Q1.
Express the number 143 in this new system.
Answer
Start from the largest landmark number that is smaller than 143. That is 53 = 125, since the next one, 625, is too big.
143 − 125 = 18
18 = 5 + 5 + 5 + 3 (three 5s, then 3 ones)
So 143 = 125 + 5 + 5 + 5 + 1 + 1 + 1
18 = 5 + 5 + 5 + 3 (three 5s, then 3 ones)
So 143 = 125 + 5 + 5 + 5 + 1 + 1 + 1
Numeral: ○ □□□ △△△
Check: 125 + 15 + 3 = 143 ✓
Check: 125 + 15 + 3 = 143 ✓
Written as base-5 digits, the same number is
143 = (1)×125 + (0)×25 + (3)×5 + (3)×1 = 1033 in base 5
Why it happens: the hexagon ⬡ (25) does not appear at all, because after taking away 125 only 18 is left and 18 < 25. In the picture form its absence is simply a symbol not drawn; in the digit form the same absence must be written as the digit 0. That one difference is the gap between the Egyptian idea and place value.
Check it yourself: no symbol appears five or more times here. If it ever did, five of them would be swapped for one of the next symbol up — which is why each base-5 digit stays between 0 and 4.