Mesopotamian base-60: place value in a sexagesimal system

What to assume they know

• Why a base alone still runs out of symbols — why a sign for every landmark cannot go on forever
• What "base n" means, and why ten is a choice not a law — the base-n definition, and regrouping at the base
• Powers written as 60², 60³, and multiplication of a two-digit number by 60
• The Indian place value system, well enough to notice what is being generalised
• Reading time as hours, minutes and seconds

What they should be able to do

• State the two marks the Mesopotamians used, and build any number from 1 to 59 from them
• Report the theories the chapter offers for the choice of 60, without ranking them
• Write a number as a sum of multiples of powers of 60, and then in positional form
• Explain why no place can hold 60 or more, and regroup an expression that does
• State the defining property of a positional or place value system in your own words
• Show, with the chapter's own examples, how uneven spacing lets one numeral be read as several numbers
• Explain what a placeholder does, and name the case it still fails to handle

Where it usually goes wrong

• "Base 60 means sixty different symbols." Two. Everything from 1 to 59 is built additively from those two inside a single place, which is why the table on Part I p.71 exists.
• "Place value was invented in India." The chapter is explicit that Mesopotamia, the Maya, China and India all used place value representations. What India added is the subject of a later topic, and overstating it here spoils that one.
• "Their system was base 60, so it was completely unlike ours." The structure is identical; only the base and the marks differ. Reading a Mesopotamian numeral from the right, place by place, is exactly what a student already does.
• "A blank space is as good as a zero." It is not, and the six-column table proves it: a gap has no fixed width, so a reader cannot count gaps. That is the whole failure.
• "The placeholder fixed it." It fixed the middle of a numeral. A number ending in an empty place was still ambiguous, because the mark was not used there.
• "Sixty was chosen for a known reason." Three named theories, an open end, and no verdict. The chapter says outright that the choice has puzzled many.
• "The chapter's diamond symbols are Mesopotamian." They are the book's own teaching notation, borrowed from Indian numerals, and the book says so on the page where it introduces them.

Questions to check understanding

• Write any number from 1 to 59 with the two marks, and read one back
• Convert a number below 216000 into sums of multiples of powers of 60, then into positional form
• Regroup an expression in which a place holds 60 or more, showing each step
• Given a numeral with an unclear gap, list every number it could be read as
• State what makes a system positional, and test a given system against it
• Explain what a placeholder is for, and give a number the Mesopotamian placeholder still could not disambiguate

Examples worth working on the board

Items marked counted were read off the printed page; the Mesopotamian marks are absent from the extracted text throughout.

• The framing (Part I p.70): the system began with different signs for different landmark numbers and only later became base-60, at which point its representation became very efficient.
• The theories for 60 (Part I p.70). The chapter names three, all offered as competing, and then closes the list with an explicit open end rather than a fourth. The three: a connection between 60 and the periods of important events, with a 30-day lunar month and the apparent circuit of the Sun given as instances; the ease of writing fractions, which the chapter declines to pursue; the collapse of an earlier landmark list — 1, 10, 60, 600, 3600, 36000, … — down to the powers of 60 alone; and others unnamed. Present them as a set; the chapter does not choose.
• The survival (Part I p.70): 1 hour = 60 minutes, 1 minute = 60 seconds.
• The two marks (Part I p.70): one mark for 1, a second for 10.
• The 1-to-59 table (Part I p.71). Counted, giving the arrangement a teacher needs: 1, 2, 3 are one, two and three upright marks in a row; 4 is three over one; 5 is three over two; 6 is three over three; 7 is four over three; 8 is four over four; 9 is three over three over three. Then 10 is a single corner mark; 11 and 12 are that mark followed by one and by two uprights; 20 and 30 are two and three corner marks; 40 and 50 stack them two-over-two and three-over-two; 59 is five corner marks beside nine uprights. The pattern to state: tens first and then units, each written additively, all inside one place.
• The chapter's borrowed landmark symbols (Part I p.71): for 60, 60² = 3600 and 60³ = 216000 the chapter draws diamonds containing the figures 1, 2 and 3, and says outright that these are Indian numerals adopted for easy recall rather than anything Mesopotamian. Keep that admission in view — it is a model of how to borrow notation honestly.
• The tablet (Part I p.71): a photograph captioned as a reproduction of a Mesopotamian tablet, ruled into a grid of cells with marks in each. Useful as atmosphere; nothing in the argument depends on it.
• 640, worked on the page (Part I p.72): 640 = 10 x 60 + 40. Written the Egyptian way it would take ten copies of the 60 symbol and four corner marks; written compactly it becomes a corner mark, the 60 symbol, and four corner marks — read as ten sixties and one forty.
• 7530, worked on the page (Part I p.72): 7530 = 2 x 3600 + 5 x 60 + 30.
• The regrouping identity, printed in full (Part I p.72): (1) x 3600 + (70) x 60 + 2 = (1) x 60² + (60 + 10) x 60 + 2 = (1) x 60² + 60² + (10) x 60 + 2 = (2) x 60² + (10) x 60 + 2. Three lines, and the whole reason no place may hold 60.
• The compact-form table (Part I p.72): 640 and 7530 each shown twice, once with the landmark symbols and once without. Both rows are needed; the argument is the difference between them.
• The reading rule (Part I p.73): rightmost group counts ones, the group to its left counts sixties, the next counts 3600s, and so on; where a power does not occur, a blank space was left.
• The historical caveat (Part I p.73): the chapter says the Mesopotamians probably did not arrive at this the way the chapter just did, and reports a suggestion that the similarity of their earlier signs for 1 and 60, plus an accidental use of them, may have led there. Keep the hedge.
• Figure it Out (Part I p.73), item 1, five numbers to write in this system: 63, 132, 200, 60, 3605. Hand over the five. As a check only: they are chosen so that 60 has an empty ones place and 3605 has an empty sixties place — the two cases the blank has to survive, and the reason the exercise is here at all rather than after the placeholder is introduced.
• The ambiguity table (Part I p.73), six columns, each with the number above and two rows of numeral beneath, labelled as the chapter's own spacing and as the Mesopotamian spacing. The numbers are 1, 60, 3600, 12, 602, 36002. Counted: the first three are all a single upright mark, distinguished only by which place it sits in; the last three are a corner mark and two uprights, distinguished only by the width of the gap between them. Six numbers, three shapes. This is the single most persuasive figure in the section.
• The placeholder (Part I p.74): later Mesopotamians assigned a mark to stand for a blank space, which the chapter compares directly to our 0 — and then notes that it was used mainly inside a numeral and not at the end, so a number like 3600 was still not written unambiguously.

Figures to have open

• The 1-to-59 chart, redrawn with the two marks clean and the stacking exactly as printed (Part I p.71). Section 3 depends on it and the printed version is small.
• The compact-versus-landmark comparison for 640 and 7530 (Part I p.72), both rows. The whole of section 7 is the difference between the two rows.
• The six-column ambiguity table (Part I p.73). This is the chapter's own figure and it is worth redrawing so the gaps can be shown opening — the same marks sliding apart and the reading changing.
• A place-strip for section 8 labelled 60³, 60², 60, 1 with a numeral dropped into it. Standard schematic.
• The tablet photograph is not needed.

Where this sits in the book

• NCERT Ganita Prakash Class 8, Part I, printed Chapter 3, "A Story of Numbers", §3.4 "Place Value Representation", subsection I. The Mesopotamian Number System, Part I pp.70–74. The placeholder paragraphs run to the top of Part I p.74, immediately before subsection II.
• Figure it Out for this subsection is at Part I p.73, one item with five parts.
• The ambiguity table is at Part I p.73 and carries no figure number.
• The chapter's SUMMARY (Part I p.81) gives the place value definition in its fifth bullet and lists the four civilisations in its sixth.
• Back-reference: the opening scene of the chapter (Part I p.48) is about this system, and the strip of nine numerals there is its 1 to 9.