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We write 2,367 without thinking about it. Chapter 3 of the new Class 8 maths book asks how people managed before our numerals existed, and why the system we use won. This post explains the chapter's five big ideas, works through the conversions it asks you to do, flags the common mistakes and ends with a practice plan.
Which chapter is this?
The contents page of Ganita Prakash, Grade 8 Part 1 (Reprint 2026-27) lists Chapter 3 as "A Story of Numbers". The book's introduction says the chapter shows how number representation evolved across times and places to reach its modern form. The Chapter 3 PDF starts with Reema finding a paper of strange symbols, and the rest of the chapter decodes them.
What were the earliest ways to write numbers?
The chapter begins with counting on body parts and with tally marks cut into bone. It describes the Ishango bone and the Lebombo bone, which carries 29 notches, as two of the oldest objects thought to record numbers. It also gives the number words of the Gumulgal people of Australia, who counted in twos: 3 was "ukasar-urapon" (2 + 1) and 4 was "ukasar-ukasar" (2 + 2).
Tallies work for small numbers, but writing 2,367 as marks is hopeless. The next idea was to give special symbols to "landmark numbers".
How do Roman numerals work, and why were they slow?
Roman numerals use symbols for landmark numbers: I = 1, V = 5, X = 10, L = 50, C = 100, D = 500, M = 1,000.
Worked example 1. The chapter breaks 2,367 into landmark numbers, taking as many of the largest as possible: 2,367 = 1,000 + 1,000 + 100 + 100 + 100 + 50 + 10 + 5 + 1 + 1. Each landmark becomes its symbol: MM, CCC, L, X, V, II, giving MMCCCLXVII. Check: 2,000 + 300 + 50 + 10 + 5 + 2 = 2,367. (In the Reprint 2026-27 PDF the numeral is printed as MMCCCLXII, which drops the V and would be 2,362. Working it out yourself catches slips like this.)
The problem is calculation. The chapter asks you to multiply CCXXXI by MDCCCLII without converting them, and explains that people using Roman numerals relied on the abacus, which only specially trained people used.
What is a base, and what is place value?
A number system whose landmark numbers are the powers of a number n (1, n, n², n³ and so on) is a base-n system. The Egyptian system used powers of 10 as landmarks, with a separate symbol for each, so it had a base but still needed a new symbol for every power.
The big step is place value: let the position of a symbol say which power it counts. The chapter shows this with the Mesopotamian base-60 system. The St Andrews MacTutor history of mathematics site describes the Babylonian system as a positional base-60 system built from just two symbols, one for 1 and one for 10, which are combined to make each "digit" from 1 to 59. Its traces survive in our clocks: 60 seconds in a minute and 60 minutes in an hour.
Worked example 2: base 60. Write 7,530 in base 60, as the chapter does.
3,600 (that is, 60²) goes into 7,530 twice: 2 × 3,600 = 7,200. Remainder 330.
60 goes into 330 five times: 5 × 60 = 300. Remainder 30.
So 7,530 = (2) × 3,600 + (5) × 60 + 30, written in three places as 2 | 5 | 30.
Why did zero matter so much?
Now try 3,605 = (1) × 3,600 + (0) × 60 + 5. The middle place is empty. Without a symbol for "nothing here", 1 | _ | 5 can be misread as 1 | 5, which is 65. The chapter explains that early Mesopotamian scribes left blank spaces, which caused this ambiguity, and that later Mesopotamians used a placeholder symbol. MacTutor confirms that later Babylonians did invent a symbol for an empty place, but did not use it at the end of a number.
The Maya of Central America also had place value and a shell-shaped zero, but the chapter points out that their third landmark was 360, not 400, so it was not a true base-20 system. MacTutor's Mayan mathematics page makes the same point.
Worked example 3: Mayan. Write 721 with landmarks 1, 20 and 360. 721 = 2 × 360 + 0 × 20 + 1. The 20s place is empty, so the shell zero is needed.
The chapter then shows the Indian system: base 10, ten symbols including 0, one digit in each place, and zero treated as a number in its own right. It credits Brahmagupta's *Brāhmasphuṭasiddhānta* (628 CE) with codifying arithmetic with zero. MacTutor's page on Indian numerals notes that the Babylonians had place value much earlier, around the 19th century BC, but in base 60, and that the Indians were the first to develop a base-10 positional system, the direct ancestor of ours.
What mistakes do Class 8 students make with this chapter?
Thinking place value means base 10. Base 60 and base 5 systems can have place value too.
Forgetting the empty place. When converting, write a 0 for every power that is not used, or the number changes.
Calling the Mayan system base 20. Its third place counted 360s.
Mixing up zero as a placeholder and zero as a number. The first fills a gap; the second can be added, subtracted and multiplied like any other number.
Treating the history boxes as dates to memorise. Focus on why each idea solved a problem the previous one had.
Worked example 4: other bases. The chapter asks you to write 25 in base 8, base 5 and base 2.
Base 8: 25 = 3 × 8 + 1, so 31.
Base 5: 25 = 1 × 25 + 0 × 5 + 0, so 100.
Base 2: 25 = 16 + 8 + 1 = 1×16 + 1×8 + 0×4 + 0×2 + 1×1, so 11001.
A practice plan
Convert five numbers into Roman numerals and back.
Write 63, 132, 200, 60 and 3,605 in base 60, as the chapter's exercise asks. Mark which need an empty place.
Write 77, 100 and 361 in the Mayan system.
Build your own base-4 system with made-up symbols and teach it to a family member.
Write 25 and 50 in base 2, then check by adding up the place values.
When a conversion goes wrong and you cannot see why, the Fermi Problem Library has a step-by-step solver that explains the reasoning behind each step. For more number puzzles in the same spirit, see our post on Ramanujan's notebooks.
Try this now: write today's date in base 2, then ask someone at home to decode it.
