how to check a multiplication without doing it again, using digit sums
Built as a memo rather than a table: the thing worth hanging it up for reads across the room, with the full reference in smaller type below it.
Three sizes in one archive — from a sheet on the fridge to a poster on the wall.
If either factor reduces to 9, the answer must reduce to 9 as well, whatever the other one gives.
Hippolytus of Rome (c. 170–235) described how the Greeks reduced a number to a “root” from 1 to 9 by adding its digits again and again, but showed no sign of knowing it could check a calculation. The earliest known text that teaches this is an Indian treatise written around 950. Around 1020 the Persian polymath Ibn Sina (Avicenna) set out the method in full, calling it the “Hindu method”, and Fibonacci later described it in his Liber Abaci.
Why the numbers can be trusted. Every countable number here is computed from data rather than typed in, and the sheet must pass 3 independent checks before it is allowed to build — how many items there are, the order they come in, and the claim in the highlighted box itself. If one fails, there is no file.
the IWF colour code and the plate count: from target weight to each side of the bar, no calculator
why a “yes” from a test that is right 99 times in 100 is wrong more than nine times in ten when one in a thousand has the thing
the first digit of a number: why a one turns up more than six times as often as a nine, and in which sets that is true
how many people it takes for two of them to share a birthday — and the same count for codes and fingerprints