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Formulas & shortcuts

World History · 34

Every formula in this course and the easiest ways to remember them, on one page.

1. The ancient world2. From the Renaissance to the revolutions3. Empires, faith and revolution in depth4. The 20th century and today5. The 20th century in depth: world wars and decolonization6. Historical analysis: method, state and economy7. Time, war and world order

1The ancient worldBeginner

What is history and how do historians work?

Open lesson

The first civilizations: Mesopotamia and Egypt

Open lesson

2From the Renaissance to the revolutionsIntermediate

The Renaissance, printing and the Age of Discovery

Open lesson

The age of revolutions: America, France and industry

Open lesson

3Empires, faith and revolution in depthIntermediate

Ancient Persia, India and China

Open lesson

The Ottoman Empire: rise, reform and fall

Open lesson

The Reformation and the wars of religion

Open lesson

The Napoleonic era and the Congress of Vienna

Open lesson

4The 20th century and todayAdvanced

The First and Second World Wars

Open lesson

The Cold War and the modern world

Open lesson

5The 20th century in depth: world wars and decolonizationAdvanced

The First World War: causes, fronts and the peace

Open lesson

The Second World War: the road, the turning points and victory

Open lesson

Decolonization: the end of colonial empires

Open lesson

6Historical analysis: method, state and economyUniversity

Historical method and historiography

Open lesson
N = N₀ · (½)^(t / T½) ⇔ t = T½ · log₂(N₀ / N)N = N₀ · (½)^(t / T½) ⇔ t = T½ · log₂(N₀ / N)
where:
  • Namount (or percentage) of C-14 left in the sample
  • N₀initial amount at the time of death (100%)
  • ttime since death, in years
  • T½half-life of C-14 ≈ 5,730 years

Where it comes from: every T½ years the amount is multiplied by ½; after t/T½ half-lives we get (½)^(t/T½). Taking logarithms gives the expression for t.

The modern state, nationalism and empires (19th–20th centuries)

Open lesson

Economic history: industrialisation, the world economy and globalization

Open lesson
Yₜ = Y₀ · (1 + g)ᵗ
where:
  • Yₜthe value after t years (for example GDP per person)
  • Y₀the starting value
  • gannual growth rate as a decimal (3% = 0.03)
  • tnumber of years

Where it comes from: each year the value is multiplied by (1 + g); over t years this multiplication happens t times. It is the same logic as compound interest at a bank.

t₂ₓ ≈ 70 / g(%)t₂ₓ ≈ 70 / g(%)
where:
  • t₂ₓyears needed for the value to double
  • g(%)annual growth in percent (write 3 for 3%)

The “rule of 70”. Derivation: (1 + g)ᵗ = 2 ⇒ t = ln 2 / ln(1 + g). For small g, ln(1 + g) ≈ g and ln 2 ≈ 0.693, so t ≈ 0.693 / g ≈ 70 / g(%).

Real = Nominal / (P / 100) Openness = (X + M) / GDP · 100%Real = Nominal / (P / 100) Openness = (X + M) / GDP · 100%
where:
  • Pprice index (100 in the base year)
  • Xexports
  • Mimports
  • GDPgross domestic product — the value of all goods and services produced in a country in a year

A real value removes inflation: it measures what money can buy, not the amount of money. Openness shows how much a country depends on world trade.

7Time, war and world orderUniversity

Calendars, periodization, the Annales school and global history

Open lesson
Δ = ⌊Y / 100⌋ − ⌊Y / 400⌋ − 2Δ = ⌊Y / 100⌋ − ⌊Y / 400⌋ − 2
where:
  • Δhow many days the Gregorian date is ahead of the Julian one
  • Ythe year (for dates from 1 March of year Y onwards)
  • ⌊ ⌋integer part (drop the fraction)

Where it comes from: in the Julian calendar every century year is a leap year, in the Gregorian only those divisible by 400. Each leap day “skipped” since the reform adds one day to the gap; the constant −2 fits the initial 10-day gap of 1582: 15 − 3 − 2 = 10. For January–February of a century year, use Y − 1.

G ≈ H − H / 33 + 622 ⇔ H ≈ (G − 622) · 33 / 32G ≈ H − H / 33 + 622 ⇔ H ≈ (G − 622) · 33 / 32
where:
  • GCommon Era (Gregorian) year
  • Hlunar Hijri year (AH)
  • 622the year of the Hijra, the start of the Hijri era
  • 33 / 3233 / 3233 lunar years ≈ 32 solar years (354.37 / 365.24 ≈ 0.97)

Where it comes from: H lunar years equal H · 0.97 solar years, and 0.97 ≈ 1 − 1/33. The result is accurate to ±1 year, because one Hijri year spans two Gregorian years.

Why do great wars begin? Analysing the causes of major wars

Open lesson
CINC = (TPOP + UPOP + IRST + PEC + MILEX + MILPER) / 6CINC = (TPOP + UPOP + IRST + PEC + MILEX + MILPER) / 6
where:
  • TPOPthe country's share of world population (0–1)
  • UPOPshare of world urban population
  • IRSTshare of world iron and steel production
  • PECshare of world primary energy consumption
  • MILEXshare of world military expenditure
  • MILPERshare of the world's military personnel

Where it comes from: the average of six shares gives equal weight to the demographic, industrial and military dimensions. The CINC values of all states add up to 1.

R = CINC(challenger) / CINC(dominant)R = CINC(challenger) / CINC(dominant)
where:
  • Rthe power ratio; power-transition studies often treat R ≈ 0.8 or more as approaching parity
  • challengerthe rising challenger
  • dominantthe dominant state

From the Congress of Vienna to the UN and the EU: a history of international organizations

Open lesson
Adopted ⇔ (Yes ≥ 9) ∧ (P5 “No” = 0)
where:
  • Yesthe number of the 15 Security Council members voting in favour
  • P5 “No”negative votes cast by the five permanent members (a veto); abstaining is not a veto

A Security Council decision on substantive (non-procedural) matters. On procedural matters 9 votes are enough and there is no veto. The practice that a permanent member's abstention is not a veto dates from the 1940s and was confirmed by the International Court of Justice in 1971.

Adopted (important question) ⇔ Yes ≥ ⅔ · (Yes + No)
where:
  • Yes + No“members present and voting”; abstentions are not counted

A General Assembly decision on important questions (peace and security, admission of new members, the budget); other questions need a simple majority. Assembly resolutions are, as a rule, recommendations.

Adopted ⇔ (States ≥ 55%) ∧ (Population ≥ 65%)
where:
  • Statesshare of member states voting in favour; with 27 members, 0.55 · 27 = 14.85 → at least 15
  • Populationthose states' share of the EU population

Qualified majority in the Council of the EU (the “double majority”, Treaty of Lisbon). A blocking minority must include at least 4 states representing more than 35% of the population; if fewer than 4 states vote against, the qualified majority is deemed reached.