Approximate Numbers Everyone Should Know

The Scale of Things

"Space is big. Really big. You just won't believe how vastly, hugely, mind-bogglingly big it is. I mean, you may think it's a long way down the road to the chemist, but that's just peanuts to space."
Douglas Adams, The Hitchiker's Guide to the Galaxy .

• Human Scale = (1.6 x ) 1 meter
  
  
• Mouse Scale = 10^-1 meter = 10 cm
• Bug Scale = 10^-2 meter = 1 cm
• Grain-of-sand Scale = 10^-3 meter = 1 mm
• Cell Scale = 10^-5 meter = 10 microns
• Atomic Scale = 10^-10 meter = 1 Angstrom
• Nuclear Scale = 10^-15 meter
• Planck Scale = (1.6 x ) 10^-35 meter (structure of space breaks down ??)
  
  
• House Scale = 10 meter
• City Block Scale = 100 meter
• Village Scale = 1000 meter = 1 km
• Metropolis Scale = 10^5 meter = 100 km
• Earth Scale = (1.3 x) 10^7 meter = 10K km
• Earth Orbit Scale = 10^11 meter = 100M km (a bit inside Venus)
• Solar System Scale = 10^13 meter = 10B km (a bit past Pluto)
• Interstellar Scale = 10^16 meter = 10T km = 1 light year
• Galactic Scale = 10^21 meter = 100K ly
• Intergalactic Scale = 10^23 meter = 10M ly (local group)
• Observable Universe Scale = 10^26 meter = 10B ly
  (This one is a bit tricky as the computed parallex/brightness distance given above does not equal the round-trip light time, which does not equal a "where is it now" extrapolation. For distances under a few billion ly, all these values are equal to below the accuracy of our best measurements).
• Universe in Planck Lengths 10^61

The River of Time

• Human event scale = 1 second
  
  
• Eye blink = 1/10 = .1 second
• Time for a yellow-jacket's wings to beat = 1/100 = .01 = 10^-2 second
• Time for sound to travel 1 foot = Time for a balloon to pop = camera flash time = 1 millisecond = 10^-3 second
• AM Radio wave oscillation time = 1 microsecond = 10^-6 second
• Time for light to travel 1 foot = Computer operation time (2000) = 1 nanosecond = 10^-9 second
• Chemical reaction transition time = .1 picosecond = 10^-13 second
• Oscillation time of visible light = (2 x ) 1 femtosecond = 10^-15 second
• Nuclear reaction transition time = 1 attosecond = 10^-18 second
• Planck Time (shortest physically meaningful period??) = (.5 x ) 10^-43 second
  
  
• Minute = (.6 x ) 10^2 second
• Hour = (3.6 x ) 10^3 second
• Day = (.86 x ) 10^5 second
• Week = (.6 x ) 10^6 second
• Year = (3.1 x ) 10^7 second
• Human lifetime = 70 (100) year = (2 x ) 10^9 second
• Civilization = 6000 years = (2 x ) 10^11 second
• Behaviorally modern humans = 50,000 years = (1.5 x ) 10^12 second
• Anatomically modern humans = 200,000 years = (.6 x ) 10^13 second
• Genus Homo = 2M years = (.6 x ) 10^14 second
• Monkey-like primates = 40M years = (1.2 x) 10^15 second
• Land vertebrates (amphibians) = (1.2 x) 400M years = (1.2 x) 10^16 second
• Complex multi-cellular life = 500M years = (1.5 x ) 10^16 second
• Simple multi-cellular life = 1B years = (3 x ) 10^16 second
• Oxygen respiration 2B years = (.6 x ) 10^17 second
• Photosynthesis = 3B? years = 10^17 second
• Life = 4B? years = (1.2 x ) 10^17 second
• Age of Earth = 4.6B years = (1.4 x ) 10^17 second
• Age of Universe = 13.7B years = (.4 x ) 10^18 second
• Age of Universe in Planck units = (2 x ) 10^61
• Current space-time volume of Visible Universe in Planck units = 10^245

Mathematics

• Pi = 3.14 (3.14159 265 3589793 23846 2643383 2795 02884)
• e = 2.7 (2.71828 1828 459 045)
• sqrt(2) = 1.4 (1.4142 1356 2373)
• sqrt(3) = 1.73 (1.732 0508 07568877)
• sqrt(10) = 3.16 (3.16 227766 0186)
• Golden Ratio (Phi) 1.62 = (1 + sqrt(5)) / 2
• All the squares from 1 to 20
  1 4 9 16 25 36 49 64 81 100 121 144 169 196 225 256 289 324 361 400
• Powers of 2 up to 12 or so
  1 2 4 8 16 32 64 128 256 512 1024 2048 4096
• P(x < 1 std) = 84% for normal distribution
• P(x < 2 std) = 98%
• P(x < 3 std) = 99.9%
• Approximate base 10 log pattern of Fibbonacci sequence
  log(2) = .3, log(3) = .5, log(5) = .7, log(8) = .9, log(13) = 1.1
  (10 points to the first person to explain this)
• Approximate ratio of circle to square = 3/4 (Pi/4 = .78540)
• Approximate ration of sphere to cube = 1/2 (Pi/6 = .52360)

Names and prefixes for big and small numbers

• 10^3, thousand, kilo, milli
• 10^6, million, mega, micro
• 10^9, billion, giga, nano
• 10^12, trillion, tera, pico
• 10^15, quadrillion, peta, femto
• 10^18, quintillion, exa, atto
• 10^21, sextillion, zetta, zepto
• 10^24, septillion, yotta, yocto

Common Unit conversions

• 1 inch = 2.54 cm.
• 1 meter = 39.4 inches (39.3701)
• 1 mile = 1.61 km (= 5280 ft.)
• 1 light year = 6 trillion miles = 10 trillion km.
• 1 liter = 1(.06) quarts = .264 gallons
• 1 tsp = 5 ml
• 1 barrel oil = 42 US gallons = 160 (159) liters
• 1 kg. = 2.2 lbs.
• 1 oz. = 28(.35) grams
• 1 lb. = 450 (453.6) grams
• 1 calorie = 4.2 joules
• 1 btu = 1000 (1055) joules
• 1 quad = 1 quadrillion (10^15) btu = 10^18 joules
• 1 megaton = 10^15 cal.
• 1 kg mc^2 = 20 megatons
• 1 horsepower = 550 ft-lbs/sec = 750 watts (746)
• 1 atmosphere = 15 psi (14.7) = 100,000 pascals = 34 ft of water
• 1 gigapascal = 150,000 psi
• 1 Celcius degree = 9/5 Farenheit degrees (exactly)
• 0 C = 273 K = 32 F
• 1 year = 31 million seconds
• Speed of light = 186,000 miles/s = 300,000 km/s
• Avogadro's number = 6.02 * 10^23 molecules/mole

The Earth

• Diameter = 8000 miles (7930) = 13,000 km. (12,800)
• 1 degree on surface = 70 miles (69.2)
• Area = 200 million square miles (198M) = 500M km^2 (515M)
• Oceans = 71% of surface
• Highest point = 29,000 (29,029 or 29,035) feet = 8850 (8848) m.
  (Surveying and GPS values don't agree. Look up uncertainty of the geoid for a discussion)
• Lowest point = -36,000 feet = -11,000 m.
• Age = 4.6 billion years
• Distance from sun = 93 million miles (avg.)
• Gravity = 32 ft/s^2 (32.2) = 10 m/s^2 (9.8) = 1 G
• Density 5.5 gm/cm^3
• Solar insolation = 1000W/m^2 (surface); 1366W/m^2 (space)

Astronomy

• Sun diameter = 860,000 miles = about 100 earths
• Sun distance = 93M miles = 1 AU
• Sun surface temperature = 5800K = 5500C = 10,000F
• Sun mean density = 1.4 gm/cm^3
• Moon distance from earth = 240,000 miles
• Moon orbital period = 29.5 days full to full; 27.3 days absolute
• Moon diameter = 2000 miles (2160) = about 1/4 earth
• Moon gravity = 1/6 earth
• Mars diameter = 4,000 miles (4200) = about 1/2 earth
• Mars distance from sun = 140M miles = 1.5 AU
• Mars orbital period = 1.9 years
• Mars gravity = 3/8 earth (.376)
• Jupiter diameter = 89,000 miles = about 10 earths (11)
• Jupiter distance from sun = 500M miles (480M) = 5 AU
• Jupiter orbital period = 12 years (11.86)
• Jupiter gravity = 2.5 earth
• Distance to nearest star = 4.3 light years = 25 trillion miles
• Diameter of Milky Way = 100,000 light years
• Stars in Milky Way = 200-400 billion (??)
• Distance to next major galaxy (Andromeda) 2.5M light years
• Age of universe = 14 billion years (13.7)

Water

• Density = 1gm/cc = 1kg/l = 1 lb/pint
• Boiling point = 100C = 212F
• Freezing point = 0C = 32F
• Heat of vaporization = 540 cal/gm
• Heat of fusion = 70 cal/gm
• Speed of sound = about 1 mile/s (4800 feet/s)

Air

• Composition: 78% N2 + 21% O2 + 1 % Ar + .04% C02, + other traces.
• Density = 1.2 gm/l = 1.2 kg/m^3 = 75 lbs/1000ft^3 = 1/800 water
• Speed of sound = 1100 ft/s = 340 m/s
• Sea-level pressure = 15 psi (14.7) = .1 Mpa
• Half pressure height = 18,000 feet, approximately exponential.

Fire

• Energy from burning carbon, oil, gas = 10,000 cal/gm
• Energy from burning wood, carbohydrates = 5000 cal/gm
• Energy content of TNT = 1000 cal/gm
• Energy content of completely fissioned Uranium = 3M * coal = 20B cal/gm
• Fissionable portion of natural Uranium = 0.7%
• Energy from E = mc^2 = 2B * coal = 20 trillion cal/gm
• Direct sunlight at earth's surface = 1000 W/m^2 = 250 cal/m^2/sec
  (1366 W/m^s in space, 150-300 W/m^2 temperate average over full year)
• Total world energy usage (2018) = 18 Terawatts = 4.3 Tcal/sec = 430 tons oil/s = 14 billion tons oil/year = 590 quads/year
• US share = 1/6
• World Petroleum consumption = 36 billion barrels / year; US share = 1/5

Melting points

• Mercury = -40C = -40F
• Water 0C
• Aluminum 650C = 1200F
• Copper 1100C (1080) = 2000F >= brass, bronze, silver, gold
• Iron 1500C (1530) = 2800F >= steel
• Tungsten 3400C (3420) = 6200F

Boiling points

• Helium = 4K
• Hydrogen 20K
• Nitrogen = 77K
• Water = 373K = 100C

Material Densities

• Wood .5 - 1 gm/cc
• Water = 1 gm/cc
• Organic compounds, Plastics = .9 - 1.5 gm/cc
• Stone, concrete, brick, glass = 2 - 3 gm/cc
• Aluminum = 2.7 gm/cc
• Titanium = 4.3 gm/cc
• Iron, steel = 8 gm/cc
• Copper = 9 gm/cc
• Lead = 11 gm/cc
• Gold, tungsten, uranium = 19 gm/cc

Common material strengths

• Aluminum alloys = 20,000 - 80,000 psi (tensile, compressive, sheer)
• Steel = 40,000 - 300,000+ psi
• Wood = 4,000 - 8,000 psi compressive, <= 20,000 psi tensile
• Concrete = 2,000 - 4,000+ psi compressive, 0 - 200 psi tensile
• Stone = 2,000 (limestone) - 20,000+ (granite) compressive
• Polystyrene = 6,000 psi
• Silk 70,000 psi
• Kevlar up to 500,000 psi tensile
• Glass fiber up to 700,000 psi tensile (but usually closer to 100,000)
• Carbon fiber up to 800,000 psi tensile
• Carbon Nanotubes: 500,000 psi macroscopic test; 9,000,000 psi microscopic test; 45,000,000 psi theoretical

Humanity

• World Population (2019) = 7.7 B
• World population (2040 estimated) = 9 B
• China (2019)= 1.4 B (1.43 B)
• India (2019) = 1.4 B (1.37 B)
• US Population (2019) = 0.329 B
• NYS Population (2019) = 19.5 M
• Greater Rochester Population = 1M
• Total ever lived = 50B - 100B+ (??), mostly since agriculture, 5000BCE
• Deaths (2019) = 60M /year
• Births (2019) = 142M /year

Economic (most figures from The Internatinal Monetary Fund

• Gross World Product (2019) $88T = $11,400 per capita.
  In purchasing power parity (PPP) = $143T = $20,400 per capita.
• EU GDP (2019) $18.7T = $37,400/capita; PPP: $19.0T = $38,100/capita
• US GDP (2019) $21.3T (1st) = $65,000 per capita (8th).
  PPP: $21.3T (2nd), $65,000 per capita, (12th),.
• China GDP (2019) = $14T (2nd) = $10,000 per capita (72nd)
  PPP: $27.5T (1st) = $19,600 per capita (75th)
• India GDP (2019) $3.0T (5th) = $2,200 per capita (145th)
  PPP: = $11.4T = $8,400 per capita (126th)
• US government expenditures (2019) = $4.5T Federal, $8.1T total.
• US government debt (2019) $22.0T;
  $16.1T publicly held, $6.3T of that is foreign, $1.1T by China, $1.06T by Japan
• US Exports (goods and services, 2018) $2.5T
• US Imports (goods and services, 2018) $3.1T

The following figures are hard to find, especially for recent years, and somewhat uncertain. Ranges represent highs and lows from different sources widely considered reputable. Some variation is due to differences in definitions (e.g. is projected social security income counted as "wealth")

• Total US Personal Income (2018) = $17.5T = $50K per capita
  Bottom 1/10 = 1%, Top 1/10 = 50%, Top 1/100 = 20%, Top 1/1000 = 10%, Top 1/10,000 = 5%.
• Total US Household Wealth (2018) = $100T = $304K per capita
  Bottom 1/3 = 0%, Bottom 1/2 = 2.0%,
  Top 1/5 = 88%-93%, Top 1/10 = 70%-78%, Top 1/100 = 32%-39%, Top 1/1000 = 22%.

Simple Formulas

• Area of circle = pi * r^2
• Area of triangle = 1/2 * base * height
• Area of a sphere = 4 * pi * r^2
• Volume of a sphere = 4/3 * pi * r^3
• Volume of a pyramid, cone etc. = 1/3 * base_area * height
• F = m * a = m * dv/dt = dp/dt (Newton's second law)
• E = 1/2 * m * v^2 (Kinetic energy)
• F = k * x (Hook's law for springs)
• p * V = n * R * T (Ideal gas law)
• F = G * m1 * m2 / r^2 (Law of universal gravitation)
• F = k * q1 * q2 / r^2 (Coulomb's law)
• V = I * R (Ohm's law)
• q = C * V; I = C * dV/dt (Capacitor law)
• V = L * dI/dt (Inductor law)

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