Q1.
Instead of grouping together 10 collections of size equal to the previous landmark number (as in the case of the Egyptian system), can we get a number system by grouping together 5 collections of size equal to the previous landmark number? Can this 5 be replaced by any positive integer?
Answer
Yes to the first question, and almost yes to the second.
Grouping in 5s. Start with 1 as the first landmark and multiply by 5 each time:
50 = 1 51 = 5 52 = 25 53 = 125 54 = 625 55 = 3125
These are all powers of 5, so we get a perfectly good base-5 system, with symbols △ □ ⬡ ○ ∿ ↑ for the six landmarks shown.
Replacing 5 by any n. The same construction works for any n: landmarks 1, n, n2, n3, … This is exactly the definition of a base-n number system, and the Egyptian system is the case n = 10.
Why it happens: one positive integer must be ruled out — n = 1. Then the landmark numbers are 1, 1 × 1, 1 × 1 × 1, … which are all just 1. Nothing new is ever created, so a “base-1 system” is nothing but tally marks. So the correct statement is: 5 may be replaced by any integer greater than 1.
Did you know? Every computer you use works in base 2, with landmark numbers 1, 2, 4, 8, 16, … and only two digits. A smaller base means fewer symbols to remember but longer numerals — the number 25 is “25” in base 10 and “11001” in base 2.