Arithmetic

Part of Mathematics

Core number operations for everyday rebuilding calculations.

Why This Matters

Arithmetic — the four operations of addition, subtraction, multiplication, and division — is the workhorse mathematics of daily life. Every trade transaction, every construction measurement, every food ration, every fuel calculation requires arithmetic. In a rebuilding world without electronic devices, these skills must be performed reliably by hand or in the head.

The difference between a community that can do arithmetic and one that cannot is the difference between organized resource management and guesswork. Can you calculate how many days your grain stores will last? Can you figure out how many bricks you need for a wall? Can you divide a harvest fairly among 23 families? These are arithmetic problems, and getting them wrong has real consequences.

Arithmetic also provides the foundation for every technical discipline. You cannot do engineering without multiplication. You cannot do chemistry without ratios and division. You cannot do accounting without all four operations working together. Mastering arithmetic unlocks everything that comes after.

Multiplication

The Multiplication Table

Memorizing products up to 10 x 10 is the single most valuable investment in mathematical literacy. Write this table on a permanent surface (carved wood, fired clay) as a community reference:

	1	2	3	4	5	6	7	8	9	10
1	1	2	3	4	5	6	7	8	9	10
2	2	4	6	8	10	12	14	16	18	20
3	3	6	9	12	15	18	21	24	27	30
4	4	8	12	16	20	24	28	32	36	40
5	5	10	15	20	25	30	35	40	45	50
6	6	12	18	24	30	36	42	48	54	60
7	7	14	21	28	35	42	49	56	63	70
8	8	16	24	32	40	48	56	64	72	80
9	9	18	27	36	45	54	63	72	81	90
10	10	20	30	40	50	60	70	80	90	100

Long Multiplication

For numbers larger than 10, use the standard algorithm:

Example: 47 x 36

    47
  x 36
  ----
   282   (47 x 6)
  141    (47 x 3, shifted one position left)
  ----
  1692

Step by step:

1. Multiply 47 by 6: 7x6=42, write 2 carry 4. 4x6=24, plus 4=28. Write 282.
2. Multiply 47 by 3 (tens digit): 7x3=21, write 1 carry 2. 4x3=12, plus 2=14. Write 1410 (shifted left because we are multiplying by 30, not 3).
3. Add 282 + 1410 = 1692.

Mental Multiplication Tricks

1. Multiply by 5: Multiply by 10 and halve. 36 x 5 = 360 / 2 = 180.
2. Multiply by 9: Multiply by 10 and subtract the original. 36 x 9 = 360 - 36 = 324.
3. Multiply by 11: Add the digits and put the sum in the middle. 36 x 11 = 3(3+6)6 = 396. (If the middle sum exceeds 9, carry to the left digit.)
4. Multiply by 25: Multiply by 100 and divide by 4. 36 x 25 = 3600 / 4 = 900.
5. Breaking apart: 47 x 36 = 47 x 30 + 47 x 6 = 1410 + 282 = 1692.

Division

Long Division

The most important algorithm for division of larger numbers.

Example: 1692 / 36

       47
   -------
36 | 1692
     144    (36 x 4 = 144)
     ---
      252
      252   (36 x 7 = 252)
      ---
        0

Step by step:

1. 36 goes into 169 how many times? 36 x 4 = 144 (fits). 36 x 5 = 180 (too big). Write 4.
2. 169 - 144 = 25. Bring down the 2 to get 252.
3. 36 goes into 252 how many times? 36 x 7 = 252 (exact fit). Write 7.
4. 252 - 252 = 0. No remainder.

Answer: 47

Division with Remainders

Real-world division rarely comes out even.

Example: 500 rations divided among 23 families

500 / 23 = 21 remainder 17

Each family gets 21 rations; 17 rations remain for the communal store or for families with more members.

Checking Division

Quotient x Divisor + Remainder = Dividend

21 x 23 + 17 = 483 + 17 = 500. Correct.

Order of Operations

When a calculation involves multiple operations, the order matters:

1. Parentheses (brackets) first
2. Multiplication and division (left to right)
3. Addition and subtraction (left to right)

Example: Total bricks = 3 walls x 200 bricks + 1 wall x 150 bricks

Calculate: 3 x 200 + 1 x 150 = 600 + 150 = 750 bricks

Not: 3 x 200 + 1 = 601 x 150 (wrong — this ignores operation order).

When in Doubt, Use Parentheses

Write out your calculation with explicit grouping: (3 x 200) + (1 x 150). This eliminates ambiguity, especially when teaching others.

Practical Calculation Patterns

Ratios and Proportions

Many practical problems are ratio problems:

If 5 workers build 1 wall in 3 days, how long for 8 workers?

Worker-days needed: 5 x 3 = 15 worker-days. Time for 8 workers: 15 / 8 = 1.875 days (about 1 day and 7 hours).

Mortar mix 1:3 (cement:sand). Need 200 kg total. How much of each?

Total parts: 1 + 3 = 4. Cement: 200 / 4 x 1 = 50 kg. Sand: 200 / 4 x 3 = 150 kg.

Percentages

Percentage means “per hundred.” Useful for expressing yields, losses, and efficiency.

• Finding a percentage: 15% of 340 = 340 x 15 / 100 = 5100 / 100 = 51
• Percentage increase: Old value 200, new value 230. Increase = 30. Percentage = 30/200 x 100 = 15%
• Percentage decrease: 200 to 170. Decrease = 30. Percentage = 30/200 x 100 = 15%

Unit Rates

Convert measurements to “per unit” for easy comparison:

Problem	Calculation	Unit Rate
150 kg wheat from 0.3 ha	150 / 0.3	500 kg/ha
12 km walked in 3 hours	12 / 3	4 km/hour
600 bricks laid by 4 workers	600 / 4	150 bricks/worker

Estimation and Mental Math

Rounding for Quick Estimates

Round numbers to the nearest convenient value before calculating:

How many planks (each 2.3 m long) from a 15.7 m log?

Quick estimate: 16 / 2.5 = about 6 planks (actual: 15.7 / 2.3 = 6.8, so 6 complete planks with waste).

Cross-Multiplication for Proportions

If a/b = c/d, then a x d = b x c.

A recipe uses 3 cups flour for 2 loaves. How much flour for 7 loaves?

3/2 = x/7 3 x 7 = 2 x x 21 = 2x x = 10.5 cups

Squaring Numbers

Useful for area calculations. Quick method for numbers near a round number:

Square of 48: 48 is 50-2. Use (a-b)^2 = a^2 - 2ab + b^2. 50^2 - 2(50)(2) + 2^2 = 2500 - 200 + 4 = 2304.

Error Prevention

Reasonableness Check

After every calculation, ask: “Does this answer make sense?”

• Building material: If your calculation says you need 50,000 bricks for a small shed, recheck.
• Food rations: If 100 kg of grain divided among 20 people gives 50 kg each, you divided wrong.
• Time estimates: If 3 workers building a fence in 5 days gives 45 worker-days, recheck (it should be 15).

Consistent Units

The most common source of arithmetic error in practical work is mixed units:

Wrong	Right
3 meters x 50 centimeters = 150	3 m x 0.5 m = 1.5 sq m
2 hours + 30 minutes = 230	2 hours + 0.5 hours = 2.5 hours
5 kg + 200 grams = 5200	5 kg + 0.2 kg = 5.2 kg

Teach Units First

Before teaching arithmetic to newcomers, establish and teach the measurement units your community uses. Arithmetic is useless if people are multiplying meters by centimeters without converting.

Double-Entry Verification

For critical calculations (food stores, building materials, trade accounts):

1. Calculate the answer
2. Have a second person calculate independently
3. Compare results
4. If they disagree, both recalculate

This is the same principle behind double-entry bookkeeping, and it catches errors that self-checking misses because a different person is likely to make different mistakes.