The long ruler
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Their number: about 10 to the power 3.
Abū Rayḥān al-Bīrūnī
Asked: How big is the Earth?
Chunked it: He climbed a hill, measured its height, then measured how far the horizon dipped below level from the top. A triangle drawn from those two numbers gives the Earth’s radius.
Their number: a radius that works out to somewhere between about 5,800 and 6,760 kilometres, depending on how long the cubit he measured in was.
What we know now: The Earth’s mean radius is 6,371 km — inside his range. One modern reading of his units puts him about 2% high.
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Their number: about 10 to the power 4.
Enrico Fermi
Asked: How strong was the blast of the first atomic bomb test?
Chunked it: About 40 seconds after it, the blast wave reached him. He dropped small scraps of paper from about six feet before, during and after it passed; they were pushed about 2.5 metres, and he worked that push into an energy.
Their number: about 10,000 tons of TNT.
What we know now: The official estimate is about 21,000 tons, and a 2021 re-analysis says about 24,800 — so a handful of paper got within a factor of about two and a half. (His paper only felt the blast wave, which is part of the total energy.) The test was kept secret: people living nearby were not warned.
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Their number: about 10 to the power 5.
Eratosthenes
Asked: How far is it all the way around the Earth?
Chunked it: At noon on midsummer the Sun shone straight down a well in Syene, but in Alexandria a rod still cast a shadow of about 7.2 degrees — one fiftieth of a circle. So the whole way round is fifty times the distance between the two cities.
Their number: about 250,000 stadia.
What we know now: Nobody knows exactly how long his stadion was (somewhere between about 160 and 210 metres), so his answer is 40,000 to 52,500 km. The real distance is about 40,075 km.
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Their number: between 10 to the power 3 and 10 to the power 8.
Frank Drake and the first SETI meeting
Asked: How many civilisations in our galaxy might we be able to talk to?
Chunked it: Drake wrote the question as a chain of pieces multiplied together: how many stars form each year, how many have planets, how many planets could hold life, and so on, down to how long a talking civilisation lasts.
Their number: somewhere between 1,000 and 100,000,000.
What we know now: Still unknown. Several of the pieces have no measurement at all yet, so the honest band is enormous: that is the point of writing it as pieces.
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Their number: about 10 to the power 63.
Archimedes
Asked: How many grains of sand would it take to fill the whole universe?
Chunked it: He started small — grains in a poppy seed, poppy seeds across a finger-width — and scaled up step by step to the size of the universe as Aristarchus pictured it. On the way he invented a way to name numbers far bigger than the Greeks had names for.
Their number: no more than 10⁶³ grains (a 1 followed by 63 zeros).
What we know now: The universe he pictured is far smaller than the one we know, so his number answers his question, not ours. The method — build a huge number from small steps — is exactly the one this app uses.