Toolbox of the mind

Here is an image of the toolbox I carry around.  It contains a selection of useful tools to do odd jobs (mainly at other people's places).  There are tools for measuring, cutting, assembling/dismantling, and so on.

I have many more tools, but mostly the ones in the box are sufficient.

I have an invisible toolbox too:  the methods and numbers that are in my memory, ready to help with decisions in certain situations.

It's that mental toolbox that is discussed here.

Why?

Today's smartphones give instant access not only to data but to instructions for how to do things.

While that is helpful and good, it also means one has to be connected to consult them.  Observing the young, it seems to me that they always instantly look to the information on the internet to find aswers or instructions.  I found this a bit worrying:  they seem to have no mental toolbox, they are completely lost without their smartphone and a fast connection.

When you want to quickly get an estimate of something, it helps to know some numbers already.

Say your car weighs 1.2 tons but has an allowed road weight of 2.0 tons.  You want to load a large number of boxes of bottles of wine.  How much can you stow (assuming there is enough room) ?  It then helps to know that a box contains 6 bottles, that each bottle has 0.75 litres of wine, that wine is very nearly the same density as water, and that the glass of the bottle on average weighs about 0.5kg.  Thus a bottle weighs 0.75+0.5 = 1.25kg, a box will weigh 6x1.25 = 7.5kg.  To be on the safe side, given the carton itself also has a weight:  say 8.0kg for a box.  You weigh 75kg, you have an allowance of 2000-75-1200 = 725 kg.  Thus you can buy 725/8 = 90.625, rounded 90 boxes.  90 boxes is probably more than your car has room for, so you are safe.

But you needed to know the approximate weight of a full bottle of wine, i.e. 1.25kg.

There are many other numbers that are useful.  This list is not exhaustive, but I hope you find it interesting.

Concepts

Statistics and probability

Units

List of values

Maths

pi (3.1415926535… )
mnemonic: the number of letters in the phrase:
English:  “See, I have a rhyme assisting my feeble brain”
French:  “que j'aime à faire connaitre un nombre utile aux sages”

Mechanics

tensile strength of steel >50kg/mm2
that is kg force, i.e. weight, which is not metric; the metric value is >500N/mm2.  If you want to know what a steel cable can suport in weight, then the kg force value is quicker to use.

a calorie = 4.186 Joule = ~4.2 Joule.
A calorie is a non-metric value of energy.
Food packages often still show a value in kCal (kilo calories) next to the value in kJ (kiloJoule).

1kWh = 3'600'00 J
the kWh is yet another non-metric unit.

energy in fossil fuels ~10'000kCal/kg, or 42'000 MJ/kg.
Most fossil fuels, from alcohol upwards, to heavy stuff, is made from carbon and hydrogen in the form of chains:

H | … - C -… | H

and therefore as soon as there are a few of these units, the burning of the bits at the end of the chain does not make much of a difference, it's all in the burning of the H-C-H units.

LED lights: ~100 lumen/W
we have living room lights that use 28W and produce 3000lumen.
Note that many manufacturers still put “W equivalent” figures on their light bulb packages.  This is confusing and misleading:  the light coming from a classic, incandescent light bulb of 60W is about 600 lumen, i.e. approximately 10 lumen per watt instead of 100.  But the impression of quantity of light you get is very much dependent also on the colour of the light (from yellowish to blueish white) and the shape of the light source (bulb, strip, flat panel, … )
LED lights are not very much more efficient than fluorescent lights, their main advantage is in the shapes they can take and the long life.
Life span of a LED source though is limited not so much by the LED technology as by that of the electronics needed to convert the 240V alternating current into the low voltage direct current the LED needs.  This conversion electronics is inside the bulb.

seconds in a day: 86'400 (mnemonic: decreasing order 8,6,4.)

solar constant: energy emitted by the sun through 1m², at the distance of the Earth: 1360W

solar panel energy conversion: 20 to 25 %

atmospheric pressure: about same as in a column of 10m of water. Also 1kg force on 1cm²

power of a human being: farm labourer needs 2500kCal per day; or 2500x4.2 = 10'500 MJ, say a normal person would use 8'640MJ, then dividing that by 86'400 seconds in a day, gives 100W. It is somewhere between 60W and 120W.

medium sized car on a flat road at reasonable speed: 40kW.

40'000km circumference of the Earth (by definition)

394'000km to the Moon

154 million km to the Sun

8 minutes for light to travel from the Sun to us

300'000km/s speed of light in a vacuum; 150'000km/s speed in a cable.

1.2kg/m3 density of air at standard temperature and pressure

4ºC temperature at which water is densest

nearly all household liquids are 1kg/litre. Oil is more like 0.75.

−273ºC absolute zero

25.4mm in an inch (by the modern definition of the inch)

1.609 km to the mile

1.852 km to the nautical mile; a nautical mile is one minute of arc on the Earth's surface; a km is one minute of a grade (100 grades in a right angle, 100 grade-minutes in a grade)

440Hz the standard middle note of the piano

16-20'000Hz the limits of the human ear (when your age, not my age)

8'760 hours in a year (menmonic:  8-7-6, descending by 1.

365.25 days in a year

75 cm standard (dinner) table height

1 bar of pressure equals 10m of water (approximately).  To get water to the faucets at the top of a 40m high building you need at least 4bars at the bottom.  A bar is very close to atmospheric pressure at sea level, where it is 1.013bars.  The bar is defined as 100'000Pascals (100'000N.m²).  In diving you get 1 bar extra pressure per 10m you go down.

21cm wavelength of hydrogen emission, therefore also thought ot be the best wavelength to listen on or to transmit on for extraterrestrial comms.

6x10²³ Avogadro's number (the 23 is the important bit)

343 m/second speed of sound at standard air pressure;  or about 3 seconds to cover 1km, therefore counting the seconds between seeing a lightning and hearing the thunder will tell you how far away the lighting struck:  divide seconds by 3.

10⁸⁰ approximate number of particles in the universe

kilo - mega - giga - tera ; milli - micro - nano - pico

210mm x 297mm size of A4 paper ( A0 is 1m², with proportions 1/√2 ; 16 pages A4 cover 1m²)

About 4'000 characters in a reasonable font size on an A4 page.

2.5m the legal height of a room; 3.0m the height of a “floor” in a building (the thickness of a floor being approx. 50cm). This lets you quickly estimate the height of a building by counting the number of floors.

8mm horizontal pitch of LEGO, 9.6mm vertical.

Law of Titius—Bode (not really a law, but a “reasonable fit”) for the distances of the planets from the sun:

To remember this: the strength of the operator goes down (power-multiplication-addition) while the number used goes up (2-3-4).

Now divide by 10 to get astronomical units (distance Sun-Earth)

0.4 0.7 1 1.6 2.8 5.2 …

and that gives the distance from the Sun of Mercury, Venus, Earth (=1 of course), Mars, the asteroid belt (2.8) and Jupiter.

Useful when going out to find mysterious monoliths.

energy in a standard, good quality AA battery. It gives 1.5V. But I don't know the energy. And mAh is NOT an energy unless one knows the voltage.

111km is a degree of latitude.

nearly all kitchen liquids are 1 kg per litre

power: human=60W;

A MAC address has 12 hexadecimal digits.  A hexadecimal digit encodes 4 bits.  Thus a MAC address is a 4×12=48 bit number.  That gives 2^48 possible MAC addresses.  How much is that in decimal?

Well, 10^0.30103 = 2  (0.30103 being the logarithm of 2 in base 10, a number surely everyone knows…)

Therefore 2^48 = (10^0.30103)^48 = 10^(0.30103×48) = 10^14.45

Or a 14 digit number.  1G=10^9, 1T=10^12.  A hundred tera = 10^14.  Or 10^14/10^10=10'000 per inhabitant of the planet.  (actually 12'000, per the table of inverses, since we are not 10 but only 8 billion people)

Other Stuff

A4 paper formats