Reference

Useful constants, equations, and notation habits.

Mental model challenges get easier when a few anchor numbers live in memory. Use these as starting points, then round aggressively enough to keep the structure visible.

Scientific notation

Scientific notation writes a number as a value between 1 and 10 multiplied by a power of ten. For example, 330 million becomes 3.3 x 10^8, 20 billion becomes 2 x 10^10, and 0.004 becomes 4 x 10^-3.

The exponent makes scale visible and simplifies rough arithmetic. When multiplying, multiply the leading numbers and add the exponents:

(2 x 10^6) x (3 x 10^4)
= 6 x 10^(6 + 4)
= 6 x 10^10

Significant digits

Significant digits communicate how precise a number deserves to look. A Fermi estimate built from rough assumptions should normally keep only one or two significant digits: write 1.9 x 10^7 or 20 million, not 19,428,571.

Round during the calculation when it makes the structure easier to see, but keep enough information to avoid shifting the answer into the wrong order of magnitude. Apparent precision is not accuracy.

Orders of magnitude

An order of magnitude is a factor of ten. The central goal is usually to identify whether the answer belongs near 10^4, 10^6, 10^8, or another power of ten. Landing within 10x of a credible answer counts as success; the assumptions explain why.

Reference pegs used in articles

These tables show reusable pegs that appear in more than one completed problem article. The full audit still tracks every used peg in peg-usage.csv, and each article count is calculated by scanning every problem-*.html page once per peg. Click any column header to sort.

Constants and conversions

Peg	Value or conversion	Articles	Importance	Why it helps
Gallons, liters, cups, and everyday liquid volume	1 gallon ~= 3.8 L ~= 16 cups	27	High	Useful for coffee, oil, fuel, pools, water, and other everyday liquid-volume estimates.
Density and mass-volume conversion	mass = density x volume; water ~= 1,000 kg/m3; air ~= 1 kg/m3	24	High	Used whenever a volume needs to become mass, or mass needs to become volume.
GW, MW, and kW conversions	1 GW = 1,000 MW = 1,000,000 kW	15	High	A recurring source of thousandfold mistakes in power problems.
Gasoline and fuel energy	gasoline ~= 1.3 x 10^8 J/gallon; jet fuel ~= 35 MJ/L	13	High	Turns unfamiliar energy totals into fuel-equivalent comparisons.
Kilowatt-hour to joules	1 kWh = 3.6 x 10^6 J	13	High	Essential for translating electricity use into joules.
U.S. households	~1.3 x 10^8 households	12	High	Useful for homes, pools, utilities, vehicles, consumer goods, and ownership rates.
U.S. population	~3.4 x 10^8 people	12	High	The default national denominator for U.S.-scale estimates.
Household electricity and average household power	~10,000 kWh/year; ~1.2 kW continuous	10	High	The bridge between grid-scale energy and a familiar home-scale comparison.
Mph to m/s	1 mph ~= 0.45 m/s	9	High	Converts road, wind, aviation, and speed-record problems into SI units.
Large power plant	~1 GW	8	High	A useful public-scale comparison for grid power.
Human lifetime and calendar-year conversions	lifetime ~= 80 years; year ~= 8.8 x 10^3 hours	6	High	Useful for turning collective human time into intuitive comparisons.
Oil barrel	1 barrel = 42 gallons ~= 159 L ~= 0.159 m3	6	High	Turns oil-market barrels into everyday volumes.
Work year and work-hours	~2,000 work-hours/year	6	High	Converts collective hours or staff-hours into worker-years.
Factory spare capacity	high-utilization plants ~= 90% to 97% of capacity	5	High	The spare margin controls how quickly shutdown backlogs can be cleared.
Hydrogen energy	~120 MJ/kg; compressed hydrogen ~= 5 MJ/L at 700 bar	5	High	High by mass, low by volume; important for hydrogen vehicle comparisons.
Resting and hard-exercise human power	resting ~= 100 W; hard exercise ~= 500 to 1,000 W	4	Medium	Useful for metabolic heat and human-performance comparisons.
Utility-scale battery module	one four-hour Megapack ~= 1 MW / 4 MWh; weight ~= 40 tonnes	4	Medium	A practical module-size peg for converting GWh-scale backup into containers, mass, and site scale.
World oil consumption	~1 x 10^8 barrels/day	4	Medium	A strong starting point for oil-market and chokepoint scale checks.
Emergency water and tanker capacity	Sphere minimum ~= 15 L/person/day; medium water tanker ~= 10,000 L	3	Medium	Turns displaced populations into daily truckload requirements for emergency water.
Full-year generation per MW	1 MW running all year ~= 8,800 MWh/year	3	Medium	Quick conversion from power-plant capacity to maximum annual generation before applying capacity factor.
Humanitarian food logistics	ration ~= 2,100 kcal/day; grain ~= 3,500 kcal/kg; truck ~= 20 tons; ship ~= 50,000+ tons	3	Medium	Useful for food-aid mass and transport scaling.
Reference river and waterfall flows	Niagara ~= 3 x 10^3 m3/s; lower Mississippi ~= 2 x 10^4 m3/s	3	Medium	Grounding comparisons for flash-flood flow estimates.
Space Shuttle stack mass	~2 x 10^6 kg at liftoff	3	Medium	A million-kilogram launch-system scale peg.
U.S. light-duty vehicle fleet	~2.7 x 10^8 light-duty vehicles	3	Medium	Useful for scaling per-car estimates to national fuel use, traffic, and emissions.
World population	~8 x 10^9 people	3	Medium	The default global denominator for worldwide scale checks.
Average car fill-up	~10 gallons	2	Medium	Turns oil flows into gas-pump equivalents.
Average U.S. light-duty fuel use	~10 gallons/week ~= 500 gallons/year	2	Medium	A compact peg for gasoline-car emissions and fuel-demand estimates.
Carbon to CO2 conversion	CO2 mass = carbon mass x 44/12 ~= 3.7	2	Medium	Prevents confusing carbon-mass emissions with CO2-mass emissions.
CPM and impression value	$1 CPM = $0.001 per impression	2	Medium	Converts tiny per-impression values into familiar ad-pricing units.
Daily half-hour annualization	30 min/day ~= 180 h/year	2	Medium	A tiny daily habit becomes a large annual time budget.
Earth surface area and land-ocean split	Earth surface ~= 5.1 x 10^8 km2; land ~= 1.5 x 10^8 km2; ocean ~= 3.6 x 10^8 km2	2	Medium	The main peg for global percentage, conservation, climate, and ocean-area problems.
Emergency department visit rate	~0.5 ED visits/person-year	2	Medium	Useful for turning a population into emergency-care volume before staffing assumptions.
Evaporative cooling and sweat	1 L evaporated sweat removes ~= 2.4 x 10^6 J	2	Medium	Connects athletic heat production to cooling by sweat evaporation.
Federal debt and interest scale	$40T debt; 1% of $1T = $10B; 1% of $40T = $400B	2	Medium	Turns debt balances and interest-rate pegs into annual carrying costs.
Gasoline density and carbon factor	gasoline ~= 0.75 kg/L; 1 gallon emits ~= 8.9 kg CO2 ~= 2.4 kg C	2	Medium	Turns gasoline volume into fuel mass, CO2, and carbon emissions.
Global ocean surface area	~3.6 x 10^14 m2	2	Medium	The denominator for sea-level-equivalent and ocean-surface spreading problems.
Jet fuel density and emissions factor	jet fuel ~= 0.8 kg/L; Jet-A/A1 emits ~= 3.16 kg CO2/kg fuel	2	Medium	Turns jet-fuel volume into mass and, when needed, combustion emissions.
Lithium carbonate and LCE	Li2CO3; 1 kg Li ~= 5.3 kg LCE; ~0.85 kg LCE/kWh	2	Medium	Connects lithium mine output to battery-pack counts.
Low-Earth-orbit speed	~8 km/s	2	Medium	Earth-centered orbital speed; distinguish it from Earth's roughly 30 km/s solar-orbit speed.
Mediterranean Sea surface area	~2.5 x 10^12 m2	2	Medium	A regional-scale area peg for Mediterranean ocean-heat calculations.
Natural-gas heat content and mass energy	natural gas ~= 1,000 Btu/ft3 ~= 50 MJ/kg	2	Medium	Converts fuel energy into gas volume or mass for data-center and power-plant estimates.
Natural-gas plant heat rate	modern combined-cycle gas ~= 6,000 to 7,000 Btu/kWh, roughly 50% to 55% efficient	2	Medium	Turns around-the-clock electric load into natural-gas fuel input.
Physician office visit rate	~3 office visits/person-year	2	Medium	The baseline peg for turning a covered population into outpatient-care volume.
Rain over land	1 inch over 1 square mile ~= 6.5 x 10^4 m3	2	Medium	Converts rainfall headlines into runoff-volume scale.
Rare-event testing trial count	n ~= 1/p for 63%; n ~= 3/p for 95%	2	Medium	A practical probability peg for rare behavior and detection problems.
Rocket propellant and dry mass fraction	first stage mostly propellant; dry mass ~= 5% to 10% of wet mass	2	Medium	Separates reusable hardware from propellant that is still burned every launch.
Semiconductor fab resource load	large fab ~= hundreds of MW to ~1 GW; millions of gallons/day	2	Medium	Frames fabs as city-scale utility loads rather than ordinary factories.
Solar mass and luminosity	solar mass ~= 2 x 10^30 kg; solar luminosity ~= 4 x 10^26 W	2	Medium	Core astronomy pegs for stellar mass and energy-output comparisons.
U.S. ton to kilogram	1 U.S. short ton ~= 2,000 lb ~= 907 kg	2	Medium	Useful when U.S. news uses tons but the calculation needs SI mass.
Water heat of vaporization	water ~= 2.3 MJ/kg to evaporate	2	Medium	Turns rejected heat into an evaporation-rate scale for cooling and water-use comparisons.
Water specific heat	~4 x 10^3 J/kg/degree C	2	Medium	The central peg for water and ocean heat-content estimates.
World annual energy demand	~6 x 10^20 J/year	2	Medium	Useful for comparing extreme physical energy totals with human civilization scale.

Equations and models

Model	Equation or structure	Articles	Importance	Why it helps
Kinetic energy	KE = 1/2 x mass x velocity^2	9	High	The core model for moving vehicles, satellites, ejecta, robots, and explosions.
Audience-hours and viewer-hours	viewer-hours = viewers x hours/viewer	7	High	Converts attendance or users into collective human attention.
Food-aid mass and transport loads	mass = people x kg/person/day x days; loads = mass / payload	7	High	Separates food need from shipping and trucking bottlenecks.
Face-search comparisons and false alerts	comparisons ~= face searches x watchlist size; false alerts ~= searches x FPIR	6	High	Distinguishes detections, searches, comparisons, alerts, and reviews.
Gravitational potential energy	E = m g h	6	High	The first-pass model for lifting water, people, aircraft, debris, or other masses through a height.
Battery backup duration	stored energy = power load x outage duration	5	High	The first calculation for backup batteries, UPS systems, and grid-storage duration.
Energy equals power times time	energy = power x time	5	High	The default conversion between rates and totals in energy problems.
Expected events and Poisson probability	lambda = rate x time; P(0) = e^-lambda; P(at least one) = 1 - e^-lambda	4	Medium	Use expected events first, then convert to zero-event or at-least-one probability.
Flow equals volume divided by time	flow = volume / time	4	Medium	The key bridge from total volume to rate or flood-wave intensity.
Per-household public cost	cost per household = annual public cost / households	4	Medium	Turns large public budget lines into household-scale comparisons.
Continuous industrial load	daily energy = average power x 24 hours	3	Medium	Useful for fabs, data centers, and other around-the-clock utility loads.
Kepler and orbital speed scaling	P^2 ~= a^3 / M; v ~= 30 km/s x sqrt(M/a)	3	Medium	Use years, AU, and solar masses for orbital Fermi estimates.
Price pass-through impact	impact = spending x exposed share x price change x pass-through	3	Medium	Turns import or tariff shocks into household-budget scale.
Testing compute and human review throughput	parallel agents = tasks x task time / time available; reviewers = tasks x review time / available time	3	Medium	Separates computation, storage, and human review bottlenecks.
Aerodynamic drag force and power	drag ~= 1/2 rho CdA v^2; power ~= 1/2 rho CdA v^3	2	Medium	At high speed, power scales with the cube of velocity.
Airliner cruise work	cruise work ~= distance x weight / (L/D)	2	Medium	A better mental model for aircraft cruise fuel than direct gravity-overcoming.
Annual interest cost	annual interest = principal debt x effective interest rate	2	Medium	The basic model for turning a debt stock into an annual carrying cost.
Area-depth and area-thickness volume	volume = area x depth or thickness	2	Medium	Used for rain, ocean layers, oil slicks, pools, ramps, and surface layers.
Capital to power capacity	MW = capital / dollars per MW	2	Medium	Turns infrastructure spending into eventual electrical load.
Combustion CO2 from fuel	CO2 = fuel energy x emissions factor, or fuel carbon x 44/12	2	Medium	Checks combustion emissions from either heat input or fuel mass.
Electrical power	P = I x V	2	Medium	A basic appliance and circuit power model.
Evacuation travel time	evacuation time = distance to safety / evacuation speed	2	Medium	Compares warning windows with realistic movement to high ground or exits.
Flood runoff flow	flow = watershed area x rain depth x runoff fraction / runoff time	2	Medium	Turns rainfall and watershed size into flash-flood discharge.
Healthcare utilization workload	clinical hours = people x visits/person-year x clinician-hours/visit	2	Medium	Turns a covered population into encounters, clinical labor time, and full-time worker equivalents.
Operational throughput	resource-days/outcome = resources x days / outcomes	2	Medium	Useful for arrests, cases, inspections, and other labor-throughput estimates.
Price-wage work-time burden	extra work = old spending / hourly pay x (price multiplier / wage multiplier - 1)	2	Medium	Converts a price-wage gap into extra hours required to buy the same item or basket.
Reserve coverage time	coverage time = available inventory / daily shortfall	2	Medium	The central model for emergency stocks, backlogs, storage buffers, and other inventory-versus-rate problems.
Reusable booster hardware saved	saved hardware = avoided boosters x dry booster mass	2	Medium	Counts reusable hardware saved, not propellant burned each launch.
Sea-level equivalent from added water	sea-level rise = added water volume / ocean surface area	2	Medium	Turns land-ice mass loss or added ocean volume into global mean sea-level rise.

As new articles are published, rerun node tools/build-reference.js so both the public repeat-peg tables and the complete usage ledger continue to reflect actual article usage.

Reference sources