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Calculators/Blog

Blog

Short pieces on the arithmetic the calculators do, and the places it usually goes wrong.

Radians Are Not an Alternative to DegreesAt a radius of 8 and an angle of one radian, the arc is exactly 8. That is not a coincidence, it is the definition, and it is why the unit exists.10 October 2026When Two Cycles Line UpAn 8 tooth gear meeting a 12 tooth gear brings the same pair of teeth together every 24 teeth. Change it to 13 and they do not meet again for 104.10 October 2026A Ten Degree Rise Is Not Fifty DegreesTen degrees Celsius is fifty degrees Fahrenheit. A rise of ten degrees Celsius is a rise of eighteen. Both statements are correct and they are about different things.10 October 2026There and Back Is Not the AverageThe trip that averages 50 km/h takes 4 hours 48 minutes. The one that actually happened takes 5 hours. The missing twelve minutes are the whole argument.10 October 2026Twice the Size Is Eight Times the StuffA 13 inch pizza is 109 per cent bigger than a 9 inch one, for 44 per cent more diameter. Nobody prices them that way, which is the opportunity.10 October 2026Both Channels Improved and the Average FellPaid went from 2.0 to 2.5 per cent and organic from 8.0 to 10.0. The site-wide conversion rate fell from 4.00 to 3.38, and nobody did anything wrong.10 October 2026Safety Stock Is Priced in VarianceQuadrupling the lead time and doubling the demand variability have exactly the same effect on safety stock: both take it from 31 units to 62.10 October 2026LTV Does Not Pay the RentA $720 lifetime value against a $240 acquisition cost is the textbook 3:1 ratio. At a 30 per cent margin that customer returns $216 and the business is down $24 before it starts.10 October 2026Ten Per Cent a Month Is Not 120 Per Cent a YearTen thousand growing at ten per cent a month reaches 31,384 in a year, not 22,000. The gap between those two numbers is the entire point of compounding.10 October 2026A Four Times ROAS Can Lose MoneyOn a 20 per cent margin the break-even ROAS is 5.00. A campaign returning 4.00 is losing $300 on every $1,500 spent, and the dashboard will call it a success.10 October 2026Km Is a Concentration, Not a RateReaching half of Vmax takes a substrate concentration equal to Km. Reaching 90 per cent of it takes nine times as much.10 October 2026Carriers Outnumber SufferersA condition that affects 0.01 per cent of a population is carried silently by 1.98 per cent of it, which is 198 carriers for every affected person.10 October 2026A Buffer Works Until It Does NotAdding acid to a balanced buffer moves the pH by 0.09. Adding nine times as much moves it by 1.28, which is fourteen times the shift for nine times the acid.10 October 2026One of These Changes With TemperatureTen grams of salt gives 0.3422 mol/L in 500 mL of solution and 0.3422 mol/kg in 500 g of solvent. The agreement is a coincidence of water, and it does not survive heating.10 October 2026Ten Tubes and Nothing Is LeftFive one-in-ten steps take a hundred million organisms per millilitre down to a thousand. Five more take it to one in a hundred millilitres.10 October 2026Half-Life Is Not Half the LifetimeAfter ten half-lives, 0.098 per cent of the original material is left. After twenty, you are counting atoms rather than measuring quantities.10 October 2026Four Gas Laws Are One EquationOne mole of gas occupies 22.4 litres at zero degrees and standard pressure. Heat it to 100 degrees in the same container and there is only 0.73 of a mole in there.10 October 2026Three Decibels Is Twice the PowerTwo identical speakers instead of one is a three decibel increase, which is barely noticeable. Ten of them is ten decibels, which sounds about twice as loud.10 October 2026Drag Quadruples When Speed DoublesGoing from 45 to 90 km/h quadruples the drag force and takes the power needed from 538 watts to 4,307. Another doubling takes it to 34,453.10 October 2026The Wire Is Part of the CircuitThe same cable carrying the same current drops 1.95 per cent of a 230 volt supply and 9.33 per cent of a 12 volt one. The wire does not know the difference. The load does.10 October 2026The Six to One Rule of Rooftop SolarThe same 8,000 kilowatt hours is worth $1,100 a year at 35 per cent self-consumption and $1,900 at 75 per cent. Same panels, same sun.10 October 2026Carbon Is Not Carbon DioxideA mature tree holding 447 kilograms of carbon has taken 1,637 kilograms of carbon dioxide out of the air. Both numbers describe the same tree.10 October 2026Printers Are Limited in Cubic MillimetresChanging the layer height from 0.2mm to 0.28mm at the same print speed takes a setting from 84 per cent of the hotend limit to 118 per cent.10 October 2026Salary Is Not the Cost of an EmployeeAn $85,000 salary costs $108,297 a year, which is $60.17 for every productive hour. The gap is not a rounding error.10 October 2026Duty Is a Bracket, Not a RateOn a typical bracket scale, paying $1,000,001 instead of $999,999 costs an extra ten cents of duty. Concessions are where the real cliffs live.10 October 2026What the Sale Has to FetchA $750,000 property bought with $20,000 of costs needs $824,131 to break even after one year, and $971,370 after ten.10 October 2026A Time-lapse Is Three NumbersTwo hours at five second intervals is 1,440 shots and sixty seconds of footage. Want twenty seconds instead? Shoot every fifteen.10 October 2026The 500 Rule Is About the SensorMount the same lens on a crop-sensor body and the longest usable exposure drops by a third. The lens has not changed. The framing has.10 October 2026Underinsurance Bites TwiceA 6.25 per cent shortfall in the sum insured costs 6.25 per cent of a small claim. On a total loss the same shortfall costs 25 per cent.10 October 2026The Gradient Is Flown Over the GroundThe same aircraft climbing at the same 700 feet a minute makes 600 feet per nautical mile into a headwind and 382 with a tailwind. Only one of those numbers is about the aircraft.10 October 2026Circles in a Square WorldA nine-block circle takes 69 blocks where the area says 63.6. By thirty-one blocks across, the error has reversed.1 October 2026Pot Odds Are the Only Ratio That MattersA flush draw is a call on the flop and a fold on the turn, at the same price, and the reason is one card.1 October 2026Pokemon EVs Come in Steps of EightFour effort points buy a stat. The next four buy nothing. Both facts come from the same pair of floors.1 October 2026Point Buy Rewards the Boring BuildThree fifteens and three eights spends all 27 points for a total modifier of +3. A flat spread gets +7 from the same budget.1 October 2026Why Speedrun Pace LiesBeing thirty seconds up at the first split is worth much less than being thirty seconds up at the last, and the projection treats them identically.1 October 2026Dice Are Not InterchangeableTwo dice of six sides and one die of twelve have almost the same average and nothing else in common.1 October 2026The 400-Point RuleFour hundred points of rating means a nine per cent chance. The system is built so that everyone ends up losing half their games.1 October 2026Four Ways to Measure a PlayerEvery ratio is an incentive. The question is not which is accurate, it is which behaviour you want to encourage in yourself.1 October 2026The Shape of a Levelling CurveMinecraft uses three different formulas stitched together, and the joins are exactly where the game wants you to stop.1 October 2026Milliseconds, Not FramesGoing from 30 to 60 fps saves 16.7 milliseconds a frame. Going from 120 to 150 saves 1.7. Both are a jump of 30.1 October 2026Why Your Pulls Feel CursedGoing fifty pulls with nothing happens about three quarters of the time. It is the normal outcome, not bad luck.1 October 2026Centimetres, Not SensitivityCopying a professional sensitivity setting into a different game is close to meaningless, and the reason is a number almost nobody knows.1 October 2026The Diminishing Returns That Are NotThe hundredth point of armour adds exactly as much survivability as the first. The tooltip will tell you otherwise.1 October 2026Crit Chance or Crit Damage? There Is an AnswerIt is a one-line piece of calculus, and it says the answer is almost always closer to even than people play it.1 October 2026DPS Is the Wrong Number for a DuelTwo weapons with identical DPS can kill at completely different speeds, and the reason is that damage arrives in whole shots.1 October 2026Three Machines, One Constant, Four Decimal PlacesThe improvements are in the fourth decimal place. That is not a criticism, it is where the problem lives.1 October 2026Astra and the Ten ProblemsThe announcement was not a benchmark score. It was ten problems that had not moved in at least a decade.1 October 2026AlphaDev Wrote Assembly Nobody Had Thought OfIt did not reduce the number of comparisons. It reduced the number of instructions, which is a different and more useful thing.1 October 2026The Other Half of FunSearch: Bin PackingCap sets make the headlines. The bin packing result is the one that runs in data centres.1 October 2026FunSearch and the Cap Set ProblemSixteen extra points, in a problem almost nobody outside combinatorics has heard of. The method is what mattered.1 October 2026Erdos Problem 728, and the Factorials That DivideThe first Erdos problem widely regarded as having been resolved more or less autonomously by a machine. The statement is three lines long.1 October 2026593 Spheres: the Kissing Number in Eleven DimensionsNewton and Gregory argued about the three-dimensional case in 1694. It took until 1953 to settle, and most dimensions are still open.1 October 2026AlphaEvolve Found the Trick Strassen MissedA saving of one multiplication out of 49 sounds like nothing. It is a change in an exponent, which is not nothing at all.1 October 2026Nine Erdos Problems, Proved in LeanThe interesting number is not nine. It is nine out of 353, and the fact that a compiler can confirm all nine in seconds.1 October 2026An 80-Year-Old Conjecture, Disproved by a MachineErdos thought the square grid was essentially the best you could do. It was not, and the construction that beat it came out of a language model.1 October 2026The 26-Payment Year, and Other Frequency TrapsTwo rents can look identical per payment and differ by thousands over a year, purely because of how often they fall due.26 September 2026A 99% Accurate Test Can Be Wrong Five Times Out of SixThe accuracy of a test tells you almost nothing on its own. What matters is how many people being tested actually have the thing.26 September 2026pH Is a Logarithm, and That Changes EverythingThe pH scale looks linear, runs from 0 to 14, and does neither of the things that appearance suggests.26 September 2026Why Salt Melts Ice but Barely Changes Pasta WaterSalt does the same thing at both ends of the temperature range. It is just far more effective at one of them.26 September 2026The Joint Is Part of the WallMaterials are sold by their own dimensions and installed at rather different ones, and the gap between the two is where estimates go wrong.26 September 2026Every Spacing Job Needs One More Than You ThinkDividing the length by the spacing gives you the gaps. It does not give you the things either side of them.26 September 2026Boost Is Not the Number That MattersTwo engines running the same boost can be doing quite different amounts of work, because boost is measured from a moving baseline.26 September 2026Pipe Size Beats Pump SizeWhen the pressure at the far end is disappointing, the instinct is a bigger pump. The arithmetic usually points at the pipe.26 September 2026Latency Is Distance, and Bandwidth Cannot Fix ItA faster link puts bits onto the wire more quickly. It does not make them travel any faster once they are there.26 September 2026Why a Roof Collects Less Rain Than the Rainfall SuggestsThe arithmetic of rainwater harvesting is unusually clean, right up to the point where you decide which area to multiply by.26 September 2026Squaring Up a Corner With the 3-4-5 TriangleYou cannot trust a corner to be square because it looks square, but you can prove it with a tape measure.21 September 2026False Precision: When Extra Decimals Are a LiePrecision is how many digits you wrote down. Accuracy is whether they are right. They are not the same thing.21 September 2026Why Bakers Weigh Flour Instead of Measuring CupsThe same cup holds very different amounts depending on what is in it and how it got there.21 September 2026Why Speeding Saves Less Time Than You ThinkThe same extra ten kilometres per hour buys less and less time the faster you are already travelling.21 September 2026Dates Are Harder Than They LookAdding a month sounds like the simplest possible calculation, right up until the month you land in is shorter than the one you left.21 September 2026Unit Conversion Mistakes That Cost Real MoneyThe dangerous conversions are the ones where both units are called the same thing.2 September 2026What BMI Can and Cannot Tell YouA single number built from two measurements cannot distinguish muscle from fat, and was never designed to.2 September 2026Small Rate Differences, Large Amounts of MoneyThe difference between 6 and 6.5 per cent is not eight per cent more interest. Over decades it is far more than that.2 September 2026Why a 50% Drop Needs a 100% Rise to Get BackA 20 per cent fall followed by a 20 per cent rise does not return you to where you started.2 September 2026Margin vs Markup: Why the Two Are Not the Same NumberThe same dollar of profit, divided by two different numbers, gives two very different percentages.2 September 2026Why Area Alone Never Tells You How Much to OrderThe measurement is the easy part. What you order is the measurement plus wastage, rounded up to whole units.2 September 2026Percentage Points Are Not Per CentWhen the thing being measured is itself a percentage, there are two honest ways to describe a change, and they give very different numbers.2 September 2026Averages That Mislead: Mean, Median and the Long TailA single unusual value can pull the mean somewhere no actual data point sits.2 September 2026Why a Longer Loan Term Costs More at the Same RateStretching a loan out does not make it cheaper. It makes each payment smaller and the total larger.2 September 2026Three Levers on a Savings Goal, and Which One MovesThe lever people reach for first is the return, which is the one they control least.2 September 2026The Sticker Price Is the Smallest Number on a CarInterest, on-road costs and fuel are all decided at purchase, and none of them appear on the windscreen.2 September 2026Snowball or Avalanche: What the Arithmetic SaysOne method wins on money, the other on momentum, and the gap between them is usually smaller than the argument about it.2 September 2026What a Small Credit Card Payment Actually CostsThe size of the payment does not change the interest rate. It changes how long the rate has to work on you.2 September 2026The Rate You Are Quoted Is Not Always the Rate You PayTwo accounts advertising the same rate can pay different amounts of interest, and neither is misleading you.2 September 2026Why a Poll of 1,000 Can Describe MillionsA national poll and a survey of one town need about the same number of responses, which sounds wrong and is not.2 September 2026What a MET Actually MeasuresThe number on the treadmill display was calculated for somebody, and possibly not for you.2 September 2026Body Fat Methods Do Not Agree, and That Is FineEvery field method is a prediction of a measurement nobody is actually taking.2 September 2026Two Formulas for Maximum Heart Rate, Two AnswersTraining zones are built on an estimate that carries about ten beats of individual variation either way.2 September 2026Why Two Calorie Calculators Give You Different NumbersThe same person, the same day, two reputable formulas, and a gap of sixty calories before anyone has picked an activity level.2 September 2026Concrete: Bags, Mix Ratios and Where the Crossover IsConcrete volume looks small on paper right up until you convert it into bags you have to carry.2 September 2026Roof Pitch: Degrees, Ratios, and the Area You Actually CoverThe roof is bigger than the building underneath it, and the difference is a fixed multiplier.2 September 2026How Unusual Is Unusual? Standard Deviations and Z-ScoresThirty points above average means nothing until you know how spread out the data is.2 September 2026Coins Have No Memory, and Neither Do Roulette WheelsThe coin does not know what it did last time, and no amount of history changes the next flip.2 September 2026What a Correlation Coefficient Does Not Tell YouA strong r means the points sit near a straight line. It does not mean one column caused the other.2 September 2026Inflation and the Money You Have Not Spent YetA balance that grows every year can still buy less every year, and the statement will not mention it.2 September 2026Adding Tax and Removing Tax Are Different SumsThe percentage going on and the percentage coming off are calculated from different bases.2 September 2026The Part of Your Power Bill That Is Not UsageComparing plans on the usage rate alone compares the smaller half of some bills.2 September 2026Business Days and Calendar Days Are Different ClocksA quarter looks like three months of work and contains rather less than that.2 September 2026Why MPG Hides How Much Fuel You Actually SaveTwo upgrades, one worth twenty extra miles per gallon and one worth ten, and the smaller one saves far more fuel.2 September 2026