4 ms·

This representation is useful to reason about products in general, but I don't see how it can be used to compute a specific product.

enriquto 6y ago
This is wildly offtopic now, but greek geometry (thales, euclid, etc) pretty much starts with the construction of a segment whose length is the product of those of two arbitrary given segments. It can be done with compasses and ruler. We've been doing effective and precise computations with numbers thousands of years before the decimal notation was even invented.
- vbarrielle 6y ago
  You're completely right, I had forgotten about constructive geometry. Thanks for pointing it out!
- laumars 6y ago
  I think you're conflating two things: formulas and numbering systems. Ancient Greeks still operated in decimal albeit they'd use the Greek alphabet (since dedicated glyphs for numbers didn't exist) but those letters followed a base-10 system. They would have done the same finger counting we did as kids. The difference between then and now is their calculations were geometric instead than algebraic, rather than them not thinking in decimal. In that regard your argument is a little like saying kids test question such as "Jennifer has 3 apples and gives one to Bob and Terry, how many does she have left" isn't decimal because the calculation is written in English rather than as a "number sentence" (as schools call them). But clearly the kids are still thinking about those numbers as decimal even if the question is not phrased as a mathematical equation. Whereas some other cultures used base-12 or base-60 (https://en.wikipedia.org/wiki/Sexagesimal https://en.wikipedia.org/wiki/Sexagesimal) which, incidentally, are also believed to originate from finger counting much like base-10 was. I believe the Romans also thought about things in decimal despite their numerals being logarithmic. However their numerals do follow the decimal orders of magnitude.

enriquto 6y ago

This is wildly offtopic now, but greek geometry (thales, euclid, etc) pretty much starts with the construction of a segment whose length is the product of those of two arbitrary given segments. It can be done with compasses and ruler. We've been doing effective and precise computations with numbers thousands of years before the decimal notation was even invented.

vbarrielle 6y ago
You're completely right, I had forgotten about constructive geometry. Thanks for pointing it out!
laumars 6y ago
I think you're conflating two things: formulas and numbering systems. Ancient Greeks still operated in decimal albeit they'd use the Greek alphabet (since dedicated glyphs for numbers didn't exist) but those letters followed a base-10 system. They would have done the same finger counting we did as kids. The difference between then and now is their calculations were geometric instead than algebraic, rather than them not thinking in decimal. In that regard your argument is a little like saying kids test question such as "Jennifer has 3 apples and gives one to Bob and Terry, how many does she have left" isn't decimal because the calculation is written in English rather than as a "number sentence" (as schools call them). But clearly the kids are still thinking about those numbers as decimal even if the question is not phrased as a mathematical equation. Whereas some other cultures used base-12 or base-60 (https://en.wikipedia.org/wiki/Sexagesimal https://en.wikipedia.org/wiki/Sexagesimal) which, incidentally, are also believed to originate from finger counting much like base-10 was. I believe the Romans also thought about things in decimal despite their numerals being logarithmic. However their numerals do follow the decimal orders of magnitude.