0.2% |
| 1 mile ≈ 1.6 km | 1.609 km | 0.6% |
| 1 gallon ≈ 3.8 liters | 3.785 liters | 0.4% |
| 1 acre ≈ 4,000 sq meters | 4,047 sq meters | 1.2% |
| °F to °C: subtract 30, divide by 2 | (F-32)×5/9 | Approximate (good enough for estimation) |
Mental Math Shortcuts:
| Shortcut | Method | Example |
|---|---|---|
| Multiply by 5 | Multiply by 10, divide by 2 | 48 × 5 = 480/2 = 240 |
| Multiply by 25 | Multiply by 100, divide by 4 | 36 × 25 = 3600/4 = 900 |
| Multiply by 9 | Multiply by 10, subtract original | 47 × 9 = 470 - 47 = 423 |
| Square numbers ending in 5 | (n)(n+1) then append 25 | 35² = 3×4=12, append 25 → 1,225 |
| Percentage tip (15%) | 10% + half of 10% | 15% of $84: $8.40 + $4.20 = $12.60 |
Chapter 10: Calculation Tools
The Abacus:
A frame with beads on rods. Each rod represents a place value. Beads moved toward the center bar are "active" (counted). A skilled abacus user can add, subtract, multiply, and divide as fast as a calculator.
Slide Rule:
Two logarithmic scales that slide against each other. Multiplication becomes addition of lengths; division becomes subtraction. Can also do squares, square roots, trigonometry, and logarithms. Accuracy: 3 significant figures (sufficient for most engineering).
| Scale | Function |
|---|---|
| C/D scales | Multiplication and division |
| A/B scales | Squares and square roots |
| S scale | Sine values |
| T scale | Tangent values |
| L scale | Logarithms |
| CI scale | Reciprocals (inverse) |
Logarithm Tables:
Before calculators, all complex multiplication was done by adding logarithms:
- log(a × b) = log(a) + log(b)
- log(a ÷ b) = log(a) - log(b)
- log(a^n) = n × log(a)
A printed table of logarithms (4-5 decimal places) enables any calculation to any precision needed for engineering.
Reference Card
MATHEMATICS ESSENTIALS:
- Pythagorean theorem: a² + b² = c² (check square corners with 3-4-5)
- Circle: circumference = 2πr, area = πr², π ≈ 3.14159
- SOH-CAH-TOA: sin = opp/hyp, cos = adj/hyp, tan = opp/adj
- Height of object = distance × tan(angle of elevation)
- Concrete volume: L × W × D (convert all to same units first)
- Pendulum period: T = 2π√(L/g), where g = 32.2 ft/s² or 9.81 m/s²
- Horizon distance (miles) = 1.17 × √(height in feet)
- When in doubt, estimate conservatively (round up for materials, round down for capacity)
This campaign provides the complete mathematical knowledge needed to design, build, and operate the technologies described in all other campaigns. Mathematics is not optional — it is the foundation upon which engineering, navigation, trade, and science rest. A community with mathematical literacy can solve any quantitative problem it encounters.