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Calculators/Blog

Blog

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

Dates 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.Why Speeding Saves Less Time Than You ThinkThe same extra ten kilometres per hour buys less and less time the faster you are already travelling.Why Bakers Weigh Flour Instead of Measuring CupsThe same cup holds very different amounts depending on what is in it and how it got there.False 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.Squaring 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.Why 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.Business Days and Calendar Days Are Different ClocksA quarter looks like three months of work and contains rather less than that.The Part of Your Power Bill That Is Not UsageComparing plans on the usage rate alone compares the smaller half of some bills.Adding Tax and Removing Tax Are Different SumsThe percentage going on and the percentage coming off are calculated from different bases.Inflation 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.What 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.Coins 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.How Unusual Is Unusual? Standard Deviations and Z-ScoresThirty points above average means nothing until you know how spread out the data is.Roof Pitch: Degrees, Ratios, and the Area You Actually CoverThe roof is bigger than the building underneath it, and the difference is a fixed multiplier.Concrete: 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.Why 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.Two Formulas for Maximum Heart Rate, Two AnswersTraining zones are built on an estimate that carries about ten beats of individual variation either way.Body Fat Methods Do Not Agree, and That Is FineEvery field method is a prediction of a measurement nobody is actually taking.What a MET Actually MeasuresThe number on the treadmill display was calculated for somebody, and possibly not for you.Why 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.The 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.What 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.Snowball 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.The 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.Three Levers on a Savings Goal, and Which One MovesThe lever people reach for first is the return, which is the one they control least.Why 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.Averages That Mislead: Mean, Median and the Long TailA single unusual value can pull the mean somewhere no actual data point sits.Percentage 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.Why 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.Margin vs Markup: Why the Two Are Not the Same NumberThe same dollar of profit, divided by two different numbers, gives two very different percentages.Why 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.Small 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.What BMI Can and Cannot Tell YouA single number built from two measurements cannot distinguish muscle from fat, and was never designed to.Unit Conversion Mistakes That Cost Real MoneyThe dangerous conversions are the ones where both units are called the same thing.Why 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.Latency 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.Pipe 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.Boost 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.Every 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.The 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.Why 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.pH 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.A 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.The 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.An 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.Nine 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.AlphaEvolve 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.593 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.Erdos 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.FunSearch and the Cap Set ProblemSixteen extra points, in a problem almost nobody outside combinatorics has heard of. The method is what mattered.The Other Half of FunSearch: Bin PackingCap sets make the headlines. The bin packing result is the one that runs in data centres.AlphaDev 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.Astra and the Ten ProblemsThe announcement was not a benchmark score. It was ten problems that had not moved in at least a decade.Three Machines, One Constant, Four Decimal PlacesThe improvements are in the fourth decimal place. That is not a criticism, it is where the problem lives.DPS 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.Crit 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.The Diminishing Returns That Are NotThe hundredth point of armour adds exactly as much survivability as the first. The tooltip will tell you otherwise.Centimetres, Not SensitivityCopying a professional sensitivity setting into a different game is close to meaningless, and the reason is a number almost nobody knows.Why Your Pulls Feel CursedGoing fifty pulls with nothing happens about three quarters of the time. It is the normal outcome, not bad luck.Milliseconds, 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.The Shape of a Levelling CurveMinecraft uses three different formulas stitched together, and the joins are exactly where the game wants you to stop.Four 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.The 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.Dice Are Not InterchangeableTwo dice of six sides and one die of twelve have almost the same average and nothing else in common.Why 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.Point 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.Pokemon 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.Pot 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.Circles 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.