Number systems gallery
A Story of Numbers26 min readLesson 5 of 8
5.1 Why a gallery helps
Each number system in this chapter solved the same problem in a different way: how to write a quantity with a small, ordered set of symbols. A gallery puts the systems side by side, so the pattern is easier to see. A base-n system uses the powers of n as landmarks. A positional system adds the idea that the position of a symbol decides which landmark it counts. Zero, when it appears, marks an empty position and also works as a number.
5.2 Base-2 and base-4
Base-2 landmarks are 1, 2, 4, 8, 16 and 32. To write 45, take the largest landmark that fits: 45 = 32 + 8 + 4 + 1, so 45 = 101101 in base 2. A zero is needed for the missing 16 and 2 places.
Base-4 landmarks are 1, 4, 16 and 64. The same number is 45 = 2 × 16 + 3 × 4 + 1, so it is written 231 in base 4. The digits in base 4 are 0, 1, 2 and 3; a digit 4 would carry into the next place.
Original question (base 2). Write 37 in base 2. 37 = 32 + 4 + 1, so the answer is 100101.
Original question (base 4). Write 100 in base 4. 100 = 1 × 64 + 2 × 16 + 1 × 4 + 0, so the answer is 1210.
5.3 Base-5 and base-7
Base-5 landmarks are 1, 5, 25, 125 and 625. For 137, one 125 leaves 12, two 5s leave 2, and two 1s finish the number, so 137 = 1022 in base 5.
Base-7 landmarks are 1, 7, 49 and 343. For 200, four 49s make 196, a zero in the 7s place keeps the position, and four 1s finish: 200 = 404 in base 7.
Original question (base 5). Write 84 in base 5. 84 = 3 × 25 + 1 × 5 + 4, so the answer is 314.
Original question (base 7). Write 100 in base 7. 100 = 2 × 49 + 0 × 7 + 2, so the answer is 202.
5.4 Base-60
The Mesopotamian system used base 60, with landmarks 1, 60 and 3600. Its place values are large, so a number needs few positions. For 3857, one 3600 leaves 257; four 60s make 240; seventeen 1s finish the number. The numeral records the coefficients 1, 4 and 17.
Original question (base 60). Write 500 in base 60. 500 = 8 × 60 + 20, so the coefficients are 8 and 20.
5.5 Roman numerals
Roman numerals use fixed-value landmarks: I stands for one, V for five, X for ten, L for fifty, C for a hundred, D for five hundred and M for a thousand. A smaller symbol before a larger one is subtracted, as in IV and IX.
Worked conversion. 1984 = 1000 + 900 + 50 + 30 + 4 = M + CM + L + XXX + IV = MCMLXXXIV.
Original question. Write 2469 in Roman numerals. 2469 = 2000 + 400 + 60 + 9 = MM + CD + LX + IX = MMCDLXIX.
5.6 Egyptian numerals
Egyptian numerals are base-10 landmarks written by repetition. There is a symbol for each power of ten, and a quantity is written by repeating each symbol as many times as its coefficient. Ten copies of one symbol regroup into one symbol of the next landmark.
Worked conversion. 2456 = 2 × 1000 + 4 × 100 + 5 × 10 + 6, so it uses two 1000-symbols, four 100-symbols, five 10-symbols and six unit symbols.
Original question. Represent 709 in Egyptian numerals. 709 = 7 × 100 + 9, so it uses seven 100-symbols and nine unit symbols, with no 10-symbol.
5.7 Mayan numerals
The Mayan system places symbols in vertical positions with values 1, 20, 360 and 7200. A dot means 1 and a bar means 5; a shell marks zero.
Worked conversion. 400 = 1 × 360 + 2 × 20 + 0, so the numeral has one dot in the 360s place, two dots in the 20s place and a shell in the units place.
Original question. Write 123 in Mayan numerals. 123 = 6 × 20 + 3, so the numeral has six dots in the 20s place and three dots in the units place.
5.8 Chinese rod numerals
Chinese rod numerals are base-10 and positional. Rods for 1 to 9 alternate between a vertical form and a horizontal form, so that neighbouring places do not merge into one long symbol. A blank position is left for a missing power of ten.
Worked conversion. 604 = 6 hundreds + 0 tens + 4 units, so it uses the rod for 6 in the hundreds place, a blank in the tens place and the rod for 4 in the units place.
Original question. Write 352 in Chinese rod numerals. 352 = 3 hundreds + 5 tens + 2 units, so the hundreds rod shows 3, the tens rod shows 5 and the units rod shows 2, with alternating orientations.
5.9 Comparing the systems
The table below summarises what each system contributes.
| System | Base or landmark type | Positional? | Zero or blank? |
|---|---|---|---|
| Base 2, 4, 5, 7 | powers of the base | yes, if written positionally | digit 0 in the position |
| Mesopotamian | powers of 60 | yes | blank or placeholder |
| Roman | fixed values 1, 5, 10, 50, 100, 500, 1000 | no | no zero |
| Egyptian | powers of 10 | no | no zero |
| Mayan | 1, 20, 360, 7200 | yes | shell for zero |
| Chinese rods | powers of 10 | yes | blank position |
The comparison shows why the Hindu system won: ten digits, place value and a zero that is both a placeholder and a number. Everything else in this gallery is a step along the way.