A Story of Numbers — Class 8 Mathematics (Ganita Prakash)

"The ingenious method of expressing every possible number using a set of ten symbols (each symbol having a place value and an absolute value) emerged in India. The idea seems so simple nowadays that its significance and profound importance is no longer appreciated." — Pierre-Simon Laplace (1749–1827)

What the book actually covers (2026-27) This chapter is a history of number systems, not a chapter on rational numbers. The pre-2025 Class 8 book had rational numbers in this slot; Ganita Prakash Chapter 3 replaces it entirely. You will meet tally marks, the Gumulgal counting system, Roman numerals, Egyptian numerals, an invented base-5 system, the Mesopotamian base-60 system, Mayan numerals, Chinese rod numerals and finally the Hindu system — and at each step you will ask one question: what is this system missing? The old rational-numbers material has been kept at the end of this page under Appendix — beyond the current syllabus, since it is useful revision but is no longer examined in this chapter.

1. Reema's Question

The chapter opens with Reema finding a scrap of paper covered in strange marks. Her father tells her they are numbers written in Mesopotamia around 4000 years ago — and that sets off the questions the whole chapter answers:

• Since when have humans been counting, and what were they counting?
• Since when have people written numbers in the modern form?
• How would the Mesopotamians have written 20? 50? 100?

Humans needed to count as far back as the Stone Age — to track livestock, food stores, goods traded, offerings made, and the passing of days so that a new moon or the onset of a season could be predicted. But they did not write those numbers the way we do.

The Indian origin, in dates

On the name. European scholars learned these numerals from the Arab world and so called them 'Arabic'. Arab scholars themselves called them 'Hindu numerals'. The word Hindu here refers to a geography and a people, not a religion. Modern usage is 'Hindu numerals', 'Indian numerals' or 'Hindu-Arabic numerals'.

2. The Mechanism of Counting

Imagine you keep a herd of cows, ten thousand years ago, with no number words and no numerals. Three natural questions arise:

1. Have all the cows returned from grazing?
2. Do I have fewer cows than my neighbour?
3. If fewer, how many more would I need to match them?

Method 1 — objects. Keep one stick for every cow. Matching each cow to exactly one stick, with no two cows sharing a stick, is called a one-to-one mapping. The collection of sticks is the number.

Method 2 — names or sounds. Use a fixed sequence of sounds, such as the letters a, b, c, …, z, and match objects to them in order. Convenient to say, but it runs out — only 26 objects can be counted.

Method 3 — written symbols. Use a fixed sequence of marks. The sequence I, II, III, IV, V, … is exactly what Europe used before the Hindu system: the Roman number system.

The two definitions to remember

A number system is a standard sequence of objects, names or written symbols that has a fixed order. Counting a collection means making a one-to-one mapping between the collection and this sequence, following its order.

The written symbols in a number system are called numerals. So 0, 1, 5, 36, 193 are numerals of the Hindu system.

The challenge

Numbers never end, so a good number system must be unending and easy to count with.

Every system in this chapter is an attempt to get both at once.

3. Some Early Number Systems

I. Body parts

Groups of people in Papua New Guinea used, and still use, a fixed sequence of body parts as their standard sequence.

II. Tally marks

Notches cut into bone or stone — the same idea as sticks, but the mark is made instead of an object added.

• The Lebombo bone (South Africa) — about 44,000 years old, 29 notches, possibly a lunar calendar. One of the oldest known mathematical artefacts.
• The Ishango bone (Democratic Republic of Congo) — 20,000 to 35,000 years old, notches arranged in columns, possibly calendrical.

III. Counting in twos — the Gumulgal

The Gumulgal of Australia had these number names:

Anything greater than 6 was simply called ras.

A genuine historical puzzle. The Bakairi of South America and the Bushmen of South Africa independently developed equivalent systems, despite no known contact and enormous distance. One theory is a shared distant ancestry, with descendants later migrating apart.

The idea that emerges: count in groups of a fixed size, and use the word for that group to build bigger numbers. Common group sizes across history have been 2, 5, 10 and 20.

Why group at all? Try glancing at a scatter of objects and naming the count without counting. Most people manage up to about 4 — beyond 5 or so, we cannot take in a collection at a glance. That limit of perception is probably what pushed people to replace every group of 5 tally marks with a single new symbol.

4. The Roman Number System

Roman numerals build a number from landmark numbers — numbers important enough to be given their own symbol.

How to write a number: take as many of the largest landmark as possible, then the next, and so on.

27 = 10 + 10 + 5 + 1 + 1 → XXVII

2367 = 1000 + 1000 + 100 + 100 + 100 + 50 + 10 + 5 + 1 + 1 → MMCCCLXVII

The subtractive shortcut. Rather than IIII for 4, write IV (one less than five). Likewise XL for 40, XC for 90, CM for 900. The Romans were not always consistent about this — 40 was sometimes written XXXX.

Why arithmetic is hard here

Addition works by pooling symbols and regrouping — but the regrouping size keeps changing: five Is make a V, but two Vs make an X; five Xs make an L, but two Ls make a C.

Worked example. CCXXXII + CCCCXIII Pool: 6 Cs, 4 Xs, 5 Is. Five Cs make a D → D + C. Four Xs → XL. Five Is → V. = DCXLV (check: 232 + 413 = 645 ✓)

Multiplication is worse still, because the product of two landmarks is usually not a landmark: V × L = 250, which has no symbol of its own. And L × D = 25,000 has to be written as M repeated twenty-five times, since there is no symbol above M.

This is why users of Roman numerals depended on an abacus, and why only specially trained people could calculate at all.

5. The Idea of a Base

I. The Egyptian system (c. 3000 BCE)

The Egyptians built their landmark numbers by a single repeated rule:

• Start with 1.
• Group ten of the current landmark → the next landmark.

This gives 1, 10, 100, 1000, 10000, … — every landmark a power of 10, each with its own symbol.

324 = 100 + 100 + 100 + 10 + 10 + 1 + 1 + 1 + 1 → three hundred-symbols, two ten-symbols, four one-symbols.

II. The same idea with a different group size

Nothing forces the group size to be 10. Group five at a time instead:

$5^0=1,5^1=5,5^2=25,5^3=125,5^4=625,5^5=3125$

143 = 125 + 5 + 5 + 5 + 1 + 1 + 1

The definition

A number system is a base-n system if (a) its first landmark number is 1, and (b) every next landmark is the current one multiplied by a fixed number n.

Its landmark numbers are then exactly the powers of n: n⁰ = 1, n¹, n², n³, …

The Egyptian system is base-10 (also called decimal). The system just built is base-5.

Why a base makes arithmetic easy

Because every landmark is a power of the same number:

$n^a\times n^b=n^{a+b}$

The product of any two landmark numbers is again a landmark number. Nothing has to be regrouped into an awkward in-between value.

Two consequences worth memorising:

• Multiplying by the base replaces every symbol by the next one up. In base 10 this is the familiar rule "append a zero".
• No symbol can appear n or more times. Ten hundreds are one thousand, so ten hundred-symbols must be exchanged for one thousand-symbol. This is exactly why the digits of base 10 run only from 0 to 9 — and exactly why carrying works in ordinary addition.

The abacus

By the 11th century even Roman-numeral users calculated on a decimal abacus: a board of lines, each line a successive power of 10, with counters placed on each line, and a counter above a line worth 5.

What the Egyptian system still lacked

Bigger and bigger numbers demand an unending supply of new symbols, one for each higher power of 10. The original problem has simply reappeared in a new form.

6. Place Value — The Final Idea

I. Mesopotamia (base 60)

The Mesopotamian, or Babylonian, system became base-60 (sexagesimal), with symbols for 1 and for 10 used to build every count from 1 to 59.

Why 60? Nobody is certain. Theories include calendar periods (a 30-day lunar month), the ease of writing fractions with a number having many divisors, and an earlier landmark sequence 1, 10, 60, 600, 3600 collapsing into powers of 60. Its legacy is still on your wrist: 60 seconds, 60 minutes.

640 = (10 × 60) + 40 7530 = (2 × 3600) + (5 × 60) + 30

The breakthrough: drop the symbols for the powers of 60 altogether, and let position say which power each group counts. The rightmost group counts 1s, the next counts 60s, the next 3600s.

A number system with a base that uses the position of a symbol to determine which landmark it counts is a positional number system, or place value system.

The defect, and the invention of a placeholder

If a power of 60 is missing, the Mesopotamians left a blank space — and spacing was inconsistent between scribes. The numeral for 60 looks like the numeral for 1. Numbers became genuinely ambiguous.

Later Mesopotamians solved this with a placeholder symbol marking an empty position — the ancestor of our 0. But they used it mainly in the middle of numbers, not at the end, so ambiguity remained.

Zero is not optional in a place value system. Once position carries meaning, an empty position must be marked, or the number cannot be read.

II. The Mayans (3rd–10th centuries CE)

In Central America, the Maya independently invented place value and a placeholder — a symbol shaped like a seashell. A dot was 1 and a bar was 5, building 1 to 19; rows were stacked with units at the bottom.

Their landmarks were 1, 20, 360. Note the anomaly: the third is 360, not 400, possibly for calendar reasons. So the Mayan system has place value and zero but is not a true base-20 system, and therefore loses the computational advantages a genuine base gives.

III. Chinese rod numerals

A base-10 place value system, developed by at least the 3rd century CE and used until the 17th. Its clever trick: alternate vertical (Zong) rods for units, hundreds and ten-thousands with horizontal (Heng) rods for tens, thousands and hundred-thousands — so adjacent positions always look different and the boundary between places is visible.

Read: 2 (Heng) 6 (Zong) 3 (Heng) 4 (Zong) = (2 × 10³) + (6 × 10²) + (3 × 10) + 4 = 2634

Like the Mesopotamians they used a blank for a skipped place, but their more uniform symbols made the blanks easier to spot. With a symbol for zero, this would have been a fully developed place value system.

IV. The Hindu number system

Base 10, place value, ten symbols 0–9.

375 → (3 × 10²) + (7 × 10) + (5 × 1) = 375

The Hindu system has had a symbol for 0 since at least 200 BCE. Because it uses exactly one digit in each position, including 0, no ambiguity can arise anywhere.

And the decisive step: in India, zero was not merely a placeholder but a number in its own right, on equal footing with the others. Aryabhata used its arithmetic properties in 499 CE; Brahmagupta codified them in 628 CE.

By introducing 0 alongside the negative numbers, Brahmagupta created what is now called a ring — a set of numbers closed under addition, subtraction and multiplication. This laid the foundations of algebra and analysis.

7. The Five Ideas, in Order

The whole chapter is this ladder:

1. Count in groups of a single number. (ukasar-ukasar-urapon)
2. Group using landmark numbers. (I V X L C M)
3. Choose the landmark numbers to be powers of one number — the idea of a base. (1, 10¹, 10², 10³, 10⁴, …)
4. Use position to say which landmark a symbol counts — place value. (1 7 2 9)
5. Introduce 0, both as a positional digit and as a number.

Each step fixes the weakness of the one before it.

8. Summary

• A number system is a standard sequence of objects, names or written symbols with a fixed order; the written symbols are numerals.
• Landmark numbers are the reference sizes a system groups by, and gives symbols to.
• A base-n system has landmark numbers that are exactly the powers of n, starting from n⁰ = 1.
• In a base system, nᵃ × nᵇ = nᵃ⁺ᵇ, so the product of two landmarks is another landmark — which is what makes multiplication tractable.
• n of any landmark make the next landmark, so no symbol may appear n or more times — the reason base-10 digits stop at 9, and the reason carrying works.
• A place value (positional) system uses a symbol's position to determine which landmark it counts. Used by the Mesopotamian, Mayan, Chinese and Indian civilisations.
• A place value system must be able to mark an empty position, so a placeholder is unavoidable.
• The Hindu number system — base 10, place value, ten digits including 0 treated as a number — originated in India around 2000 years ago, spread worldwide, and is considered one of humanity's greatest inventions.

9. Quick Reference — Writing a Number in Any System

Appendix — beyond the current syllabus

The material below was the content of the old Class 8 Chapter 3 (rational numbers), replaced entirely in Ganita Prakash. It is not examined in this chapter under the 2026-27 syllabus. It is kept here because rational-number arithmetic remains assumed knowledge elsewhere in Class 8, and because it is directly useful revision before Class 9's Number Systems.

The family of numbers

• Natural numbers (N): 1, 2, 3, … — counting numbers, no zero, no negatives.
• Whole numbers (W): N together with 0.
• Integers (Z): … −3, −2, −1, 0, 1, 2, 3 … — makes subtraction always possible.
• Rational numbers (Q): all numbers p/q with p, q integers and q ≠ 0 — makes division (except by 0) always possible.
• Irrational numbers: cannot be written as p/q; decimal expansion is non-terminating and non-repeating (√2, π, e).
• Real numbers (R): all rationals together with all irrationals.

Standard form

p/q is in standard form when p and q share no common factor other than 1, and q is positive. 6/8 → 3/4; 5/−7 → −5/7.

Properties

• Commutative under + and ×, but not under − or ÷.
• Associative under + and ×, but not under − or ÷.
• Distributive: a × (b + c) = a × b + a × c.
• Identities: 0 for addition, 1 for multiplication.
• Inverses: −a for addition; 1/a for multiplication (a ≠ 0).

Operations

• Add / subtract: same denominator → operate on numerators. Different → take the LCM first. 1/2 + 1/3 = 3/6 + 2/6 = 5/6.
• Multiply: numerators together, denominators together. 2/3 × 4/5 = 8/15.
• Divide: multiply by the reciprocal. 2/3 ÷ 4/5 = 2/3 × 5/4 = 5/6.

Density

Between any two rational numbers there are infinitely many others — take the average repeatedly. Between 1/4 and 1/2: (1/4 + 1/2) ÷ 2 = 3/8.

Decimal expansions

A rational p/q in standard form terminates exactly when q's only prime factors are 2 and 5.

• 1/8 = 0.125 (q = 2³, terminates)
• 3/20 = 0.15 (q = 2² × 5, terminates)
• 1/3 = 0.333… (q = 3, repeats)
• 2/7 = 0.285714285714… (block of six repeats)

Later Indian mathematicians

• Bhaskara II (1114–1185) — author of Lilavati, a mathematics text in verse.
• Madhava of Sangamagrama (c. 1340–1425) — founder of the Kerala School; infinite series for π, sine and cosine, anticipating calculus.
• Srinivasa Ramanujan (1887–1920) and Manjul Bhargava (Fields Medal, 2014) in modern times.