Reference

Useful constants, equations, and notation habits.

Fermi problems 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	17	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	14	High	Used whenever a volume needs to become mass, or mass needs to become volume.
Kilowatt-hour to joules	1 kWh = 3.6 x 10^6 J	9	High	Essential for translating electricity use into joules.
U.S. households	~1.3 x 10^8 households	9	High	Useful for homes, pools, utilities, vehicles, consumer goods, and ownership rates.
Gasoline and fuel energy	gasoline ~= 1.3 x 10^8 J/gallon; jet fuel ~= 35 MJ/L	8	High	Turns unfamiliar energy totals into fuel-equivalent comparisons.
Household electricity and average household power	~10,000 kWh/year; ~1.2 kW continuous	8	High	The bridge between grid-scale energy and a familiar home-scale comparison.
U.S. population	~3.4 x 10^8 people	8	High	The default national denominator for U.S.-scale estimates.
GW, MW, and kW conversions	1 GW = 1,000 MW = 1,000,000 kW	7	High	A recurring source of thousandfold mistakes in power problems.
Large power plant	~1 GW	5	High	A useful public-scale comparison for grid power.
Mph to m/s	1 mph ~= 0.45 m/s	5	High	Converts road, wind, aviation, and speed-record problems into SI units.
Rare-event testing trial count	n ~= 1/p for 63%; n ~= 3/p for 95%	5	High	A practical probability peg for rare behavior and detection problems.
Factory spare capacity	high-utilization plants ~= 90% to 97% of capacity	4	Medium	The spare margin controls how quickly shutdown backlogs can be cleared.
Hydrogen energy	~120 MJ/kg; compressed hydrogen ~= 5 MJ/L at 700 bar	4	Medium	High by mass, low by volume; important for hydrogen vehicle comparisons.
World population	~8 x 10^9 people	4	Medium	The default global denominator for worldwide scale checks.
Human lifetime and calendar-year conversions	lifetime ~= 80 years; year ~= 8.8 x 10^3 hours	3	Medium	Useful for turning collective human time into intuitive comparisons.
Resting and hard-exercise human power	resting ~= 100 W; hard exercise ~= 500 to 1,000 W	3	Medium	Useful for metabolic heat and human-performance comparisons.
Work year and work-hours	~2,000 work-hours/year	3	Medium	Converts collective hours or staff-hours into worker-years.
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.
CPM and impression value	$1 CPM = $0.001 per impression	2	Medium	Converts tiny per-impression values into familiar ad-pricing units.
Full-year generation per MW	1 MW running all year ~= 8,800 MWh/year	2	Medium	Quick conversion from power-plant capacity to maximum annual generation before applying capacity factor.
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.
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.
Oil barrel	1 barrel = 42 gallons ~= 159 L ~= 0.159 m3	2	Medium	Turns oil-market barrels into everyday volumes.
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. light-duty vehicle fleet	~2.7 x 10^8 light-duty vehicles	2	Medium	Useful for scaling per-car estimates to national fuel use, traffic, and emissions.
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	6	High	The core model for moving vehicles, satellites, ejecta, robots, and explosions.
Audience-hours and viewer-hours	viewer-hours = viewers x hours/viewer	5	High	Converts attendance or users into collective human attention.
Energy equals power times time	energy = power x time	4	Medium	The default conversion between rates and totals in energy problems.
Expected events and Poisson probability	lambda = rate x time; P(at least one) = 1 - e^-lambda	4	Medium	Use expected events first, then convert to probability.
Food-aid mass and transport loads	mass = people x kg/person/day x days; loads = mass / payload	4	Medium	Separates food need from shipping and trucking bottlenecks.
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.
Administrative workload and completion time	staff-hours = cases x hours/case; weeks = staff-hours / worker capacity	2	Medium	Separates theoretical batch processing from individual review and exception handling.
Airliner cruise work	cruise work ~= distance x weight / (L/D)	2	Medium	A better mental model for aircraft cruise fuel than direct gravity-overcoming.
Capital to power capacity	MW = capital / dollars per MW	2	Medium	Turns infrastructure spending into eventual electrical load.
Continuous industrial load	daily energy = average power x 24 hours	2	Medium	Useful for fabs, data centers, and other around-the-clock utility loads.
Evacuation travel time	evacuation time = distance to safety / evacuation speed	2	Medium	Compares warning windows with realistic movement to high ground or exits.
Face-search comparisons and false alerts	comparisons ~= face searches x watchlist size; false alerts ~= searches x FPIR	2	Medium	Distinguishes detections, searches, comparisons, alerts, and reviews.
Flow equals volume divided by time	flow = volume / time	2	Medium	The key bridge from total volume to rate or flood-wave intensity.
Kepler and orbital speed scaling	P^2 ~= a^3 / M; v ~= 30 km/s x sqrt(M/a)	2	Medium	Use years, AU, and solar masses for orbital Fermi estimates.
Per-household public cost	cost per household = annual public cost / households	2	Medium	Turns large public budget lines into household-scale comparisons.

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