## 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
• 10^18, 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 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 (29,028 or 29,035) feet = 8850 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 (2008) = 15-16 Terawatts = 4 Tcal/sec = 400 tons coal/s = 12 billion tons coal/year = 500 quads/year
• US share = 25% = 100 tons coal/s
• World Petroleum consumption = 30 billion barrels / year; US share = 25%

### 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
• 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 (2016) = 7.4 B
• World population (2040 estimated) = 9 B
• China (2016)= 1.4 B
• India (2016) = 1.3 B
• US Population (2016) = 0.324 B
• NYS Population (2016) = 19.8 M
• Greater Rochester Population = 1M
• Total ever lived = 50B - 100B+ (??), mostly since agriculture, 5000BCE
• Deaths (2014) = 57M /year
• Births (2014) = 143M /year

### Economic (most figures from The World Bank

• Gross World Product (Nominal, 2014) = \$78T = \$10,700 per capita.
In purchasing power parity (PPP) = \$107T = \$16,100 per capita.
• EU \$18.4T, PPP \$19.1T
• US GDP (2015) = \$17.9T = \$56,000 per capita (10/185).
PPP: \$17.9T, \$56,000 per capita, (5/185),.
• China GDP (Currency, 2015) = \$11T = \$8,000 per capita (72/185)
In purchasing power parity (PPP) = \$19.5T = \$14,100 per capita (84/185)
• India GDP (Currency, 2015) \$2.1T = \$1,600 per capita (140/185)
In purchasing power parity (PPP) = \$8.0T = \$6,200 per capita (122/183)
• US government expenditures (2015) = \$3.7T Federal, \$6.4T total.
• US government debt (2016) \$19.5T;
\$14T publicly held, \$6.1T of that is foreign, \$1.4T by China
• US Exports (goods and services, 2015) \$2.3T
• US Imports (goods and services, 2015) \$2.8T

The following figures are hard to find, especially for recent years, and somewhat uncertain.

• Total US Personal Income (2015) = \$15.4T = \$48K 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 (2015) = \$87T = \$272K per capita
Bottom 1/3 = 0%, Bottom 1/2 = 2.0%,
Top 1/5 = 88%, Top 1/10 = 75%, Top 1/100 = 35%, Top 1/1000 = 22%.
• And just for fun, check this out.

### 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)