The Egyptian system, and what a landmark number is for

What to assume they know

• Roman numerals: grouping a number into tens, fives and ones — landmark numbers, and greedy grouping from the largest down
• Powers of ten, written and read as 10², 10³, up to 10⁷
• Reading a number's digits and knowing what each place is worth
• Adding the digits of a number, for the length argument in section 8

What they should be able to do

• State the Egyptian rule for generating landmark numbers and apply it to produce the first five
• Explain why the rule forces every landmark to be a power of ten
• Name the eight powers the chapter gives signs to, and the largest number the set can reach
• Write a given number as a sum of powers of ten and then as a run of Egyptian signs
• Read an Egyptian numeral back into a Hindu numeral by counting each sign
• Predict how many signs a number's Egyptian numeral will need, before writing it
• Say what a landmark number is doing that a plain tally is not

Where it usually goes wrong

• "The Egyptians had a place value system." They did not. The signs may be written in any arrangement, and the numeral means the same thing — position carries nothing. The two printed numerals in item 2 are laid out in rows precisely to make that visible.
• "You have to write the signs from largest to smallest." Convention, not arithmetic. Rearranging the signs of a printed Egyptian numeral does not change its value, which is exactly what place value will later stop being true.
• "Ten was picked because it is a nice round number." It is round because it was picked. The chapter's next subsection replaces it with 5 and everything goes through, which is the argument that ten is a choice.
• "A landmark number is just a big number." It is a number the system gives a new basic sign to and bundles by. The definition is functional, and the chapter states it that way at Part I p.58.
• "More signs in the numeral means a bigger number." 1111 needs four signs and 784 needs nineteen. The count of signs is the digit sum, not the value.
• "Reading a numeral means reading the top row first." In item 2(i) an arch sits at the end of the bottom row. Sort by sign, not by position.

Questions to check understanding

• Generate the landmark numbers of the Egyptian system from the stated rule
• Convert a number up to five digits into Egyptian signs and back
• Read a numeral whose signs are deliberately out of order
• State, without writing the numeral, how many signs a given number will need
• Explain why every Egyptian landmark is a power of ten, from the rule alone
• Say what the largest number writable with the eight printed signs is, and why

Examples worth working on the board

Items marked counted were read off the printed page image; the Egyptian signs never appear in the extracted text at all, so every numeral below was read from the page.

• Date and framing (Part I p.61): a written system developed by the Egyptians around 3000 BCE, presented as still using landmark numbers, with the novelty located entirely in which landmarks it uses.
• The pebble construction (Part I p.61, text and artwork). First landmark 1. Bundle ten collections of the previous landmark; that bundle's size is the next landmark. So 1, then 10, then 10 x 10 = 100, and onward. The drawing beneath makes the powers geometric: counted, a single pebble, then a slanting line of ten, then a filled square array, then a coloured cube. Point, line, square, cube — the first four powers of ten, one dimension at a time. This is the cleanest visual in the section and the chapter does not comment on it.
• The eight signs (Part I p.62, a labelled row). Powers printed above the signs: 1, 10, 10², 10³, 10⁴, 10⁵, 10⁶, 10⁷. The signs, described rather than reproduced: a single upright stroke; an arch open at the bottom; a coil; a tall hooked stem; a curved tapering form; a small creature with a marked eye; a kneeling figure with raised arms; a small circle with rays. Eight signs, and the set stops there.
• The stated procedure (Part I p.62): count the number into groups of the landmark numbers, beginning from the largest landmark below it, then write the signs — the same greedy rule as the Roman section, with a different landmark list.
• 324, worked on the page (Part I p.62): 324 = 100 + 100 + 100 + 10 + 10 + 4, written as three coils, two arches and four strokes. Both the sum and the numeral are printed.
• Figure it Out item 1 (Part I p.62). Write these in Egyptian signs: 10458, 1023, 2660, 784, 1111, 70707. Hand over the six numbers.
• Figure it Out item 2 (Part I p.62), two numerals to read back. Described exactly as printed, because the layout is part of the task — (i) a top row of two coils; then a row of three arches; then another row of three arches; then a bottom row of six strokes followed by one more arch. (ii) a top row of four hooked stems; then a row of three coils; then a bottom row of two strokes followed by two arches. Note that in (i) the arches are split across three rows and one of them sits after the strokes, so a student who reads row by row rather than sign by sign will get it wrong. That trap is the exercise. Added readings, as a check only: (i) totals two hundreds, seven tens and six ones; (ii) totals four thousands, three hundreds, two tens and two ones.
• The length rule. Worked by me, not printed: because each sign stands for one power and no power is needed ten or more times, the number of signs in an Egyptian numeral is exactly the sum of the number's digits. On the six numbers of item 1 that gives 18, 6, 14, 19, 4 and 21 signs. 1111 is the cheapest of the six and 70707 the dearest, which is not the order of their sizes — a good thirty seconds of video.

Figures to have open

• The four-dimension pebble figure — dot, line of ten, ten-by-ten square, ten-by-ten-by-ten cube — with each labelled by its power. This is the chapter's own artwork (Part I p.61) and it is the best argument in the section, so redraw it cleanly and add the labels the page leaves off.
• The eight-sign chart with powers above (Part I p.62). Redraw the signs as simple line shapes; they are ancient forms, but the printed rendering is the book's and should not be traced.
• Item 2's two numerals, drawn in the printed row layout, so the sorting trap in section 7 is real. Layout matters here; keep it.
• A sign-count bar for section 8: six bars, one per number of item 1, heights 18, 6, 14, 19, 4, 21. An added construction.

Where this sits in the book

• NCERT Ganita Prakash Class 8, Part I, printed Chapter 3, "A Story of Numbers", §3.3 "The Idea of a Base", subsection I. The Egyptian Number System, Part I pp.61–62. The Figure it Out set of two items is at Part I p.62.
• Back-reference: landmark numbers are defined at Part I p.58, in §3.2 IV.
• Forward pointers: the base-5 variant and the definition of base-n begin on the same page, Part I p.62, and belong to What "base n" means, and why ten is a choice not a law; the ceiling of the sign set at 10⁷ is argued at Part I p.69 and belongs to Why a base alone still runs out of symbols.
• The chapter's SUMMARY (Part I p.81) states the base-n definition; it does not mention Egypt by name.