Mathematics

Why This Matters

Mathematics is the language that makes engineering possible. Without it, you cannot divide land fairly, calculate how much grain will feed your community through winter, design a roof that will not collapse, or build a water system with the correct pipe diameter. Every advanced technology described in this guide depends on mathematical principles that you can learn and teach with nothing more than a stick and a patch of dirt.

Number Systems

Tally Marks: Start Here

The simplest counting system requires no training. Make one mark per item:

| | | |    = 4
|||| |     = 5 (gate tally — cross every fifth mark)
|||| |||| |||| |||  = 18

Gate tallies (crossing every fifth mark) make it easy to count by fives. This works for counting livestock, days, votes, or inventory up to a few hundred. Beyond that, you need positional notation.

Positional Notation (Base-10)

The system we use today: ten digits (0-9) where position determines value.

Position	Name	Value
Rightmost	Ones	x 1
Second	Tens	x 10
Third	Hundreds	x 100
Fourth	Thousands	x 1,000

The number 3,527 means: 3 thousands + 5 hundreds + 2 tens + 7 ones.

Why Zero Matters

Zero is the most important invention in mathematics. Without zero as a placeholder, you cannot tell 15 from 105 from 1,050. Ancient Romans used MCCCV for 1305 — try multiplying that. Zero makes arithmetic possible. Teach it as the first concept.

Writing Numbers

If you have no paper, scratch numbers in clay tablets, carve into wood, or write in sand for teaching. Any ten distinct symbols work — you do not need to use Arabic numerals. What matters is consistent position and a zero placeholder.

Basic Arithmetic

Addition and Subtraction

These are the foundation. Every person in your community should master them:

Column addition:

  347
+ 285
-----
  632    (7+5=12, write 2 carry 1; 4+8+1=13, write 3 carry 1; 3+2+1=6)

Subtraction with borrowing:

  503
- 267
-----
  236    (3-7: borrow, 13-7=6; 9-6=3; 4-2=2)

Practice daily until these are automatic. Speed matters when calculating inventories, rations, and construction measurements.

Multiplication

Multiplication is repeated addition, but knowing the times tables by heart saves enormous time.

Multiplication table (essential up to 12 x 12):

x	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

Copy this table onto a board and hang it in your school or workshop.

The 9s Trick

To multiply any single digit by 9: hold up all 10 fingers. Lower the finger at the position of the number you are multiplying. The fingers to the left are the tens digit, fingers to the right are the ones. Example: 9 x 7 — lower finger 7 — you see 6 fingers left, 3 right = 63.

Division

Division is the inverse of multiplication. Long division is essential for splitting resources fairly:

  156 / 12 = 13
  12 goes into 15 once (12), remainder 3
  Bring down 6 → 36
  12 goes into 36 three times (36), remainder 0
  Answer: 13

Fractions and Ratios

Working With Fractions

Fractions appear constantly: half-rations, quarter-turns, three-eighths of a log.

Key operations:

• Adding fractions: Find a common denominator. 1/3 + 1/4 = 4/12 + 3/12 = 7/12
• Multiplying fractions: Multiply tops, multiply bottoms. 2/3 x 3/4 = 6/12 = 1/2
• Dividing fractions: Flip the second and multiply. 3/4 / 2/3 = 3/4 x 3/2 = 9/8

Ratios and Proportions

Ratios let you scale recipes and construction plans:

• Mortar mix is 1:3 (lime:sand). For 20 kg total, you need 5 kg lime + 15 kg sand
• A map at 1:1000 means 1 cm on paper = 10 meters on ground
• If 5 workers dig 3 meters per day, 15 workers dig 9 meters per day (direct proportion)

Cross-multiplication for solving proportions: If 4 bricks weigh 12 kg, how much do 25 bricks weigh?

4/12 = 25/x
4x = 300
x = 75 kg

Geometry: Areas and Volumes

Area Formulas You Need

Shape	Formula	Example
Rectangle	length x width	5m x 3m = 15 m2
Triangle	(base x height) / 2	(6m x 4m) / 2 = 12 m2
Circle	pi x radius x radius (pi = 3.14)	3.14 x 2m x 2m = 12.56 m2
Trapezoid	(top + bottom) / 2 x height	(3m + 5m) / 2 x 4m = 16 m2

Measuring Irregular Areas

For oddly shaped fields or plots:

1. Divide the area into rectangles and triangles
2. Calculate each piece separately
3. Add them together
4. For curved boundaries, approximate with many small triangles

Volume Formulas

Shape	Formula	Use
Box/rectangular	l x w x h	Water tanks, rooms, bins
Cylinder	pi x r2 x h	Barrels, wells, pipes
Cone	(pi x r2 x h) / 3	Grain piles, funnels
Sphere	(4/3) x pi x r3	Rarely needed in practice

Practical Volume: The 1-Meter Cube

A cube 1 meter on each side holds exactly 1,000 liters (1 cubic meter). This is your reference point. A water tank 2m x 1m x 0.5m holds 1,000 liters. A barrel 0.5m diameter and 1m tall holds about 196 liters.

The Pythagorean Theorem

This is the single most useful mathematical relationship for builders:

In any right triangle: a2 + b2 = c2

Where c is the longest side (hypotenuse) and a and b are the two shorter sides.

The 3-4-5 Rule: Creating Perfect Right Angles

This is used every day on construction sites worldwide:

1. Mark a point where you want a right angle
2. Measure 3 units along one direction, mark it
3. Measure 4 units along the perpendicular direction, mark it
4. The diagonal between those two marks should be exactly 5 units
5. If it is not 5, adjust the angle until it is

Other right-angle triples:

• 5-12-13
• 8-15-17
• 7-24-25

Use Real Measurement Tools

“Eyeballing” a right angle always introduces error. Even 2 degrees off square means a wall 3 meters tall will be 10 cm out of plumb at the top. Always use the 3-4-5 method or a large square for foundations and framing.

Finding Unknown Distances

Problem: You need to span a river but cannot measure across. You can see a tree on the far bank.

1. Stand directly across from the tree (point A)
2. Walk 30 meters along the bank (point B)
3. Turn 90 degrees and walk inland until the tree and point A form a line at 45 degrees
4. The distance you walked inland equals the river width

This is basic triangulation — the foundation of all surveying.

Trigonometry Basics

Trigonometry relates angles to side lengths in triangles. You need three ratios:

For a right triangle with angle A:

• Sine (sin) = opposite side / hypotenuse
• Cosine (cos) = adjacent side / hypotenuse
• Tangent (tan) = opposite side / adjacent side

Remember: SOH-CAH-TOA (Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent)

Essential Trigonometry Values

Angle	Sin	Cos	Tan
0	0	1	0
30	0.50	0.87	0.58
45	0.71	0.71	1.00
60	0.87	0.50	1.73
90	1	0	undefined

Copy this table and keep it in your workshop. These five angles cover most practical needs.

Measuring Heights of Tall Objects

Problem: How tall is that tree?

1. Walk away from the tree until you can see the top at a 45-degree angle (use a simple clinometer — a protractor with a hanging weight)
2. At 45 degrees, tan = 1, so the tree height equals your distance from it plus your eye height
3. If you stand 20 m away and your eye is at 1.7 m: tree height = 20 + 1.7 = 21.7 m

For other angles: height = distance x tan(angle) + eye height.

Surveying and Standard Units

Establishing Standard Units

Your community needs agreed-upon measurement standards. Base them on reproducible physical references:

Unit	Standard	Approximate Metric
Foot	Length of a specific reference stick	30 cm
Pace	One full stride (left foot to left foot)	1.5 m
Rod	The reference stick (carry one always)	5 m
Chain	20 rods or a physical chain of known length	100 m
Acre	Area a team of oxen plows in one day	~4,000 m2

Make a Master Rod

Carve a straight, hardwood rod exactly as long as your chosen “rod” unit. Mark subdivisions on it (halves, quarters, eighths). Make copies. Store the master in a dry, protected place. All measurements in your community reference this one physical standard.

Chain Surveying

To map land using only a chain (or rope of known length) and stakes:

1. Choose a baseline — a straight line across your site. Measure it carefully.
2. From each end of the baseline, measure the distance to every corner of the property
3. Every point is now defined by two distances from two known points — you can plot it on paper
4. Use the 3-4-5 method to establish perpendicular offsets for features near the baseline

Leveling

Finding true horizontal is critical for water systems, foundations, and roads:

Water level (ancient method):

1. Fill a long, clear tube (animal intestine or bamboo) with water
2. Hold both ends upright — water finds its own level
3. Mark where water sits at each end — those marks are at exactly the same elevation
4. Works over any distance the tube can reach

A-frame level:

1. Build an A-shaped frame from straight sticks
2. Hang a plumb bob from the apex
3. Place the feet on the ground and mark where the string crosses the crossbar
4. Swap the feet positions and mark again
5. The midpoint between the two marks is true level

Practical Math for Construction

How Much Material Do You Need?

Brick wall example:

• Wall: 5 m long, 2.5 m high, one brick thick (25 cm)
• Standard brick: 25 cm x 12 cm x 8 cm
• Bricks per square meter of wall face (with 1 cm mortar joints): approximately 50
• Wall face area: 5 x 2.5 = 12.5 m2
• Bricks needed: 12.5 x 50 = 625 bricks
• Add 10% for waste: 688 bricks

Roof timber example:

• Roof span: 6 m, pitch: 30 degrees
• Rafter length: span/2 / cos(30) = 3 / 0.87 = 3.45 m per rafter
• Rafters spaced 60 cm apart, roof length 8 m: 8 / 0.6 + 1 = 15 pairs = 30 rafters
• Total rafter wood: 30 x 3.45 m = 103.5 meters of timber

Record Keeping With Numbers

Write down every calculation. Keep a dedicated record book or clay tablet set for:

• Inventory counts (grain, tools, animals, building materials)
• Measurements for ongoing construction projects
• Trade records (what was exchanged, quantities, dates)
• Harvest records (yield per field, year over year)

This data lets you plan, predict, and improve. A community that records and analyzes its numbers makes better decisions than one that guesses.

What’s Next

With mathematics as your tool, you can advance to:

• Structural Engineering — calculate loads, spans, arch geometry, and foundation sizes
• Precision Measurement — build instruments for accurate surveying and navigation
• Electrical Theory — Ohm’s law, power calculations, and circuit design all require arithmetic fluency

Mathematics — At a Glance

Number system: Base-10 with zero placeholder. Teach positional notation first. Arithmetic: Master addition, subtraction, multiplication (tables to 12x12), and long division. Fractions: Common denominator for adding, flip-and-multiply for dividing. Cross-multiply for proportions. Areas: Rectangle = l x w, Triangle = (b x h)/2, Circle = 3.14 x r2 Volumes: Box = l x w x h, Cylinder = 3.14 x r2 x h Pythagorean theorem: a2 + b2 = c2. Use 3-4-5 for right angles on every build. Trig: sin/cos/tan — memorize values for 30, 45, 60 degrees. Height = distance x tan(angle). Surveying: Chain survey from a baseline, water level for elevation, A-frame for plumb. Master rod: One physical standard for all community measurements. Mark subdivisions, make copies.