Q1.
Can we represent this more compactly?
Answer
Yes — and the way we do it is the birth of place value.
The number on page 71 was 640, grouped as
640 = (10) × 60 + 40
Following the Egyptian idea we would draw the 60-symbol ten times over and then four 10-wedges — fourteen symbols in all. Instead, write the numeral for ten in the 60s slot and the numeral for forty in the ones slot:
< | <<<<
read as “ten 60s and one 40” — exactly what the equation says
read as “ten 60s and one 40” — exactly what the equation says
The same for 7530:
7530 = (2) × 3600 + (5) × 60 + 30
Check: 7200 + 300 + 30 = 7530 ✓
Numeral: YY | YYYYY | <<<
Check: 7200 + 300 + 30 = 7530 ✓
Numeral: YY | YYYYY | <<<
Once each slot has its own place, the symbols marking which power of 60 it is are no longer needed at all — the position says it. Drop them, and the numeral is as compact as it can be.
Why it happens: the compression works because no power of 60 can ever occur 60 or more times. If it did, sixty of them would be regrouped into one of the next power — the book shows this with (1) × 3600 + (70) × 60 + 2 = (2) × 602 + (10) × 60 + 2. So each slot holds a number between 0 and 59, which a handful of wedges can always write. A bounded slot is what makes place value possible.