3.4 Place Value Representation

I. The Mesopotamian (Babylonian) system

The Mesopotamians eventually settled on a base-60 scheme, the sexagesimal system, relying on only two marks: one for 1 and one for 10, combined to make every number from 1 to 59.

Why 60? No one is certain. Guesses point to their 30-day lunar month, the apparent yearly journey of the Sun, the convenience of writing fractions (60 splits many ways), and the way their early landmark run 1, 10, 60, 600, 3600, \dots eventually boiled down to the pure powers of 60. Whatever the cause, base 60 still rings in our clocks: 1 hour = 60 minutes, 1 minute = 60 seconds.

Let us borrow Indian numerals to label their landmark numbers:

1,\quad 60,\quad 60^2 = 3600,\quad 60^3 = 216000,\; \dots

To write a number, group it into these powers. For example:

75 = (1)\times 60 + 15, \qquad 8590 = (2)\times 3600 + (23)\times 60 + 10.

Now the brilliant move. Rather than drawing a symbol for each power of 60, simply write the counts side by side in fixed positions: the rightmost group counts the 1s, the next group to its left counts the 60s, the next counts the 3600s, and onward. A power that is absent is shown by a blank. This is the birth of place value, a symbol’s meaning depends on where it sits.

The Fourth Big Idea: Place Value A positional (place value) system is a base system that uses the position of each symbol to tell you which power of the base it counts. This is the summit of the story of numbers: it lets us write every number in the endless sequence using only finitely many symbols, never needing to invent new ones forever.

When grouping into powers of 60, no power may appear 60 or more times, sixty of one power bundle into one of the next:

(1)\times 3600 + (70)\times 60 + 2 = (2)\times 60^2 + (10)\times 60 + 2.

But the Mesopotamian system was not yet flawless. With only blanks to flag an empty position, scribes could not keep the spacing steady, so the same marks could be read more than one way (is that gap one empty place or two?). Later Mesopotamians patched this with a placeholder symbol for an empty position, an early ancestor of our 0. Even then, they mostly used it in the middle of a number, not at the end, so a number such as 3600 stayed ambiguous.

Figure it Out

Represent these in the Mesopotamian (base-60) system, written as counts of each power of 60: (i) 75   (ii) 145   (iii) 250   (iv) 120   (v) 3666.

    1. 75 = (1)\times 60 + 15 → one 60, then 15.
    1. 145 = (2)\times 60 + 25 → two 60s, then 25.
    1. 250 = (4)\times 60 + 10 → four 60s, then 10.
    1. 120 = (2)\times 60 + 0 → two 60s, then a blank (zero) ones.
    1. 3666 = (1)\times 3600 + (1)\times 60 + 6 → one 3600, one 60, then 6.

II. The Mayan system

In Central America, the Maya (flourishing around the 3rd–10th centuries CE) independently devised a place value system with a placeholder for zero, drawn like a seashell. They used a dot for 1 and a bar for 5 to assemble the numbers 1 to 19, then stacked positions vertically: the bottom group counts 1s, the group above counts 20s, the next counts 360s, and so on.

Oddly their third landmark is 360, not 400 (which a true base-20 system would demand), probably tied to their 360-day calendar. Because it is not a pure base, Mayan arithmetic loses the smoothness a true base offers, but their place value and their zero-placeholder were a big step forward.

Figure it Out

Represent these in the Mayan system (give the count in each position, 1s, 20s, 360s): (i) 85   (ii) 144   (iii) 380   (iv) 745.

    1. 85 = (4)\times 20 + (5)\times 1 → 4 in the 20s place, 5 in the 1s place.
    1. 144 = (7)\times 20 + (4)\times 1 → 7 in the 20s place, 4 in the 1s place.
    1. 380 = (1)\times 360 + (1)\times 20 + (0)\times 1 → 1, then 1, then shell.
    1. 745 = (2)\times 360 + (1)\times 20 + (5)\times 1 → 2, then 1, then 5.

III. The Chinese rod numerals

The Chinese used rod numerals (in use by at least the 3rd century CE, lasting until the 17th), a true base-10 place value system. The digits 1 to 9 were laid out as small arrangements of rods, and the system alternated two styles, zong (vertical) and heng (horizontal), at successive places, so neighbouring places could be told apart. For example,

\text{(4 zong)(2 heng)(7 zong)(1 heng)} = (4)\times 10^3 + (2)\times 10^2 + (7)\times 10 + 1 = 4271.

Like the Mesopotamians, the Chinese left a blank for an empty place; the alternating styles made those blanks easier to catch. With a symbol for zero, this would have been a fully grown place value system, and it is remarkably close to our own.

IV. The Hindu (Indian) number system

At last we reach the system you already use. It is base 10, with ten symbols 0, 1, 2, \dots, 9, and it is a place value system. Reading 486:

486 = (4)\times 10^2 + (8)\times 10 + (6)\times 1.

The Hindu system has carried a symbol for 0 since at least 200 BCE. Its decisive stroke is that 0 is not just a placeholder, it is a number in its own right, standing equal with the rest. Because every position holds exactly one digit (and an empty position is filled by 0), the scheme has no ambiguity whatsoever, which is why the whole planet adopted it.

Zero: The Idea That Changed Everything Aryabhata (Aryabhatiya, 499 CE) computed elaborately with the ten-symbol scheme and used the properties of 0 (that 0 + a = a and 0 \times a = 0). Brahmagupta (Brahmasphutasiddhanta, 628 CE) then set down the arithmetic of 0 as a full number, alongside negative numbers. In modern language he had described a ring, a set of numbers closed under addition, subtraction, and multiplication. These ideas grew into the bedrock of algebra and analysis. The discovery of 0 together with the Indian place value system is among the greatest and most far-reaching inventions in all of human history, underpinning modern science, computing, accounting, and engineering.

Did You Know? Binary, base 2, runs every computer you have ever touched. A switch is either off (0) or on (1), so a base-2 place value system fits electronics perfectly. The German polymath Gottfried Leibniz wrote about binary arithmetic in 1703, long before electronics existed; centuries later it became the silent language of every phone and laptop. So the same four ideas, grouping, landmark numbers, base, and place value, that the ancients chiselled into bone and clay now hum inside the chip in your pocket.

Five Stages in the Story of Numbers 1. Count in groups of a single size (Gumulgal: ukasar-ukasar-urapon). 2. Group using landmark numbers (Roman: I V X L C M). 3. Choose powers of a number as landmark numbers, the idea of a base (1, 10, 10^2, 10^3, \dots). 4. Use position to denote the landmark number, the idea of a place value system. 5. Treat 0 as a positional digit and as a number, completing the Hindu number system.

Try the place-value explorer below to see how each digit of a number contributes its landmark value.

Place Value Explorer (base 10)

Type a number and see how each digit is multiplied by its landmark number (a power of 10) and added up.

Figure it Out

  1. Why did the Chinese alternate zong and heng symbols? If only zong were used, how would 51 look, and how else might that numeral be misread when spaces are unclear?
  2. Build a base-2 place value system using “ukasar” and “urapon” as the two digits. How does it compare with the Gumulgal counting-in-twos scheme?
  3. Where in daily life, and in which professions, do the Hindu numerals and 0 matter? How might life differ if zero had never been invented?
  4. We probably use base 10 because we have 10 fingers. With only 8 fingers we might use base 8. Write the base-10 numeral 29 in base 8, base 6, and base 2.
  5. Convert the base-8 numeral 455_{(8)} into base 10, and then write that same value in base 6.
  1. The zong (vertical) and heng (horizontal) styles alternate so that neighbouring places look different and a reader can see where one place ends and the next begins. With only zong and unclear spacing, “5 1” could be misread as 6, 51, or 15, exactly the ambiguity a clear zero/placeholder removes.
  2. With urapon = 0 and ukasar = 1, base 2 gives $1 = $ uk, $2 = $ uk-ur, $3 = $ uk-uk, $4 = $ uk-ur-ur, \dots Both this base-2 system and the Gumulgal system use only two landmark numbers, but the base-2 system uses place value (so it can reach any number compactly), while Gumulgal merely repeats the group name and halts at 6.
  3. Zero and the Hindu numerals prop up commerce, accounting, measurement, science, computing (binary is base 2!), and communication. Without zero and place value, arithmetic with large numbers, and so most modern technology, would have been far harder to build. (Open-ended.)
  4. 29 = (3)\times 8 + 5 = \mathbf{35}_{(8)};   29 = (4)\times 6 + 5 = \mathbf{45}_{(6)};   29 = 16 + 8 + 4 + 1 = \mathbf{11101}_{(2)}.
  5. 455_{(8)} = 4\times 64 + 5\times 8 + 5 = 256 + 40 + 5 = \mathbf{301}. Then 301 = 1\times 216 + 2\times 36 + 2\times 6 + 1 = \mathbf{1221}_{(6)}.