Lightning distance
Count the seconds from flash to thunder, divide by 3 for kilometres (by 5 for miles).
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Mercator squares the stretch
On a Mercator map, area is exaggerated by 1/cos²(latitude) — four times at 60°, fifteen times at 75°.
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Air cools as you climb
Temperature drops about 6.5 °C for every 1,000 m of altitude.
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Pressure halves every ~5.5 km
Atmospheric pressure falls by half roughly every 5,500 m you climb.
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Ten metres of water = one atmosphere
Every 10 m of seawater adds about 1 atmosphere of pressure.
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Boiling point falls with altitude
Water boils about 1 °C cooler for every 300 m of altitude.
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Dew point is the comfort number
Dew point above 20 °C feels oppressive to most people; below 10 °C feels dry.
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Fog watch
Clear night + calm air + temperature within ~2.5 °C of the dew point = expect fog.
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Solar panels: 3–5 sun-hours a day
1 kW of panels yields roughly 3–5 kWh per day in temperate climates, averaged over the year.
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The sun moves 15° per hour
The sun (and the stars) sweep 15 degrees of sky per hour.
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"Feels like" switches formula, not just number
Below about 10°C, wind chill drives "feels like." Above about 27°C, humidity (heat index) does. In between, it’s just the thermometer.
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Frostbite risk is about minutes, not degrees alone
At wind-chill near −28 °C, frostbite can start in under 30 minutes on exposed skin; near −40 °C, in under 10.
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Most time zones are whole hours — not all
A handful of zones sit on the half-hour or even quarter-hour: India (+5:30), Nepal (+5:45), parts of Australia (+9:30, +8:45).
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A 100W device left on all month costs about $10
100 W running continuously for a month is roughly 72 kWh — about $10 at a typical $0.15/kWh rate.
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At noon the sun is 90° minus your latitude, plus the sun’s declination, above the horizon
At solar noon the sun stands 90° minus your latitude plus the sun’s declination above the horizon: add 23° in June, nothing at the equinoxes, subtract 23° in December.
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Solar noon is later by 4 minutes for every degree you are west of your time zone’s meridian
Solar noon is 12:00 plus 4 minutes for every degree of longitude west of the middle of your time zone, plus or minus up to 16 minutes through the year, plus an hour in summer time.
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The Moon rises about 50 minutes later each day
The Moon rises about 50 minutes later each day, so tomorrow’s moonrise is about three-quarters of an hour after today’s.
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The Moon is highest about an hour after the Sun for every 15° it is east of it
The Moon is highest about one hour after the Sun for every 15° it is east of the Sun: a new moon at noon, a first quarter at 6 pm, a full moon at midnight and a last quarter at 6 am.
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There are between four and seven eclipses every year
In any calendar year there are at least four and at most seven eclipses, solar and lunar together, and they come in two or three clusters about six months apart.
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A supermoon is at most 8% wider than an average full moon
The Moon is 14% wider at its nearest than at its farthest, but a supermoon is only up to 8% wider than an average full moon, which you cannot see without the two side by side.
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Where you charge beats what you drive
Home off-peak to public rapid is roughly a tenfold difference in price per kWh. No two mainstream EVs differ in efficiency by even twofold — so the socket sets your running cost, not the car.
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The tide moves in twelfths
From high water the tide falls 1, 2, 3, 3, 2 then 1 twelfth of its range in each sixth of the time. Never more than 2.6% of the range away from the truth — but only if the fall really does take six equal hours.
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Each magnitude step is ten times the shaking and thirty times the energy
One step of magnitude is 10 times the ground motion and about 32 times the energy. Two steps is 100 times the motion and 1,000 times the energy, so it takes a thousand magnitude 5s to match one magnitude 7.
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Count the seconds between the jolts and multiply by eight
Count the seconds between the first jolt and the heavier shaking behind it, and multiply by 8: that is roughly how many kilometres away the quake began.
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Come up no faster than 30 feet (9 metres) a minute
Ascend at no more than about 9–10 m (30 ft) a minute, slower in the last few metres, and pause for about three minutes at 5 m (15 ft) on any dive deeper than about 10 m. Your dive computer and training set the real figure.
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Air reaches its oxygen limit at about 57 metres
The deepest safe depth for a gas is 10 × (1.4 ÷ oxygen fraction − 1) metres: about 57 m on air and about 34 m on nitrox 32.
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Radiators decide it, not the cold
A radiator gives about a third of its rated output at 45 °C flow instead of 70 °C. Emitter size, not outdoor temperature, is what stops a heat pump working.
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A party wall loses nothing
Only count the surfaces with cold on the other side. A mid terrace has half the exposed wall of a detached house the same size, and needs roughly a third less heat.
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The bolt-torque shortcut
Tightening torque ≈ 0.2 × preload × bolt diameter (dry steel).
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Torque mostly fights friction
Roughly 90% of tightening torque is lost to friction; only ~10% actually stretches the bolt.
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Deflection limits: span/250
Keep beam deflection under span/250 generally, span/360 where people will notice a bouncy floor.
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Depth beats width, cubed
Double a beam’s depth and its deflection drops 8-fold. Double its width and it only halves.
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Keep water at 1–2.5 m/s
Size pipes so water flows between about 1 and 2.5 m/s.
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One pipe size up fixes most pressure problems
Friction pressure drop scales with 1/d⁵ — a modest increase in diameter slashes losses.
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Laminar below 2,300
Pipe flow is laminar below Re ≈ 2,300 and reliably turbulent above ~4,000.
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Steel grows 1 mm per metre per 85 °C
A metre of steel expands about 1 mm when heated ~85 °C.
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A stretched O-ring is a thinner O-ring
Cord diameter falls as 1/√(1+stretch). Fit a ring with 5% stretch and it arrives 2.4% thinner than the groove was cut for — so the squeeze you get is always under the squeeze you asked for.
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O-ring squeeze: 20–30% static
Compress an O-ring 20–30% of its cross-section for static seals, only 10–20% for dynamic.
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Double the load, an eighth of the life
Ball-bearing fatigue life scales with 1/load³.
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Safety factor 2 — or 4 with people involved
Use a safety factor around 2 for well-known static loads, 4 or more for dynamic loads, uncertainty, or human safety.
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A litre of water is a kilogram
1 L of water ≈ 1 kg ≈ 10 N of weight. A cubic metre is a tonne.
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Miles per gallon hides how much you save
Fuel saved goes as the difference of the RECIPROCALS. Taking a 10 mpg vehicle to 20 saves 5 gallons per 100 miles; taking a 30 mpg car to 60 saves 1.7 — the same doubling, a third of the fuel.
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An inch is exactly 25.4 mm
Many “round” imperial–metric pairs are exact by definition (1 in = 25.4 mm, 1 lb = 0.453 592 37 kg). Others are measured.
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Density and viscosity are independent
Mercury is 13× denser than water yet flows just as easily; honey is barely denser than water yet far more viscous.
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Diesel compresses harder because it doesn’t pre-mix
Diesel engines run compression ratios of 16–22:1; petrol engines top out around 8–12:1.
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Every gear mesh flips the direction
One meshed gear pair reverses rotation direction. An even number of stages ends up spinning the same way as the input; an odd number reverses it.
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A cylinder is weaker coming back, and faster for it
Retract works on the annulus, not the bore: the rod’s cross-section is missing from the piston face. A 100 mm bore with a 50 mm rod gives 75% of the force on the way back — and 133% of the speed for the same pump flow.
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On retract, the return line carries more oil than the pump sends
Retracting, the small annulus end takes the pump flow while the big bore end empties. A 100 mm bore with a 50 mm rod expels about 53 L/min while the pump delivers 40 — so the return line is the one that has to be bigger.
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Cycle a spring near its own resonance and it stops behaving like a spring
A compression spring has a surge frequency of its own, ½·√(k/m) with its full active mass. Work it above about a thirteenth of that and the coils stop moving together, pass a wave along the spring instead, and clash.
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A spring usually runs out of travel before it runs out of strength
Solid height is the wire diameter times the total coil count. Once the coils touch, the spring is a solid bar — the load path goes rigid with no warning, and the stress check that said it was fine no longer applies.
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One planetary set, three ratios — pick which member you hold
Hold the ring and a sun-in, carrier-out set reduces by 1 + ring/sun. Hold the sun instead and the same metal gives 1 + sun/ring. Hold the carrier and it reverses, at ring/sun. Same three gears, three different answers.
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Ring minus sun must be even, and the planets must divide
The planet has (ring − sun) ÷ 2 teeth, so an odd difference asks for half a tooth and cannot be built. Separately, sun + ring must divide by the number of planets or they will not space equally.
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Below 17 teeth, the cutter eats the tooth root
A standard full-depth gear needs about 2/sin²α teeth before the cutter starts undercutting the root — 17 at a 20° pressure angle, but 32 at 14.5° and only 12 at 25°.
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A 4-stroke camshaft turns at half crank speed
Because four strokes take two full crankshaft revolutions, the camshaft that opens each valve exactly once per cycle must turn at exactly half the crankshaft’s speed — never the same speed.
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A diesel wins on compression ratio, not on its cycle
At the SAME compression ratio the Diesel cycle is worse than the Otto one — 51% against 61% at 10.5:1. It wins in the world by running 18:1, which a petrol engine cannot.
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A Wankel rotor turns at a third of its shaft
The rotor turns once for every three turns of the eccentric shaft — but it has three faces, so each rotor still fires once per shaft revolution: a two-stroke firing rate with nothing reciprocating.
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It is piston speed that limits an engine, not rpm
Mean piston speed is 2 × stroke × rpm. Production engines sit under about 20 m/s, competition engines run 20–25, and past roughly 25 m/s cast pistons stop surviving — so a redline means nothing until you know the stroke.
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Spark advance is a clock, not an angle
Combustion takes roughly a fixed time — call it 2 ms — so the advance needed to put peak pressure just after TDC grows in direct proportion to rpm: about 12° at 1,000 rpm and 72° at 6,000.
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Double the wire, sixteen times the spring rate
A helical spring’s stiffness scales with wire diameter to the fourth power — doubling it multiplies the rate by 16.
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Double the piston diameter, quadruple the force
In a hydraulic cylinder, force scales with the piston diameter squared, not the diameter itself.
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The groove, not the ring, usually fails first
A circlip’s rated thrust load is set by the groove wall in the shaft or bore yielding under the ring’s edge, not by the ring itself breaking.
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Every circlip rating assumes a corner you cannot machine
Published circlip thrust ratings assume a perfectly sharp groove corner. Whatever radius your cutting tool leaves comes straight off the load-bearing face — subtract it from the groove depth before you believe the number.
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A circlip never holds anything still
The standard groove is deliberately wider than the ring in it — by 0.10 mm at most sizes and 0.15 mm at the largest. The retained part can always move that far, even with everything made exactly to nominal.
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Double the speed, quadruple the braking distance
Braking distance goes as the square of speed. Twice as fast is four times the distance under braking, because it is four times the kinetic energy for the same tyres to turn into heat.
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To go supersonic, the pipe has to get wider
Below the speed of sound, squeezing a duct speeds the gas up — the garden-hose intuition, and it is right. Above the speed of sound it is exactly backwards: squeezing slows the flow, and only a WIDENING duct keeps it accelerating.
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Axial load is free until it is not
Below a threshold e (roughly 0.22–0.44 of the radial load) thrust costs a deep-groove bearing nothing. Above it, the equivalent load jumps to 0.56Fr + YFa and life falls with the cube.
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A degree warmer is a third of a percent thinner
At constant pressure, air loses about 0.34% of its density for every degree Celsius it warms near room temperature — about 15% between a −5 °C winter day and a 35 °C summer one.
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Each pH unit is a tenfold change in the hydrogen-ion concentration
pH is minus the log of the hydrogen-ion concentration in mol/L, so each unit is a factor of ten: pH 3 is ten times more acidic than pH 4 and a thousand times more than pH 6, and 0.1 mol/L of a strong acid is pH 1.
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A weak acid is at about half its pKa minus half the log of its concentration, and moves half a unit per tenfold dilution
The pH of a weak acid of concentration c is about (pKa minus log c) divided by 2, so 0.1 mol/L ethanoic acid (pKa 4.76) is pH 2.88 with only 1.3 percent of its molecules ionised, and a tenfold dilution moves it half a unit, not a whole one.
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A buffer holds the pH within one unit of the pKa of its weak acid
A buffer works within one pH unit of the pKa of its weak acid, where pH = pKa + log([salt]/[acid]): a 0.1 mol/L ethanoate buffer sits at pH 4.76, and 10 mmol of hydrochloric acid moves it only to 4.58, against a fall from 7 to 2.00 in water.
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Neutral water is pH 7.0 only at 25 degrees C; hot water is not acidic
Pure water is neutral at pH 7.0 only at 25 °C: its ionisation grows with temperature, so neutral is about 7.5 at 0 °C, 6.8 at body temperature and 6.1 at 100 °C, and water at 100 °C is not acidic.
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A mole of any gas fills about 24 litres at room temperature and pressure
A mole of any gas fills about 24 litres at 20 °C and 1 atm (22.4 litres at 0 °C), so a litre of gas at room conditions is about 0.04 mol and weighs about 0.04 times its molar mass in grams.
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In a sealed container a gas’s pressure goes with the kelvin temperature, not the Celsius one
The absolute pressure of a sealed gas is proportional to its temperature in kelvin: heating a tyre from 20 °C to 50 °C raises its absolute pressure by 10 percent, and doubling it takes 586 K, which is 313 °C.
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At one temperature lighter gas molecules are faster, as one over the square root of their mass
At a given temperature every gas molecule has the same average kinetic energy, so its speed goes as one over the square root of its mass: at 20 °C the rms speed is 1,905 m/s for hydrogen, 1,352 for helium, 511 for nitrogen and 408 for carbon dioxide, and a helium balloon leaks 2.6 times as fast as a nitrogen one.
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Common gases well above their critical temperature are ideal to about 1 percent below ten atmospheres
Nitrogen and oxygen at 20 °C are within about 1 percent of ideal below 10 atm (Z = 0.993 for nitrogen at 10 atm), but carbon dioxide, nearer its critical temperature, is 4.5 percent low there (Z = 0.955).
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A mole of a substance weighs its formula mass in grams
A mole of a substance weighs its formula mass in grams, so 18.0 g of water, 58.4 g of salt and 180 g of glucose each hold 6.022×10²³ molecules.
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To make a solution weigh concentration times volume times molar mass, and add water up to the volume
To make a solution of c mol/L in a volume V litres, weigh c × V × M grams of solute and add water up to the final volume: 250 mL of 0.1 mol/L sodium chloride is 1.46 g.
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Concentrated hydrochloric acid is about 12 mol/L: ten times the percent times the density over the molar mass
The concentration of a stock reagent is 10 × mass percent × density ÷ molar mass: 37 percent hydrochloric acid is 12.0 mol/L, 98 percent sulfuric acid is 18.4 and 70 percent nitric acid is 15.7.
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Diluting keeps the moles: c1 V1 equals c2 V2, and the acid goes into the water
Diluting does not change the moles, so c₁V₁ = c₂V₂: to make a litre of 1 mol/L from 12 mol/L hydrochloric acid take 84 mL of the stock and make it up to a litre, adding the acid to the water.
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The numbers in a balanced equation are ratios of moles, not of grams
The coefficients of a balanced equation are ratios of moles, not masses: in N₂ + 3 H₂ → 2 NH₃, 28.0 g of nitrogen needs 6.05 g of hydrogen, not 84 g, and makes 34.0 g of ammonia.
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The limiting reagent has the smallest moles divided by its coefficient, not the smallest mass
The limiting reagent is the reactant with the smallest number of moles divided by its coefficient, not the smallest mass: 16 g of methane burnt in 48 g of oxygen is limited by the oxygen, 0.75 against 0.997.
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The percentage yield is the mass you get over the most the limiting reagent allows
The theoretical yield is the most product the limiting reagent allows and the percentage yield is the actual mass over it: 28 g of nitrogen and 8 g of hydrogen can make 34.0 g of ammonia, so getting 27.2 g is an 80 percent yield.
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Atom economy is the mass of the wanted product over the mass of all the reactants, fixed by the equation
The atom economy, the mass of the wanted product as a percentage of the mass of all the reactants in the balanced equation, is fixed by the equation: making iron by Fe₂O₃ + 3 CO → 2 Fe + 3 CO₂ has 46 percent, making ammonia has 100.
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Electron pairs get as far apart as they can
Count the things around the central atom — bonds and lone pairs alike — and put them as far apart as they will go. Two is a line, three a flat triangle, four a tetrahedron at 109.5°, six an octahedron.
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A motor asks for six to eight times its running current before it turns
Locked-rotor current is typically 6–8× full-load current. A 7.5 kW 400 V motor runs at 14 A and starts at 85–113 A.
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Current follows voltage, torque follows voltage squared
Reduce a motor’s terminal voltage to a fraction a of nominal and it draws a times the current but produces a² times the torque.
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Double the voltage, quadruple the power
For a fixed resistance, power scales with voltage squared — doubling voltage gives four times the power.
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The ampacity table is a best case, not a rating
Published ampacity assumes 30 °C ambient and no more than three current-carrying conductors together. Six conductors in a 45 °C roof space leaves about 65% of the printed figure — enough to turn a correct 12 AWG answer into an undersized one.
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90 °C wire does not buy you 90 °C amps
The current you may draw is capped by the lowest-rated part of the path — almost always a 60 °C or 75 °C terminal, not the cable. But you still derate from the cable’s own 90 °C column, which is what makes the better insulation worth buying.
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Ampacity and voltage drop are two different checks
A wire can be perfectly safe from an overheating standpoint and still starve a device of voltage over a long run — checking only the breaker rating misses the second failure mode entirely.
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A poor power factor means paying for current you don’t use productively
Power factor below 1.0 means some current flows just to sustain a magnetic field (in motors, transformers) without doing useful work — utilities often bill industrial customers extra for this "reactive" demand.
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Transformers don’t multiply power, only trade voltage for current
Step up the voltage and the current drops by the same ratio — a transformer changes the balance between volts and amps, never the total power available (minus small real-world losses).
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Resistor bands read digit-digit-multiplier
On a 4-band resistor: first two bands are significant digits, third is the power-of-ten multiplier, fourth is tolerance.
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Preferred values are spaced by their own tolerance
The E-series steps by 10^(1/n) per decade, sized so that consecutive tolerance bands overlap. E24 steps by 1.10 and carries 5%, which spans 1.11 — so every value in the decade is already covered, and a 5k resistor would be indistinguishable from the 5.1k next to it.
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A resistor’s power rating assumes free air, and your board is not free air
Work out V squared over R, then pick a part rated for at least twice it. 5 V across 220 ohms is 0.114 W — arithmetic says an eighth-watt part, practice says a quarter.
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Half a lead-acid battery is not yours to use
Usable capacity is about 50% of a lead-acid nameplate and 80–90% of lithium — and the Peukert effect takes more again at high draw.
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Guess the direction, let the algebra correct you
Assign any current direction you like. If the answer comes out negative, the current flows the other way — the magnitude is still right.
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Delta needs a third of the capacitance
For the same kVAR, a delta-connected bank needs one third the capacitance of a star one — because reactive power scales with the square of the voltage each capacitor sees.
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A crash is good news right up until you stop saving
Run the same run of returns over your saving years and your retirement and it gives opposite answers. While contributing, the bad years arriving first leave you better off. While withdrawing, they are the thing that ends the portfolio.
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At 3% a year, prices double in a working generation
Divide 72 by the inflation rate to get the doubling time. At 3% — close to the long-run UK and US average — prices double in 24 years, so cash left alone that long buys half as much.
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The Rule of 72
Money doubles in about 72 ÷ (interest rate %) years.
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Double the tax, round up
In the US, doubling the sales tax on the receipt gives a quick tip estimate.
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114 triples, 144 quadruples
Divide 114 by the rate for tripling time, 144 for quadrupling.
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Ten years earlier ≈ twice the pot
At typical market returns, starting to invest 10 years earlier roughly doubles your final balance.
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Half a point of charges ≈ a tenth of the pot
Every extra half a point a year in pension charges takes about a tenth off a 30-year pot.
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The 4% rule
A retirement pot can sustain withdrawals of about 4% a year — so you need roughly 25× your annual spending.
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28/36 for housing debt
Keep housing costs under 28% of gross income, and all debt payments under 36%.
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Each 1% of rate ≈ 10% of buying power
A one-point rise in mortgage rates cuts what you can borrow for the same payment by roughly 10%.
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At 6%, interest ≈ the house itself
On a 30-year mortgage above ~5.5%, total interest roughly equals — then exceeds — the amount borrowed.
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20 / 4 / 10 for car buying
Put 20% down, finance for no more than 4 years, and keep total car costs under about 10% of gross income.
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PCP balloon is deferred cost, not a discount
A low UK PCP monthly usually means a large optional final payment (GFV) is waiting at the end — not that the car is cheaper.
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50/30/20 as a first budget
Aim 50% of take-home pay at needs, 30% at wants, 20% at saving and debt paydown.
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"Rent is throwing money away" ignores what the down payment could be doing
A renter who actually invests the cash a buyer would tie up in a down payment isn’t losing that money — they’re compounding it somewhere else instead of in home equity.
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Avalanche saves more money, snowball keeps you motivated
Paying off the highest-interest debt first (avalanche) minimises total interest paid — paying off the smallest balance first (snowball) clears accounts faster and keeps momentum, even though it usually costs more overall.
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Mid-market is not what you get at the bank
ECB reference rates are mid-market snapshots — a high-street FX quote usually costs you more both ways.
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A UK student loan is a tax, not a debt
In the UK you repay 9% of income above a threshold for a fixed number of years and the rest is cancelled. If you will not clear it before the write-off, overpaying buys you nothing.
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Every tuning rule is an opinion about one ratio
Ziegler–Nichols, Cohen–Coon and lambda tuning all read the same three numbers off a process — gain, time constant and dead time — and differ only in how close to the stability edge they are willing to sit. On a τ=2 s, θ=0.5 s loop they give Kp of 4.8, 5.6 and 1.0 respectively.
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Without dead time there is barely a tuning problem
Dead time is what creates a gain ceiling. Remove it and every standard tuning formula divides by zero, because there is no longer a phase limit to respect — the difficulty was never the lag, it was the delay.
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A stop-and-go wave travels backwards at 15-20 km/h
The cars go forwards; the jam goes backwards, at roughly 15 to 20 km/h — near enough the same whatever the speed limit is.
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Sample at least twice the fastest signal
To capture a signal faithfully, sample above 2× its highest frequency — in practice 5–10×.
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Aliases fold around half the sample rate
A tone above fs/2 reappears mirrored below it — 900 Hz sampled at 1 kHz reads as 100 Hz.
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Find the edge, then back off
Tuning by hand: raise proportional gain until the loop just oscillates, then halve it.
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Read the oscillation
Slow rolling oscillation = too much integral. Fast jitter = too much derivative amplifying noise.
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One τ is 63%; five is settled
A first-order system covers 63% of a step in one time constant and is effectively settled after 4–5.
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Dead time beats PID
When transport delay exceeds the process time constant, plain PID will disappoint.
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No natural damping means P alone can’t save you
A system with zero built-in damping — like a ball rolling on a frictionless beam — can’t be stabilised by proportional control alone, no matter the gain.
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Each pole costs 20 dB a decade and 90° of phase
Every additional real pole in a transfer function adds another −20 dB/decade of rolloff and, eventually, another −90° of phase lag.
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A loop that only overshoots on big steps is winding up, not mistuned
Change the size of the step. If a small change behaves perfectly and a large one overshoots badly, the problem is actuator saturation, not tuning.
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Stop integrating while the actuator is at its stop
Freeze the integral whenever the output is saturated and the error would drive it further in. One condition, no new parameter, and it cannot destabilise the loop.
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Cycling is worst at half load, not on the coldest day
On-off cycling peaks at about 50% duty. On the coldest day the unit runs nearly continuously and barely cycles at all.
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One step test gives you all three numbers
Step the input and wait. The delay before anything moves is L, the total change over the input change is K, and the time from the end of the delay to 63.2% of the final value is τ.
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The ratio of dead time to time constant decides everything
Below L/τ ≈ 0.2 PID is easy and forgiving. Past about 1.0 it will disappoint no matter how you tune it.
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2¹⁰ ≈ 10³
Every 10 bits multiplies by ~1,000 — kibi and kilo differ by just 2.4%.
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Each refresh rate step buys less than the last
Frame time is the reciprocal of refresh rate, so equal steps in Hz are not equal steps in latency — 60 to 120 Hz saves 8.3 ms per frame, 240 to 480 Hz saves only 2.1 ms.
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Each subnet bit halves the hosts
Every bit moved from host to network halves the addresses — a /24 has 254 usable.
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RAID is not a backup
RAID protects against disk failure — deletion, ransomware and fat fingers replicate instantly.
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3-2-1 backups
Keep 3 copies of anything that matters, on 2 different media, with 1 off-site.
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The ×100 latency ladder
RAM ~100 ns, SSD ~100 µs, spinning disk ~10 ms — each rung roughly 100× slower.
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Every nine costs 10×
Each extra "nine" of uptime — 99% → 99.9% → 99.99% — costs roughly an order of magnitude more effort.
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Big disks want dual parity
Beyond ~4 large drives, prefer RAID 6 over RAID 5 — rebuilds are when second failures strike.
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A word is five characters
Every typing speed ever quoted is characters divided by five, divided by minutes. Not average words — a fixed five characters, spaces included.
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Your ISP plan is a ceiling, not a promise
A "500 Mbps" plan means up to 500 Mbps under ideal conditions — not a guaranteed floor. Peak-hour congestion, Wi-Fi, and distance from the local exchange all pull real speeds below the advertised number.
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Striped data is every n-th bit, not the next n bits
Interleaving deals bits round its consumers one at a time. Lane i gets positions i, i+n, i+2n — so what one consumer sees is scattered through the word, never a contiguous field.
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Bits to bytes costs a factor of eight
Divide advertised Mbps by 8 for ideal MB/s — a 100 Mbps link is about 12.5 megabytes per second at full throttle.
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Latency limits bandwidth, not just response time
TCP will not send more than one window before it hears back, so throughput is capped at window divided by round-trip time no matter what the line is rated at. A 64 KB window over a 10 ms hop is 52 Mbps; over a 150 ms ocean crossing it is 3.5 Mbps.
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Never underestimate the bandwidth of a van
A 1 TB drive posted next-day moves data at an effective 93 Mbps, so it beats any connection slower than that. At 20 TB the courier is doing 1.9 Gbps.
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Sixty pixels per degree is where pixels vanish
20/20 vision resolves one arcminute, and a degree holds 60 of them — so about 60 pixels per degree is the point where a pixel grid stops being visible. Past it, more resolution changes nothing you can see.
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A word is worth two characters
One more random character buys log₂(95) = 6.6 bits. One more Diceware word buys log₂(7776) = 12.9 — nearly twice as much, and you can remember it.
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Crack time means nothing without naming the attacker
The same password takes minutes or millennia depending on who is guessing — ten tries a second at a login form against 10¹² a second on a leaked fast hash.
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Hohmann is the cheapest transfer — until the ratio passes about 12
Two burns on opposite sides of an ellipse is the minimum-energy route between circular orbits, and LEO to geostationary costs about 3.9 km/s over five and a quarter hours. Past a radius ratio of roughly 11.94, a three-burn bi-elliptic route out to a distant apoapsis is genuinely cheaper.
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A narrower gap spreads light more, not less
Diffraction angle goes as wavelength over aperture, so halving the opening doubles the spread — squeezing a beam tighter makes it fan out wider, not narrower.
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Geostationary runs on the sidereal day, not the 24-hour one
The altitude falls straight out of the period: match Earth’s rotation and the orbit radius is fixed at about 42,164 km, or 35,786 km up. But the period to match is the sidereal day, 86,164 s — using 86,400 puts the answer roughly 77 km wrong.
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The peak wavelength is about 2,900 divided by the temperature
A hot object shines brightest at a wavelength, in micrometres, of about 2,900 divided by its temperature in kelvin.
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Double the temperature, sixteen times the radiation
Double the absolute temperature of a surface and it radiates sixteen times as much power.
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After ten half-lives about a thousandth is left
Seven half-lives take a radioactive sample below 1%, and ten leave about a thousandth.
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The half-life is about seven-tenths of the mean life
The half-life is about 0.7 of the mean life; the mean life is about 1.44 half-lives.
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Double the length, halve the note; close one end, drop an octave
Halve the length of a string or pipe and its note goes up an octave; close one end of a pipe and it drops an octave.
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Two close notes beat at the difference of their frequencies
Two notes close in pitch beat at a rate equal to the difference of their frequencies: 440 Hz and 444 Hz beat four times a second.
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A magnifying glass magnifies about 25 divided by its focal length in centimetres
A magnifying glass of focal length f centimetres magnifies about 25 / f times: a 10 cm lens gives about 2.5, and a shorter, more curved lens gives more.
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Lens power in dioptres is 1 over the focal length in metres, and powers add
The power of a lens in dioptres is 1 divided by its focal length in metres, and thin lenses placed together add their powers.
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It takes about 4.2 kJ to warm a kilogram of water by one degree
It takes about 4.2 kJ to warm a kilogram of water by one degree, so a litre from 20 to 100 °C takes about 335 kJ, a tenth of a kilowatt-hour.
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Boiling a kilogram of water away takes five times the energy of heating it to the boil
Turning a kilogram of boiling water into steam takes 2,256 kJ, about 5.4 times the 418 kJ it takes to heat it from 0 to 100 °C.
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At room temperature a dull surface loses as much heat by radiation as by still-air convection
At room temperature a dull surface loses about as much heat by radiation as by convection in still air, each worth roughly 6 W for every square metre and every degree.
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A hundred millimetres of mineral wool takes a wall from a U-value above 3 to about 0.4
Each 100 mm of mineral wool adds a thermal resistance of about 2.5 m²K/W, enough to take a single leaf of brick from a U-value of about 3.4 to about 0.36.
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A straight wire makes a field of 0.2 microtesla times the current in amperes over the distance in metres
A long straight wire makes a magnetic field of about 0.2 µT times the current in amperes divided by the distance in metres, so 1 A at 1 cm makes 20 µT, about the strength of the Earth’s own field.
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A magnetic compass points to a magnetic north that is some degrees from true north, more than ten in some cities
A compass points to magnetic north, not true north: to walk a true bearing set the compass to the true bearing minus the declination (east positive), which in 2026 is +1.2° in London, +12.8° in Sydney and −12.5° in New York.
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In a magnetic field an electron circles 28 billion times a second for every tesla, however fast it goes
A charged particle in a magnetic field goes round a circle at a rate set by the field and the particle alone, 28 GHz per tesla for an electron and 15 MHz per tesla for a proton, and a faster one simply goes round a bigger circle.
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A coil turning in a magnetic field makes a peak voltage of N times B times A times its angular speed
A coil of N turns and area A turning in a field B makes a sine wave of peak voltage N B A times its angular speed in radians a second, so 100 turns of 100 cm² turning at 50 revolutions a second in 0.1 T peak at 31 V, which a meter reads as 22 V.
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A tenth of lift coefficient per degree
A thin wing gains about 0.1 of lift coefficient for every degree of angle of attack, until it stalls.
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ζ = 1 is critical damping
When the damping ratio hits 1, the system returns to rest as fast as possible without overshoot — less damps and it rings; more and it crawls.
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Photon energy is 1240 over the wavelength
A photon’s energy in electronvolts is 1240 divided by its wavelength in nanometres. Green light at 550 nm is 2.25 eV.
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A metre swings in two seconds
A 1 m pendulum takes almost exactly 2 s per full swing.
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Pendulum period: √length, not mass
Quadruple a pendulum’s length and it swings half as fast. Mass doesn’t matter at all.
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Falling: 5 metres in the first second
A dropped object falls ~5 m in the first second and gains ~10 m/s of speed every second.
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Braking distance goes as speed squared
Braking distance scales with the square of speed — 20% faster needs 44% more room, and twice the speed needs four times.
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45° for range — in a vacuum
Without air drag, 45° maximizes projectile range; with drag, the real optimum is nearer 30–40°.
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The slip angle is the friction coefficient
A block starts sliding when tan(θ) exceeds µ — so a 45° ramp needs µ ≥ 1 to hold.
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A marble always beats a hoop, and mass has nothing to do with it
A rolling body accelerates at g sin(theta) divided by (1 + k), where k says how far its mass sits from the axis. A hoop gets exactly half the acceleration of a frictionless slide, a solid sphere keeps five sevenths — and mass and radius cancel out entirely.
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A ramp saves force, never energy
A ramp divides the force by up to 1 over sin(theta), and multiplies the distance by exactly the same factor. The work is unchanged — and once friction is involved a ramp actively costs energy, the more so the gentler it is.
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Cooling closes the gap in equal fractions
Cooling has a half-life: each equal time-slice closes the same fraction of the temperature gap.
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Ice water beats freezer air
An ice-water bath chills a can 5–10× faster than freezer air.
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Smash factor tops out near 1.5
Ball speed rarely exceeds about 1.5× clubhead speed, even on a perfect driver strike.
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Four minutes a night, two hours a month
A star rises about four minutes earlier every night — two hours a month, and a full turn over a year.
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Square the year, cube the distance
For anything going round the Sun, the year squared equals the distance cubed — in Earth years and astronomical units, with no constant to look up. A planet four times further out takes eight times as long to get round.
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Everybody measures light at the same speed
Measure a light beam and you get 299,792,458 m/s — standing still, running towards it, or chasing it at nine tenths of that yourself. Every other strange thing in relativity is forced by keeping that one number fixed.
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Orbits are ellipses, not circles, and speed isn’t constant
A satellite moves fastest at its closest approach (perigee) and slowest at its farthest point (apogee) — a circular orbit is just the special case where those two points coincide.
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Most of a rocket’s mass is fuel it will burn on the way up
The Tsiolkovsky rocket equation shows that reaching a given velocity needs a propellant mass fraction that grows exponentially with the target delta-v — doubling the required velocity change can multiply the fuel needed several times over, not just double it.
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Elastic collisions conserve both momentum and energy — real ones rarely do
Two equal masses colliding elastically exchange velocities completely — the moving ball stops, the stationary one takes off at the exact same speed. Any energy lost to heat or sound (an inelastic collision) breaks this clean exchange.
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A boom only lands if you outrun the sound at the ground
A shock reaches the ground only if the aircraft is faster than the speed of sound AT THE GROUND, not merely at its own altitude. At 11 km on a standard day that means above about Mach 1.15; slower than that, the boom bends back up before it lands.
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The framing sets the depth of field, not the lens
At the same subject size in the frame, depth of field is set by the F-NUMBER and the format — the focal length very nearly cancels out.
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The drop height decides the argument
Gravity accelerates everything equally. Whether you can SEE that depends entirely on how far you drop it — short drops hide air resistance, long drops exaggerate it.
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A blade grinder is set by bed depth, not by time
A blade grinder has no gap to grind to, so the only thing deciding the size is how well the beans circulate past the blade — and that is governed by how many BEANS DEEP the pile is, not by how many grams are in it. One or two beans deep circulates; five is a wall.
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Thickness, not weight
A steak’s cooking time is set by how thick it is, not how heavy it is — and it rises faster than the thickness does. Double the thickness and you more than double the time.
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Paint coverage assumes one coat on a primed wall
A litre of paint claiming 12-14 m² coverage means one thin, even coat on primed drywall — dark-to-light changes, bare plaster, or textured surfaces routinely need a second coat, doubling real consumption.
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Mixing paint colours by volume, not by eye
A target colour from two base paints is a ratio problem, not a guess — the same two paints in a different ratio produce a completely different, unrepeatable shade.
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A colour looks different on four walls than on a tester patch
Light bounces. Every wall throws its own colour onto every other surface, so four walls of a colour read stronger and warmer than the A5 patch you judged it by.
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Tap drill size is major diameter minus pitch
For a metric thread, the drill bit for tapping is simply the major diameter minus the thread pitch — M6×1.0 needs a 5.0mm hole, no lookup table required.
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Coffee ratios are about weight, not scoops
A tablespoon of coffee varies by 4-8g depending on grind size and roast — the same "two scoops" can span nearly a 2x difference in actual coffee weight.
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Wet concrete needs less volume than the dry ingredients that make it
Loose sand and aggregate have air gaps between the particles — mixing needs roughly 1.54 times the dry volume of materials to produce one unit volume of finished, settled concrete.
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Water controls concrete strength more than the mix ratio does
Strength falls roughly exponentially with the water-cement ratio. Going from 0.50 to 0.60 is about two extra litres in a 25 kg bag — invisible in a mixer — and costs around a quarter of the final strength; 0.70 costs nearly half.
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Cold does not weaken concrete, it slows it down
Concrete gains strength by chemical reaction, and the reaction follows temperature. At 5 degrees C a pour takes roughly twice as long to reach a given strength as it does at 20 degrees C — but it does get there.
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Egg cooking time scales with mass to the two-thirds power, not linearly
A jumbo egg doesn’t need proportionally more time than a medium one to reach the same doneness — heat has to penetrate through the whole egg, and that penetration time scales with mass^(2/3), not mass itself.
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A naive timer drifts, a correct one can’t
Adding up tick intervals compounds every small delay into permanent error — computing elapsed time as now minus a fixed start point means a single slow tick costs you nothing, because the next measurement is still anchored to the same origin.
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Multiplication binds tighter than addition
2 + 3 × 4 is 14, not 20. Multiplication and division are resolved before addition and subtraction, whatever order you typed them in.
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SOH-CAH-TOA is one triangle, memorised three ways
Sine, cosine and tangent are just three different ratios of the same right triangle’s sides — knowing any one angle and any one side determines every other side and angle.
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Area scales with the square, volume with the cube
Double every dimension of a shape and its area quadruples while its volume grows eightfold — the exponent, not the shape, decides how fast size compounds.
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A column of the periodic table is a prediction
Elements in the same group share the number of electrons in their outermost shell, so they react alike — which is what lets you predict the behaviour of an element you have never handled from the one above it.
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Atoms get smaller as you add electrons
Moving left to right along a period, atoms shrink even though electrons are being added — lithium is 167 picometres across and fluorine, six electrons later, is 42.
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Advantage is worth most at even odds
Rolling twice and keeping the higher result is not worth a flat bonus — it is worth nothing when you were certain either way, and a full 25 percentage points when the outcome was a coin flip.
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A long run of heads is not evidence of anything
In a run of n coin flips the longest streak of the same face lands near log₂n — about six or seven in a hundred flips — so a streak that looks rigged is usually the ordinary case.
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The mean is not the middle when data is skewed
A handful of extreme values can drag the average well away from where most of the data actually sits — the median, not the mean, is the more honest "typical value" for skewed distributions like income or home prices.
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A graph is just the equation, drawn
Every point on the curve of y=f(x) is a coordinate pair that makes the equation true — nothing more mysterious than that.
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Percent of means multiply by hundredths
X% of Y is (X/100) times Y — and "Y plus X%" is Y times (1 + X/100).
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Halving the effect quadruples the sample
Sample size scales with 1/effect². Detecting a 5% lift instead of a 10% one costs four times the traffic, not twice.
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“No significant difference” usually means “too small to tell”
An underpowered test cannot distinguish a real effect from none, so a null result from one carries almost no information.
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Week 1 is the week holding the first Thursday
ISO 8601 weeks run Monday to Sunday, and week 1 is the week containing the year’s first Thursday — the same as the week containing 4 January. Not simply the week containing 1 January.
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Five sevenths, then take off the holidays
Working days ≈ calendar days × 5/7, minus the public holidays in between. Over a whole number of weeks it is exact; otherwise it is never out by as much as a day and a half.
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The hour hand keeps moving
Angle = |30H − 5.5M|. At 3:30 that is 75°, not 90° — the hour hand has already drifted halfway to the 4.
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Rest three minutes between heavy sets
Heavy strength work needs roughly three to five minutes between sets — hypertrophy work sixty to ninety seconds, and endurance work thirty to sixty. The rest is prescribed by the goal of the set, not by how recovered you feel.
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Estimate a one-rep max from five reps or fewer
Rep-max formulas agree closely at low reps and fall apart at high ones — within about five kilos of each other at five reps on a hundred-kilo lift, but seventy-seven kilos apart at twenty.
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Keep your waist under half your height
Waist circumference should be less than half your height. At 175 cm that is 87 cm; at 160 cm it is 80 cm.
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BMI and waist disagree in both directions, and that is the point
A muscular 180 cm person at 95 kg is BMI-obese and waist-healthy. A sedentary 180 cm person at 78 kg is BMI-healthy with a 95 cm waist and a ratio of 0.53.
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BMI cannot tell muscle from fat
A lean, heavily muscled athlete can score "overweight" or even "obese" on BMI despite having very low body fat — the formula only ever sees total mass and height, never composition.
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Doubling your race distance roughly doubles your time, plus a bit more
A 25-minute 5K predicts almost exactly a 52-minute 10K, not a 50-minute one — endurance pace slows down as distance increases, not just because a race is longer but because fatigue compounds.
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Activity level is an average, not your best day
Three hard gym sessions plus a sedentary desk job the other four days averages out to lightly active, not moderately active — overestimating activity is the single most common reason a calorie estimate ends up wrong.
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Measure your own sweat rate; the guidelines fit almost nobody
Sweat rates run from 0.5 to 2.5 L/h between people doing the same work. Weigh yourself before and after one session and you know yours; no published figure can tell you.
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220 minus age is an average, not your number
It is a population mean with a spread of roughly ±10 beats, so for about a third of people it is out by more than 10 — and every training zone moves with it.
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Half the world lives on under about nine dollars a day
Half the world’s people live on less than about $9 a day each, in prices where a dollar buys what it does in the US: about $3,300 a year per person. $30 a day per person is above four in five.
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The same height is a different percentile in every country
A man of 178 cm is taller than about 84% of the world’s men, but only 27% of Dutch men and 98% of Indian men. Say which pool before you say how tall.
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There is no number, and the answer is do not
Alcohol leaves the blood at roughly 0.015 g/100 mL an hour and nothing speeds that up — but no calculation of it tells you when you are fit to drive. The only reliable rule is not to drink and drive at all.
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60 dB plus 60 dB is 63 dB
Decibels do not add. Two equal sounds together are 3 dB louder, four are 6, ten are 10 — and a sound 10 dB quieter than another adds less than half a decibel to it.
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Twice the distance costs 6 dB — unless it is a road
A point source spreads over a sphere, so doubling your distance quarters the intensity: −6 dB. A line of traffic spreads over a cylinder instead, and doubling your distance from it buys only 3 dB.
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Tune up to the note, never down
Arrive at the pitch from BELOW. If you overshoot, drop well under and come back up.
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Move the fewest notes between chords
Do not play every chord in root position. Choose the inversion whose notes are nearest the chord you just played — most of them should not move at all.
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Learn five landmarks, not one sentence
Know a handful of notes on sight — the line the clef is named after, middle C, the outside lines — and read everything else as one or two steps from the nearest one.
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A beat is a time, not an event
Never fire a click when a timer goes off. Work out WHEN each beat should sound, hand that time to the audio hardware in advance, and let a late loop be late.
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A seven-note scale uses each letter exactly once
Assign the letters A to G in order from the root, then work out what accidental each one needs. The pitch decides the sound; the letter decides the name.
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A barre presses everything it crosses
One finger laid across a fret stops EVERY string it spans. A shape that wants an open string underneath the barre is not awkward, it is impossible.
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