A Story of Numbers

1. Reema's curiosity — why a "story" of numbers?

The chapter opens with Reema finding a scrap of paper covered in strange symbols. Her father explains they are numbers written about 4000 years ago in Mesopotamia (a region in west Asia, mostly present-day Iraq). This sparks the chapter's journey: how did the idea of a number, and the way we write it, evolve across time and geography into today's neat, efficient form?

Humans counted even in the Stone Age — livestock, food, goods traded, ritual offerings, days till the next full moon. But they did not use the digits we use today. The modern way of writing numbers — the digits 0 through 9 — originated in India about 2000 years ago. The chapter traces the path that led there.

2. The mechanism of counting

Imagine a Stone-Age herder asking: "Did all my cows return? Do I have fewer than my neighbour?" — without any number names. Three methods solve this:

• Method 1 — objects: keep one stick (or pebble) per cow. The pile of sticks is the count.
• Method 2 — sounds/names: recite a fixed order of letters $a,b,c,\dots,z$, one per object. Drawback: only 26 objects.
• Method 3 — written symbols: a fixed sequence of marks, e.g. the Roman $I, II, III, IV, V,\dots, X,\dots, XX$.

The key idea behind all three is a one-to-one mapping: pair each object with exactly one item of the standard sequence (no item used twice). A sequence used like this is a number system; the written marks in it are numerals (e.g. $0,1,5,36,193$ are Hindu numerals).

3. Some early number systems

I. Body parts. A group in Papua New Guinea uses fingers, wrist, elbow, shoulder, etc. (up to 27 points) as their standard sequence.

II. Tally marks. The oldest written method — a notch cut on bone or cave wall for each object. The Lebombo bone (South Africa, ~44,000 years old, 29 notches) and the Ishango bone (DR Congo, 20,000–35,000 years old) are among the oldest mathematical artefacts — possibly lunar calendars.

III. Counting in twos. The Gumulgal (Australia) built names from 2s: $urapon=1$, $ukasar=2$, then $3=2+1$ (ukasar-urapon), $4=2+2$ (ukasar-ukasar), and so on. Remarkably, the Bakairi (South America) and the Bushmen (South Africa) — far apart, no contact — built the very same "count-in-2s" system, hinting at common ancestors.

4. The Roman numerals — landmark numbers

The Roman system was used in Europe for centuries (evolved from the Greek system, ~8th century BCE). It groups into 10s, then 5s, then 1s, and gives special symbols — called landmark numbers — to certain values:

To write a number, write it as a sum of landmark numbers, taking as many of the largest as possible first. Sometimes a subtractive shortcut is used: $4=IV$ (one less than 5), $40=XL$, $9=IX$ — though usage was not always consistent ($40$ was sometimes $XXXX$).

Drawback: arithmetic — especially multiplication and division — is very clumsy in Roman numerals (Romans used an abacus for sums), and you cannot write arbitrarily large numbers without inventing new symbols.

5. The idea of a base — Egyptian (base-10)

The Egyptian system (~3000 BCE) refined the landmark idea beautifully: every landmark is 10 times the one before it. Group 10 ones → the landmark 10; group ten 10s → $10\times10=100$; and so on. So the landmarks are exactly the powers of 10:

Each power has its own picture-symbol (a stroke for 1, a heel-bone for 10, a coil for 100, etc.). A number is written by repeating these symbols.

Because all landmarks are powers of one number, multiplying landmarks just adds the powers ($10\times100=1000$). This makes computation far easier than in the Roman system — the great advantage of a base.

6. Base-$n$ in general — base-5, base-7, base-2…

Nothing forces the group size to be 10. Group in 5s instead: landmarks become $5^{0}=1,\ 5^{1}=5,\ 5^{2}=25,\ 5^{3}=125,\dots$ — a base-5 system. The rule for any system:

Why a base helps arithmetic: the product of any two landmarks is again a landmark (e.g. $10\times10^{2}=10^{3}$), because adding powers stays in the family. This is exactly why multiplication is easy with a base but hard with Roman numerals.

7. Shortcoming of the Egyptian system

The Egyptian system handled numbers up to a crore ($10^{7}$) and computed reasonably well. But for ever-larger numbers you would need an unending sequence of new symbols for higher powers of 10 — the original "numbers never end" problem comes back. Also no symbol ever appears 10 or more times (10 of them regroup into the next landmark). The fix is the next great idea: place value.

8. Place value — Mesopotamian / Babylonian (base-60)

Ancient Mesopotamia (Babylon) used a base-60 (sexagesimal) system — landmarks $1, 60, 60^{2}=3600, 60^{3},\dots$ It used a vertical wedge $\textbf{Y}$ for 1 and a corner wedge $\textbf{<}$ for 10, building every value from 1 to 59. (Its legacy survives in time: $1$ hour $=60$ min, $1$ min $=60$ s.)

The breakthrough: instead of a separate symbol for each power, use position. The rightmost block counts the 1s, the next block the 60s, the next the $3600$s, and so on.

Defect: Mesopotamians left a blank space for a missing power. Blanks were hard to size consistently, so $60$, $3600$ etc. could be confused. Later they added a placeholder symbol for the blank — an early ancestor of 0 — but they used it mainly in the middle, never at the end (so they could not cleanly write what we call $3600$).

9. Place value elsewhere — Mayan (base-20) and Chinese (rod, base-10)

Mayan (Central America, 3rd–10th century CE): "almost" base-20, with landmarks $1, 20, 20\times18=360, 20^{2}\times18=7200,\dots$ (the odd $360$ likely tied to their calendar). Symbols: a dot $=1$, a bar $=5$, and a shell $=0$ (placeholder), written vertically — lowest block = 1s, next = 20s, next = 360s.

Chinese rod numerals (by 3rd century AD, used till 17th): a base-10 place value system using rod patterns. Zong rods mark units/hundreds, Heng rods mark tens/thousands (alternating to make positions clear). E.g. the numeral reading $2,6,3,4$ by position $=2\times10^{3}+6\times10^{2}+3\times10+4=\mathbf{2634}$. With a zero symbol it would have been a fully developed place value system.

10. The Hindu number system — and the triumph of zero

Finally, the system we use everywhere today: the Hindu (also Indian or Hindu-Arabic) number system — a base-10 place value system with ten digits $0,1,2,\dots,9$.

History grounded in the chapter:

• Ancient texts like the Yajurveda Samhita named powers of 10 — eka (1), dasha (10), shata (100), sahasra (1000), ayuta (10,000) — up to $10^{12}$.
• Earliest known use of ten digits with a 0 (a dot): the Bakhshali manuscript (~3rd century CE). Aryabhata (~499 CE) computed scientifically with the system.
• It reached the Arab world by ~800 CE; Al-Khwarizmi (whose name gives "algorithm") popularised it (On the Calculation with Hindu Numerals, ~825). From there Fibonacci (~1200) urged Europe to adopt it.
• Zero as a number (not just a placeholder) was given its arithmetic rules by Brahmagupta (628 CE) — using 0 and negatives he created what we now call a ring, the foundation of algebra.

Because 0 is a full digit and each position holds a single digit, the system writes every number unambiguously with just 10 symbols and computes efficiently. Laplace praised it as "foremost among useful inventions." Note: "Hindu" here means a people/geography, not a religion.

11. The five big ideas — the whole story in one box

Place value systems were used by the Mesopotamian (Babylonian), Mayan, Chinese and Indian civilisations; only the Indian system, with a true zero, became the world standard.

12. NCERT "Figure it Out" — solved

Roman (p.59): $1222=MCCXXII$; $\ 2999=MMCMXCIX$; $\ 302=CCCII$; $\ 715=DCCXV$.

Roman products (p.60): $V\times L=CCL\,(250)$; $\ V\times D=MMD\,(2500)$; $\ VII\times IX=LXIII\,(63)$.

Gumulgal arithmetic (urapon=1, ukasar=2):

• $9+7=16$: nine = ukasar-ukasar-ukasar-ukasar-urapon, seven similar; sum $16=$ ukasar $\times8$.
• $9-6=3=$ ukasar-urapon.
• $9\times4=36=$ ukasar repeated 18 times.
• $16\div4=4=$ ukasar-ukasar.

Egyptian (p.62): e.g. $1023=10^{3}+10+10+1+1+1$ (one coil-of-1000, two heels, three strokes). Numbers like $1111=10^{3}+10^{2}+10+1$ use one symbol of each of four powers.

Why Hindu beats Roman (p.61): only 10 symbols (vs ever-new Roman symbols), a true place value with 0, so any number is written unambiguously and arithmetic (especially $\times,\div$) is easy.

13. Common mistakes to avoid

• Confusing landmark numbers ($I,V,X,L,C,D,M$ or powers of $n$) with the base — the base is the fixed multiplier $n$; the landmarks are its powers.
• Thinking a base must be 10 — any positive integer $n\ge2$ can be a base; base-10 just matches our 10 fingers.
• Forgetting the regrouping limit: in base-$n$ no power may appear $n$ or more times (10 ones → one ten; 60 sixties → one $60^{2}$).
• Saying the Mesopotamian/Mayan system is "fully" place-value — it lacked a clean end-of-number zero, causing ambiguity.
• Treating 0 as "nothing only" — in the Indian system it is also a number with its own arithmetic (Brahmagupta).
• Writing Roman additively where a subtractive form is expected — $9=IX$ (not $VIIII$), $40=XL$.

14. Quick revision checklist

• Number system = fixed-order standard sequence; written symbols = numerals.
• Grouping → landmark numbers → powers of a base $n$ → place value → zero.
• Base-$n$ landmarks: $1,n,n^{2},n^{3},\dots$; base-10 = decimal.
• Egyptian = base-10 (no place value); Babylonian = base-60; Mayan ≈ base-20; Chinese rod = base-10; Hindu = base-10 place value with 0.
• Product of two landmarks of a base is another landmark → easy multiplication.
• Indian zero: dot in Bakhshali (~3rd c. CE); Aryabhata (499); Brahmagupta gave 0 its arithmetic (628).