The Rule Book

Every rule, why it works, and the Greeks behind it. Then take its quiz.

Rule 1

Place value: every spot is worth ten times more

In 352, the 3 means 3 hundreds, the 5 means 5 tens, the 2 means 2 ones.

Why it works

We count in groups of ten because we have ten fingers. When you fill a spot with ten, you trade them for one in the next spot to the left. That's why carrying and borrowing work: you're just trading ten small pieces for one big piece, or the other way round.

🏛️ The Greek connection

Archimedes wrote The Sand Reckoner to show that numbers never run out — he invented a way to name numbers big enough to count every grain of sand that could fill the universe, by stacking groups on groups, just like place value.

Examples

352 = 300 + 50 + 210 ones = 1 ten10 tens = 1 hundred0.3 = 3 tenths

Watch out

A zero still holds a spot: 305 is not 35

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Rule 2

Order doesn't matter for + and ×

3 + 5 = 5 + 3, and 4 × 6 = 6 × 4. (But not for − or ÷!)

Why it works

Picture 4 rows of 6 pebbles. Turn the page sideways and it's 6 rows of 4 — same pebbles, same count. Adding works the same way: two piles combined are the same no matter which one you pour in first. Subtracting and dividing are different, because taking away depends on what you start with.

🏛️ The Greek connection

Pythagoras and his students counted with pebbles (the Greek word is psephoi) laid out in shapes. Their 'oblong numbers' were rectangles of pebbles — and anyone could see a rectangle has the same count whichever way you turn it.

Examples

7 + 2 = 2 + 73 × 8 = 8 × 3(2 + 3) + 4 = 2 + (3 + 4)

Watch out

10 − 3 ≠ 3 − 1012 ÷ 4 ≠ 4 ÷ 12

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Rule 3

Multiplying is making a rectangle

6 × 4 means 6 groups of 4 — or a rectangle 6 long and 4 wide.

Why it works

Multiplying is fast adding: 6 × 4 is 4 + 4 + 4 + 4 + 4 + 4. Draw it as a rectangle and you can split it into easier pieces: 7 × 12 is 7 × 10 plus 7 × 2. That trick (splitting a rectangle) is how you do big multiplication in your head.

🏛️ The Greek connection

Euclid's Elements (around 300 BC) talks about multiplication entirely as rectangles. Book II shows that one big rectangle equals the smaller rectangles you cut it into — the same rule you use for 7 × 12 = 70 + 14.

Examples

7 × 12 = 7 × 10 + 7 × 2 = 845 × 18 = 5 × 20 − 5 × 2 = 90Square: 5 × 5 = 25

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Rule 4

Order of operations

Brackets first, then × and ÷, then + and −. Left to right inside each step.

Why it works

Multiplying makes groups, and you have to finish building a group before you can add it to anything. In 3 + 4 × 2, the '4 × 2' is one package of 8 — so the answer is 11, not 14. Brackets are how you say 'build this package first.'

🏛️ The Greek connection

Greek geometers wrote everything in words, and phrases like 'the rectangle on AB and BC, together with the square on CD' made it clear which pieces were whole packages. Our brackets do the same job with fewer words.

Examples

3 + 4 × 2 = 11(3 + 4) × 2 = 1420 − 6 ÷ 2 = 1710 − 3 + 2 = 9

Watch out

Same-level steps go left to right: 10 − 3 + 2 is 9, not 5

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Rule 5

A fraction is a part out of a whole

3/4 means the whole was cut into 4 equal parts and you have 3 of them.

Why it works

The bottom number says how many pieces the whole was cut into; the top says how many you have. Multiply top and bottom by the same number and you just cut each piece smaller — so 1/2 = 2/4 = 4/8. To add fractions, the pieces must be the same size first (same bottom).

🏛️ The Greek connection

Pythagoras found that a string cut in the ratio 1/2 sounds an octave higher, 2/3 sounds a 'fifth', and 3/4 a 'fourth'. To the Greeks, simple fractions were literally the music of the world — the Greek word logos meant both 'ratio' and 'reason.'

Examples

1/2 = 2/4 = 4/81/4 + 2/4 = 3/46/8 simplifies to 3/41/3 of 12 = 4

Watch out

1/2 + 1/3 is NOT 2/5 — make same-size pieces first: 3/6 + 2/6 = 5/6

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Rule 6

Decimals are fractions of ten

0.5 = 5/10, 0.25 = 25/100. Line up the points when you add.

Why it works

The spots after the decimal point keep the place-value pattern going: tenths, hundredths, thousandths. Money is the easiest example — $3.47 is 3 dollars, 4 dimes (tenths of a dollar) and 7 pennies (hundredths). Lining up the points makes sure you add tenths to tenths.

🏛️ The Greek connection

The Greeks did their precise astronomy in fractions, and later the scholar Hypatia of Alexandria taught and edited those astronomy tables. Decimals came much later, but they finish the job Archimedes began: a single place-value system that goes as big — or as small — as you need.

Examples

0.5 = 1/20.25 = 1/40.75 = 3/41.2 + 0.35 = 1.55

Watch out

0.5 is bigger than 0.45 — compare spot by spot, not by length

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Rule 7

Percent means 'out of 100'

40% means 40 out of every 100. 10% of anything: slide the point one spot left.

Why it works

Percent turns every comparison into the same size bottom — 100 — so it's easy to compare. 50% is half, 25% is a quarter, 10% is a tenth. To find 20%, find 10% and double it. To find a sale price, take the discount off the original.

🏛️ The Greek connection

Athens charged a 2% tax — the 'pentekoste', meaning 'one-fiftieth' — on goods coming through its harbor at Piraeus. Merchants had to figure out a fixed share of every cargo, just like you do with percent.

Examples

50% = 1/225% = 1/410% of 80 = 820% of 80 = 16

Watch out

A 50% raise then a 50% cut does NOT get you back where you started

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Rule 8

Time counts in sixties

60 seconds = 1 minute, 60 minutes = 1 hour, 24 hours = 1 day.

Why it works

Time doesn't count in tens, so you carry at 60, not 10. 2:45 plus 30 minutes is 3:15, not 2:75. Why 60? Because 60 splits evenly so many ways — halves, thirds, quarters, fifths, sixths — so a half hour (30), quarter hour (15) and third of an hour (20) are all whole numbers.

🏛️ The Greek connection

Greek astronomers Hipparchus and Ptolemy took the Babylonian count-by-60 system and used it to divide the circle and the hour. Ptolemy's 'first small parts' and 'second small parts' became our minutes and seconds.

Examples

2:45 + 30 min = 3:1590 minutes = 1 h 30 minquarter past 4 = 4:151/3 hour = 20 min

Watch out

Noon is 12:00 p.m.; midnight is 12:00 a.m.

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Rule 9

Make change by counting up

To find change from $5 for $3.65, count up: 3.65 → 3.75 → 4.00 → 5.00 = $1.35.

Why it works

Subtracting across zeros (5.00 − 3.65) is where people slip. Shopkeepers count up instead: add coins to reach the next round number, then bills to reach what was paid. Every step is easy, and the steps add up to the change.

🏛️ The Greek connection

Euclid's oldest method for comparing two numbers works by repeatedly taking the smaller away from the bigger. Greek market traders in the agora made change with coins called obols and drachmas — 6 obols to a drachma — so counting up in steps was an everyday skill.

Examples

$10 − $6.25: 6.25 → 7.00 (0.75) → 10.00 (3) = $3.75$1 − 0.35 = $0.65

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Rule 10

Area covers, perimeter goes around

Area = length × width (square units). Perimeter = add every side.

Why it works

Area counts how many 1-foot squares fit on a floor — that's why a 12 by 10 room is 120 square feet. Perimeter is the walk around the edge. Flooring, paint and carpet use area; trim and fencing use perimeter. Odd shapes are just rectangles stuck together. When buying materials, always round up.

🏛️ The Greek connection

Geometry means 'earth-measuring' in Greek. Thales is said to have measured the height of a pyramid using its shadow and a scale — the same idea as a blueprint where 1 inch stands for 4 feet. Heron of Alexandria wrote a whole book, the Metrica, on measuring fields and buildings.

Examples

12 ft × 10 ft = 120 sq ftPerimeter of 12 × 10 = 44 ft1 in = 4 ft, so 5 in = 20 ft10 × 12 × 1/3 = 40 cu ft

Watch out

Doubling both sides of a room makes the area 4 times bigger, not 2

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Rule 11

Borrowing costs interest

You pay back what you borrowed plus interest. Longer loans cost more in total.

Why it works

A lender charges interest as rent on money. Each month, interest is charged on what you still owe, and your payment covers that interest plus a bit of the loan. Early on, most of a payment is interest. A bigger down payment or a shorter loan means far less interest overall.

🏛️ The Greek connection

Athens had bankers called trapezitai — 'table men', from the tables they worked at in the market. The banker Pasion, a freed slave, became one of the richest men in Athens lending money. Solon's reforms around 594 BC cancelled crushing debts after farmers were being sold into slavery for them.

Examples

10% down on $18,000 = $1,800Simple: $10,000 × 5% × 3 = $1,500Monthly rate = yearly ÷ 12Interest = payment × months − loan

Watch out

A lower monthly payment can cost MORE overall if the loan is longer

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Rule 12

Compounding: interest earns interest

Each period, multiply by (1 + rate). Money doubles in about 72 ÷ rate years.

Why it works

With compounding, this year's interest joins your money, so next year you earn interest on it too. That's why growth starts slow and then speeds up — time matters more than the amount. The Rule of 72 gives a quick estimate of doubling time. It works against you on debt, too.

🏛️ The Greek connection

The Greek word for interest, tokos, also meant 'offspring' — money giving birth to more money. Aristotle disliked it for exactly that reason, calling it the most unnatural way to make money. Compounding is offspring having offspring.

Examples

$1,000 at 10%: 1,100 → 1,210 → 1,33172 ÷ 8% = 9 years to double$500 × 1.07^20 ≈ $1,934.84Monthly: rate ÷ 12, years × 12

Watch out

The Rule of 72 is an estimate, not exact

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