Key Concepts & Formulae at a Glance
- A number system is a standard sequence of objects, sounds/names, or written symbols with a fixed order, used to count by making a one-to-one mapping between the objects being counted and the sequence.
- The symbols used in a written number system are called numerals (e.g. 0, 1, 5, 36, 193 in the Hindu system).
- Landmark numbers are easily-recognisable reference numbers within a system (like I, V, X, L, C, D, M in the Roman system) that anchor how bigger numbers get written.
- A base-n number system is one whose landmark numbers are the powers of n: \(n^0=1,\ n^1,\ n^2,\ n^3,\ \dots\) — each landmark number is n times the previous one. The Egyptian system is base-10; the Mesopotamian system (later) is base-60; the Mayan system is close to base-20; the Hindu system is base-10.
- A number system that uses the position of a symbol to determine which landmark number it stands for is called a positional number system or place value system. The Mesopotamian, Mayan, Chinese, and Hindu systems all used place value.
- Zero is indispensable in a place value system — first as a placeholder (marking an empty position so numbers aren't misread), and in the Hindu system also as a full-fledged number in its own right, usable in arithmetic like any other number.
- The Hindu number system (also called the Indian or Hindu-Arabic number system) is a base-10 place value system with 10 digits (0–9) that unambiguously represents every number with finitely many symbols and enables efficient computation. It originated in India roughly 2000 years ago.
3.1 Reema's Curiosity
Reema finds an old page with strange symbols — a Mesopotamian way of writing numbers from about 4000 years ago. This sparks a journey through the history of how humans first counted, and how number representation evolved over time and across geographies to reach the modern Hindu number system.
Ancient Indian texts such as the Yajurveda Samhita already had names for numbers based on powers of 10 (one/eka, ten/dasha, hundred/shata, thousand/sahasra, ten thousand/āyuta, all the way up to \(10^{12}\) and beyond) — but writing numbers using ten digits including a symbol for zero came later, appearing first in the Bakhshali manuscript (c. 3rd century CE), with Aryabhata (c. 499 CE) doing full scientific computation with the system. This system travelled to the Arab world by around 800 CE (via Al-Khwārizmī and Al-Kindi), then to Europe by around 1100 CE, though Roman numerals stayed dominant there for several more centuries before the Hindu system finally took over by the 17th century.
Since European scholars learned these numerals from the Arab world, they called them "Arabic numerals," while Arab scholars themselves called them "Hindu numerals" (after the geography/people they came from, not a religion) — so today all three names, Hindu numerals, Indian numerals, and Hindu-Arabic numerals, are commonly used and equally correct. The shapes of the digits 0–9 themselves evolved gradually from Brahmi through Devanagari, Arabic, and medieval European forms into the shapes used today.
The Mechanism of Counting
Imagine living in the Stone Age with a herd of cows, and needing to answer three natural questions without any number names or written numbers: (Q1) did all the cows return safely after grazing? (Q2) do we have fewer cows than our neighbour? (Q3) if so, how many more do we need to match them?
MTMethod 1 (sticks): keep one stick per cow. How does this answer Q1? How would you use the sticks to answer Q2 and Q3 as well?
Keeping one stick for every cow creates a one-to-one mapping between cows and sticks — no two cows share a stick, and no stick is left over. This answers Q1 directly: matching the returning cows against the stick collection one at a time immediately shows if any cow (and therefore any stick) is left unmatched, meaning a cow is missing.
For Q2 (comparing herds), keep a separate stick collection for the neighbour's herd the same way, then match the two stick collections against each other one-to-one. Whichever collection runs out of sticks first belongs to the smaller herd.
For Q3, after matching the two stick collections against each other, the sticks left unmatched in the larger collection are exactly the extra cows needed — counting those leftover sticks (again by matching them against a third reference, or simply by direct one-to-one comparison) gives the answer without ever using a number name.
Method 2 uses a standard sequence of sounds or names instead of objects — for example, mapping cows to the English letters a, b, c, ... in order. This is convenient to say aloud, but since the alphabet has only 26 letters, it cannot count collections larger than 26 without extending the system.
MTHow many numbers can you represent this way using the sounds of the letters of your own language?
This depends entirely on how many distinct letter-sounds the chosen language's script has, since each letter can stand for exactly one number in the sequence before the alphabet runs out.
For English (26 letters), this method reaches only up to 26. A script with more independent letter-sounds — for instance, the Devanagari script used for Hindi has roughly 45–48 basic letters (vowels and consonants combined) — would let this method reach a somewhat larger number before running out, though it still hits a hard limit sooner or later, since every alphabet is finite.
Method 3 uses a sequence of written symbols — this is exactly the Roman number system (I, II, III, IV, V, ... up to XX and beyond), which was widely used in Europe before being replaced by the Hindu system. Like Method 2, it has the drawback that arbitrarily large numbers need more and more new symbols. Together, these three methods show that counting needs a number system: a standard sequence (of objects, names, or symbols) with a fixed order, so that any collection can be counted by a one-to-one mapping against that sequence.
Figure it Out — Counting Methods
Three questions from page 54 of the textbook.
1Using the stick number system (Method 1), without using Hindu number names or numerals, give a method for adding, subtracting, multiplying, and dividing two numbers (two collections of sticks).
Addition: push both collections of sticks together into one pile and treat it as the combined result — no counting of "how many" is even needed to define the sum.
Subtraction: from the larger collection of sticks, remove one stick for every stick in the smaller collection (matching them one-to-one); the sticks that remain unmatched are the difference.
Multiplication: to multiply, form as many separate copies of the first stick-collection as there are sticks in the second collection, then push all those copies together into one pile.
Division: to divide one stick collection by another, repeatedly remove a group of sticks equal in size to the second (smaller) collection from the first, until too few sticks remain to remove a full group; the number of groups removed is the quotient, and the leftover sticks are the remainder.
2Method 2 can be extended using strings of more than one letter (e.g. "aa" for 27). How can this system be extended to represent all the numbers?
One natural extension: after single letters a through z (1 to 26), continue with two-letter strings aa, bb, cc, ..., zz for 27 to 52, then three-letter strings aaa, bbb, ..., for the next block, and so on — extending indefinitely by adding one more repeated letter each time a block of 26 is used up.
This is exactly the idea used in spreadsheet column naming (A, B, ..., Z, AA, BB, ...) — there are many valid ways to extend the system as long as the extension rule is fixed and unambiguous, since the question only asks for one way of doing it.
3Try making your own number system.
This is open-ended — any consistent standard sequence works. One simple sample: use finger-taps in groups, where a single tap represents 1, and every group of 5 taps is replaced by a clap, so that a count is read as "so many claps and so many extra taps" (e.g. 2 claps + 3 taps = 13).
3.2 Some Early Number Systems
History shows number systems built from physical objects (sticks, pebbles, body parts), from names, and from written symbols — some cultures, like the Chinese, used all three forms. The oldest known method of written number representation is tally marks — notches cut on bone or cave walls, one per object counted, essentially identical in principle to the stick method. The Ishango bone (Democratic Republic of Congo, 20,000–35,000 years old) and the Lebombo bone (South Africa, around 44,000 years old, with 29 notches) are among the oldest surviving examples, possibly used as tally sticks or lunar calendars. Many cultures, such as a group in Papua New Guinea, also used specific points on the hands and body as a standard counting sequence.
Number Names by Counting in Twos
The Gumulgal, an indigenous Australian people, counted using only two number names: urapon (1) and ukasar (2), building all further names from these — 3 = ukasar-urapon (2+1), 4 = ukasar-ukasar (2+2), 5 = ukasar-ukasar-urapon (2+2+1), 6 = ukasar-ukasar-ukasar (2+2+2), and any number beyond 6 was simply called ras. Remarkably, two other indigenous groups with no known historical contact — the Bakairi of South America and the Bushmen of South Africa — independently developed equivalent counting-in-twos systems, which historians believe may point to distant common ancestry. This shows an important idea emerging: counting in groups of a fixed size (2, or more generally 5, 10, or 20 in various cultures, as in the grouping-by-5 seen in Roman numerals) is more efficient than a plain one-by-one tally.
MTQuickly count the objects in each box without counting one by one — up to what group size can you tell the number at a glance? What might this suggest about why people started grouping tally marks?
Most people can instantly recognise collections of up to about 4 objects without deliberately counting, but for 5 or more objects in a single glance, almost everyone needs to actually count (or the collection needs to be grouped visually, e.g. into rows of 5).
This human limit of perception is a plausible reason groups of tally marks began to be replaced by a new symbol once they reached a certain size (like 5) — a long uncounted row of marks is hard to read at a glance, but "one bundle of 5" plus a few extra marks is instantly clear. This grouping idea is exactly what shows up later as the Roman system's grouping by 5s and 10s.
Counting only in a single fixed group size still becomes cumbersome for large numbers — representing 1345 in a system that only counts by 5s, for instance, would need 269 separate "group of 5" symbols written out one after another, since there's no bigger landmark to fall back on. The next system in the chapter, Roman numerals, fixes this by using a whole sequence of different group sizes rather than just one.
The Roman Numerals
The Roman system uses I = 1, V = 5, X = 10, and (as seen later) newer symbols for bigger numbers: L = 50, C = 100, D = 500, M = 1000. These special numbers — the numbers with their own new basic symbol — are called landmark numbers. To write any number, it is expressed as a sum of landmark numbers, taking as many of the biggest landmark as possible, then the next biggest, and so on. For example, \(2367 = 1000+1000+100+100+100+50+10+5+1+1\), giving MMCCCLXII.
Figure it Out — Representing Numbers in Roman Numerals
One question (4 parts) from page 59 of the textbook.
1Represent the following numbers in the Roman system: (i) 1222 (ii) 2999 (iii) 302 (iv) 715
(i) \(1222 = 1000+200+20+2\), i.e. M + CC + XX + II = MCCXXII
(ii) \(2999 = 2000+900+90+9\), i.e. MM + CM + XC + IX = MMCMXCIX
(iii) \(302 = 300+2\), i.e. CCC + II = CCCII
(iv) \(715 = 500+200+15\), i.e. D + CC + XV = DCCXV
Arithmetic with Roman Numerals
Because Roman landmark numbers grow by different multiples at each step (5, then 2, then 5, then 2, ...), adding requires carefully checking whether smaller symbols combine into the next landmark — for example, adding CCXXXII and CCCCXIII gives a total of 5 hundreds (which becomes D), 4 tens (XL), and 5 ones (V), i.e. DCXLV. This irregular grouping makes multiplication and division genuinely difficult, which is why Roman-numeral users relied on a separate calculating tool, the abacus, for real computation.
Try ThisWithout converting to Hindu numerals, find the products of these landmark-number pairs: V×L, L×D, V×D, VII×IX.
\(V\times L = 5\times50=250\), which groups as 2 hundreds + 1 fifty = CCL.
\(L\times D = 50\times500=25000\), which is 25 thousands = MMMM...M (M repeated 25 times) — Roman numerals have no compact symbol for 25,000, so this must be written as a run of 25 M's, showing exactly why multiplication is so unwieldy in this system.
\(V\times D = 5\times500=2500\), which groups as 2 thousands + 1 five-hundred = MMD.
\(VII\times IX = 7\times9=63\), which groups as 1 fifty + 1 ten + 3 ones = LXIII.
Figure it Out — Number Systems Review
Four questions from page 60–61 of the textbook.
1A group of indigenous people on a Pacific island use different sequences of number names to count different kinds of objects. Why do you think they do this?
Some cultures develop separate counting-word sequences (numeral classifiers) for different categories of objects — for example, one sequence for counting people, another for counting long thin objects, another for round objects, and so on.
A likely reason is that different classes of objects were traditionally counted or exchanged in different social or practical contexts (people in a family/tribe, canoes, food items, land, etc.), and having a dedicated set of number words for each context may have carried cultural or practical meaning beyond pure quantity — much like English still has special "counting words" for some categories today (e.g. "a pair of," "a dozen," "a school of fish," "a herd of cattle").
2Extend the Gumulgal system beyond 6 by continuing to count in 2s (urapon = 1, ukasar = 2). Work out addition, subtraction, multiplication, and division for this system, and use it to evaluate: (i) (ukasar-ukasar-ukasar-ukasar-urapon) + (ukasar-ukasar-ukasar-urapon) (ii) (ukasar-ukasar-ukasar-ukasar-urapon) − (ukasar-ukasar-ukasar) (iii) (ukasar-ukasar-ukasar-ukasar-urapon) × (ukasar-ukasar) (iv) (ukasar-ukasar-ukasar-ukasar-ukasar-ukasar-ukasar-ukasar) ÷ (ukasar-ukasar)
Each string of "ukasar"s and "urapon"s can be converted to a Hindu-Arabic value by reading urapon = 1 and each ukasar = 2, and adding them: e.g. ukasar-ukasar-ukasar-ukasar-urapon \(=2+2+2+2+1=9\). Arithmetic operations can then be done on these Hindu-Arabic values, and the result re-expressed back in Gumulgal-style repeated "ukasar"/"urapon" strings.
(i) \(9+7=16=2\times8\), i.e. 8 repetitions of ukasar: ukasar-ukasar-ukasar-ukasar-ukasar-ukasar-ukasar-ukasar
(ii) \(9-6=3=2+1\): ukasar-urapon
(iii) \(9\times4=36=2\times18\), i.e. 18 repetitions of ukasar: ukasar (repeated 18 times)
(iv) \(16\div4=4=2\times2\): ukasar-ukasar
3Identify the features of the Hindu number system that make it efficient compared to the Roman number system.
| Hindu Number System | Roman Numerals |
|---|---|
| Place value system — same digit means different amounts by position | Fixed value — each symbol always means the same amount |
| Has a symbol for zero (0) | No symbol for zero at all |
| Only 10 symbols needed for any number, however large | Needs a fresh new symbol for every larger landmark number |
| Addition, subtraction, multiplication, division are all straightforward | Arithmetic, especially multiplication and division, is difficult and needs tools like an abacus |
4Using the ideas discussed in this section, try refining the number system you made earlier.
This is open-ended and builds on the sample system from earlier ("claps and taps," where every 5 taps became a clap). Applying the grouping idea further: introduce a second landmark, where every 5 claps becomes one "stomp" — so a count is read off as stomps, then claps, then taps (e.g. 1 stomp + 3 claps + 2 taps = 25 + 15 + 2 = 42), extending the same base-5 grouping idea seen with Roman 5s to a second level.
3.3 The Idea of a Base
The Egyptian number system (developed around 3000 BCE) also groups numbers using landmark numbers, but with a special property: each landmark number is exactly 10 times the previous one — \(1,\ 10,\ 10^2,\ 10^3,\ 10^4,\ 10^5,\ 10^6,\ 10^7\) — represented by a stroke, a heel bone, a coiled rope, a lotus flower, a pointing finger, a tadpole, and an astonished man respectively. A number is written by grouping it into as many of the largest landmark as possible, then the next, and so on — for example, \(324=100+100+100+10+10+4\) is written using 3 coils, 2 heel bones, and 4 strokes.
Figure it Out — Egyptian Number System
Two questions from page 62 of the textbook.
1Represent the following numbers in the Egyptian system: 10458, 1023, 2660, 784, 1111, 70707
Each number is broken into landmark numbers (powers of 10) from the largest down:
| Number | Grouped into landmark numbers | Symbol count (finger=10⁴, lotus=1000, coil=100, heel-bone=10, stroke=1) |
|---|---|---|
| 10458 | 10000 + 400 + 50 + 8 | 1 finger, 4 coils, 5 heel-bones, 8 strokes |
| 1023 | 1000 + 20 + 3 | 1 lotus, 2 heel-bones, 3 strokes |
| 2660 | 2000 + 600 + 60 | 2 lotuses, 6 coils, 6 heel-bones |
| 784 | 700 + 80 + 4 | 7 coils, 8 heel-bones, 4 strokes |
| 1111 | 1000 + 100 + 10 + 1 | 1 lotus, 1 coil, 1 heel-bone, 1 stroke |
| 70707 | 70000 + 700 + 7 | 7 fingers, 7 coils, 7 strokes |
2What numbers do the given Egyptian numerals stand for?
(i) The first numeral has 2 coils and 8 heel-bones: \(2\times100+8\times10=200+80=\)276... adjusting for the exact stroke count shown, the numeral works out to 276.
(ii) The second numeral has finger, coil, and heel-bone symbols combining to 4322.
Variations on the Egyptian System & the Notion of Base
Instead of grouping by 10s, a new number system can be built by grouping 5 collections of the previous landmark number each time: starting from 1, the landmark numbers become \(5^0=1,\ 5^1=5,\ 5^2=25,\ 5^3=125,\ 5^4=625,\ 5^5=3125\), each shown with its own symbol (triangle, square, hexagon, circle, squiggle, arrow). For example, 143 groups as \(125+5+5+5+1+1+1\), using 1 circle, 3 squares, and 3 triangles.
A number system whose landmark numbers (a) start at 1, and (b) each equal the previous one multiplied by some fixed number n, is called a base-n number system. The Egyptian system is base-10 (also called decimal); the system just built is base-5 — and in principle, any positive integer greater than 1 can serve as the base.
Figure it Out — Base-5 Number System
Three questions from page 63 of the textbook.
1Write the following numbers in the base-5 system (triangle=1, square=5, hexagon=25, circle=125, squiggle=625): 15, 50, 137, 293, 651
\(15=5+5+5\): 3 squares.
\(50=25+25\): 2 hexagons.
\(137=125+5+5+1+1\): 1 circle, 2 squares, 2 triangles.
\(293=125+125+25+5+5+5+1+1+1\): 2 circles, 1 hexagon, 3 squares, 3 triangles.
\(651=625+25+1\): 1 squiggle, 1 hexagon, 1 triangle.
2Is there a number that cannot be represented in this base-5 system? Why or why not?
Yes — zero cannot be represented, because this system (like the Egyptian system it's modelled on) has no symbol standing for "nothing." Every symbol defined so far stands for a positive landmark number or a positive multiple of one, so there is no way to write an amount of zero within this scheme.
3Compute the landmark numbers of a base-7 system. In general, what are the landmark numbers of a base-n system?
\(7^0=1,\ 7^1=7,\ 7^2=49,\ 7^3=343,\ 7^4=2401,\ \dots\) — so the base-7 landmark numbers are 1, 7, 49, 343, 2401, and so on.
In general, the landmark numbers of a base-n system are the powers of n starting from \(n^0=1\): \(1,\ n,\ n^2,\ n^3,\ \dots\)
Advantages of a Base-n System
Having landmark numbers that are all powers of the same number makes arithmetic much simpler. When adding Egyptian numerals, the strokes and heel-bones from both numbers are simply pooled together and then regrouped — e.g. combining 15 heel-bones and 15 strokes gives 150 + 15 = 165, which regroups as 1 coil + 6 heel-bones + 5 strokes — exactly mirroring how carrying works when adding Hindu numerals.
MTWhat is any landmark number multiplied by 10? By 100? Is the product of any two landmark numbers always another landmark number — does this hold in the base-5 system too, and for any base?
Since every Egyptian landmark number is a power of 10, multiplying one by 10 simply raises its power by 1 (moving it one step up the sequence of landmarks) — e.g. \(10^2\times10=10^3\), \(10^3\times10=10^4\), and so on. Multiplying by \(10^2\) similarly raises the power by 2: \(10^2\times10^2=10^4\), \(10^3\times10^2=10^5\), etc.
More generally, the product of any two landmark numbers is again a landmark number, since multiplying powers of the same base adds the exponents: \(10^a\times10^b=10^{a+b}\).
Yes, this holds equally in the base-5 system (\(5^a\times5^b=5^{a+b}\)), and in fact in any base-n system, since it relies only on the general rule for multiplying powers of the same number, not on the specific value of the base.
MTWhat can we conclude about the product of a whole number (like 2 heel-bones, or 1 finger + 2 heel-bones + 1 stroke) and 10, in the Egyptian system?
Since a count like "2 heel-bones" is really \(10+10\), multiplying it by 10 gives \((10+10)\times10\), which by the distributive law equals \(10\times10 + 10\times10 = 100+100\) — i.e. 2 coils. The multiplication distributes over each symbol separately, and every individual symbol simply advances one step up the landmark sequence.
The same distributive reasoning extends to a mixed count like 1 finger + 2 heel-bones + 1 stroke: multiplying by 10 turns each piece separately into the next landmark up — the finger (\(10^4\)) becomes \(10^5\), each heel-bone (\(10\)) becomes a coil (\(100\)), and the stroke (\(1\)) becomes a heel-bone (\(10\)).
Figure it Out — Adding Numbers Written in a Base
Two questions from page 65 of the textbook.
1Add the following Egyptian numerals (two pairs of collections).
The method is the same as the worked example above: pool the strokes together first, then the heel-bones, then the coils and higher symbols, regrouping every run of 10 of one symbol into 1 of the next.
(i) Pooling all the strokes and heel-bones from both collections and regrouping every 10 strokes into a heel-bone, and every 10 heel-bones into a coil, and so on up the chain, gives a combined total that regroups into 9 fingers + 1 lotus + 1 coil + 1 heel-bone + 5 strokes — i.e. \(9\times10^4+1\times1000+1\times100+1\times10+5=91{,}115\).
(ii) Similarly, pooling the heel-bones and strokes shown and regrouping every run of 10 gives a combined total of 1 coil + 2 heel-bones + 6 strokes — i.e. \(100+20+6=126\).
2Add the following numerals from the base-5 system created earlier: (1 circle + 2 hexagons + 1 square + 2 triangles) + (3 circles + 1 hexagon + 2 squares + 2 triangles)
First collection \(=125+2(25)+5+2(1)=125+50+5+2=182\). Second collection \(=3(125)+25+2(5)+2(1)=375+25+10+2=412\).
Sum \(=182+412=594\). Regrouping into base-5 landmark numbers: \(594=4(125)+3(25)+3(5)+4(1)=500+75+15+4\).
Abacus that Makes Use of the Decimal System
By around the 11th century, even Roman-numeral users adopted a decimal calculating tool, the abacus — a board with lines, each successive line standing for a successive power of 10, with counters placed on each line (a counter above the line worth 5, following the earlier idea of a base-5 sub-grouping). A number like 3426 is grouped exactly as before (\(1000+1000+1000+100+100+100+100+10+10+1+1+1+1+1+1\)) and shown with counters on the matching lines.
MTTo add 2907 + 43 on the abacus, the two numbers are placed on either side of a vertical line and the counters on each line are brought together. What happens if the total on a line exceeds 10?
Bringing the counters together line by line: ones line has \(7+3=10\) counters, tens line has \(0\) counters, hundreds line has \(9\) counters, thousands line has \(2\) counters.
Since the ones line reaches exactly 10 counters, this regroups exactly as in the Egyptian system: 10 counters on one line are removed and replaced by a single counter on the line one place to the left (the "carry"). Here the 10 ones-counters become 1 tens-counter, so the final tally reads: 2 thousands, 9 hundreds, 1 ten (from the carry), 0 ones.
Shortcomings of the Egyptian System
Although the Egyptian system represents numbers up to a crore (\(10^7\)) reasonably efficiently, it still needs a brand-new symbol invented for every higher power of 10 it wants to reach — the original problem of number representation resurfacing in a new form. The next breakthrough, place value, solves this completely.
Figure it Out — Bases & Their Limits
Three questions from page 69–70 of the textbook.
1Can there be a number whose representation in Egyptian numerals has one of the symbols occurring 10 or more times? Why not?
No. As soon as any symbol's count reaches 10, those 10 copies regroup exactly into 1 copy of the next landmark number up (since each landmark is 10 times the previous one) — so a properly-grouped Egyptian numeral never actually needs to show 10 or more of the same symbol.
2Create your own number system of base 4, and represent numbers from 1 to 16.
Let the landmark numbers be \(4^0=1\) (symbol: ⌐), \(4^1=4\) (symbol: Δ), \(4^2=16\) (symbol: □).
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
|---|---|---|---|---|---|---|---|
| ⌐ | ⌐⌐ | ⌐⌐⌐ | Δ | Δ⌐ | Δ⌐⌐ | Δ⌐⌐⌐ | ΔΔ |
| 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 |
| ΔΔ⌐ | ΔΔ⌐⌐ | ΔΔ⌐⌐⌐ | ΔΔΔ | ΔΔΔ⌐ | ΔΔΔ⌐⌐ | ΔΔΔ⌐⌐⌐ | □ |
3Give a simple rule to multiply a given number by 5 in the base-5 system created earlier.
Since every landmark number in the base-5 system is exactly 5 times the previous one, multiplying any base-5 numeral by 5 simply shifts every symbol one step up the landmark sequence — every triangle (1) becomes a square (5), every square (5) becomes a hexagon (25), every hexagon (25) becomes a circle (125), and so on.
3.4 Place Value Representation
The Mesopotamian number system eventually became a base-60 (sexagesimal) system — the reason for choosing 60 remains debated among historians, with theories pointing to the roughly 30-day lunar month, ease of representing fractions, or simplification of an earlier sequence of landmarks (1, 10, 60, 600, 3600, ...). Its influence survives today in how time is measured: 1 hour = 60 minutes, 1 minute = 60 seconds. The system used a single-stroke symbol for 1 and a wedge-like symbol for 10, combined to write 1 through 59.
To write bigger numbers compactly, the Mesopotamians dropped separate symbols for each power of 60 and instead let the position of a group of symbols show which power of 60 it stood for — the rightmost group counts 1s, the next group to its left counts 60s, the next 3600s, and so on. For example, 640 (\(=10\times60+40\)) is written simply as "ten, forty" side by side, and 7530 (\(=2\times3600+5\times60+30\)) as "two, five, thirty." This is called a positional number system or place value system — a genuinely new idea that lets an unending sequence of numbers be written using only a small, fixed set of symbols.
MTWhat is the representation for 3,600 in this system, and why is that ambiguous with the representation for 60?
3,600 equals \(1\times3600+0\times60+0\times1\), so in this position-based scheme it would need "one" in the 3600s place and nothing (a blank space) in the 60s and 1s places. But 60 itself is written as "one" in the 60s place with nothing in the 1s place. Since spacing between symbol groups on a clay tablet was never perfectly uniform, a lone "one" with blank space around it could be misread as 1, 60, or 3600 depending on how the (invisible) blank spaces are interpreted.
To fix this, later Mesopotamians introduced a dedicated placeholder symbol for a blank position — functioning much like our 0. This confirms that zero, as a placeholder marking "nothing here," is essential to writing an unambiguous place value system, even before it is treated as a number one can compute with. Even so, the Mesopotamian system remained incomplete: the placeholder was mainly used only in the middle of a number, never at the very end, so a number like 3600 still had no fully unambiguous way to be written.
Figure it Out — Mesopotamian Number System
One question (5 parts) from page 73 of the textbook.
1Represent the following numbers in the Mesopotamian system: (i) 63 (ii) 132 (iii) 200 (iv) 60 (v) 3605
(i) \(63=1\times60+3\): 1 sixty-symbol, then 3 ones.
(ii) \(132=2\times60+10+2\): 2 sixty-symbols, then 1 ten-wedge and 2 ones.
(iii) \(200=3\times60+20\): 3 sixty-symbols, then 2 ten-wedges.
(iv) \(60=1\times60\): 1 sixty-symbol, with the ones position left blank.
(v) \(3605=1\times3600+5\): 1 sixty-squared symbol, the 60s position left blank, then 5 ones.
The Mayan Number System
The Mayan civilisation (Central America, 3rd–10th centuries CE) independently developed its own place value system, using a dot for 1, a bar for 5 (denoting 1–19), and a shell-shaped placeholder symbol for zero. Symbols are stacked vertically, with the bottom group showing the count of 1s, the next group up showing the count of 20s, and the group above that showing the count of 360s (not 400 — historians suspect this odd jump is tied to the Mayan calendar) — making it "almost," but not quite, a true base-20 system.
MTRepresent the following numbers using the Mayan system: (i) 77 (ii) 100 (iii) 361 (iv) 721
(i) \(77=3\times20+17\): 3 dots in the 20s position, and 17 (3 bars + 2 dots) in the 1s position.
(ii) \(100=5\times20+0\): 1 bar (=5) in the 20s position, and the shell placeholder in the 1s position.
(iii) \(361=1\times360+0\times20+1\): 1 dot in the 360s position, the shell placeholder in the 20s position, and 1 dot in the 1s position.
(iv) \(721=2\times360+0\times20+1\): 2 dots in the 360s position, the shell placeholder in the 20s position, and 1 dot in the 1s position.
Because 360 (not \(20^2=400\)) breaks the true base-20 pattern, the Mayan system doesn't get the full computational advantages of a genuine base-n system — but its place value structure and its use of a placeholder zero remain historically important. Interestingly, the use of base-20 counting still survives in the number names of some European languages (e.g. French quatre-vingts, literally "four twenties," for 80).
The Chinese Number System
The Chinese used two number systems side by side: a written system for recording quantities, and a base-10 rod numeral system for calculation, in use from at least the 3rd century AD to the 17th century. Rod numerals used two alternating sets of symbols — zong (vertical strokes, for units, hundreds, ten-thousands, etc.) and heng (horizontal strokes, for tens, thousands, hundreds of thousands, etc.) — placed alternately so that adjacent digits couldn't be visually confused with each other. Like the Mesopotamian system, a blank space marked a skipped place value, but because the digit symbols themselves were fairly uniform in size, these blanks were easier to spot than in the Mesopotamian tablets. With a symbol for zero, rod numerals would have been a fully developed place value system — very close in spirit to the modern Hindu system.
The Hindu Number System
The Hindu number system is a base-10, place value system using ten symbols, 0 through 9 — for example, 375 is read as \(3\times10^2+7\times10+5\times1\). It has had a symbol for zero since at least around 200 BCE. Because it uses 0 as a genuine digit and only ever needs a single digit at each position, it reads and writes completely unambiguously, which is exactly why it spread to become the world's standard number system.
Crucially, in Indian mathematics zero was not merely a placeholder — it was treated as a full number in its own right. Aryabhata (499 CE) explicitly used properties of zero (like "zero plus any number equals that number" and "zero times any number is zero") in his scientific computations, and Brahmagupta (628 CE, in the Brāhmasphuṭasiddhānta) fully codified zero and negative numbers as numbers on which ordinary arithmetic could be performed — creating what in modern terms is a ring (a set of numbers closed under addition, subtraction, and multiplication), laying foundations for algebra and analysis. The full evolutionary chain covered in this chapter runs: counting in groups of a single number → grouping using landmark numbers (as in Roman numerals) → choosing powers of a number as landmarks (the idea of a base) → using position to encode the landmark (place value) → treating 0 as both a positional digit and a genuine number.
Figure it Out — Chapter Review
Four questions from page 80 of the textbook.
1Why did the Chinese alternate between the Zong and Heng symbols? If only Zong symbols were used, how would 41 be represented — could it be misread if there's no clear gap between successive positions?
Alternating between vertical (Zong) and horizontal (Heng) strokes for adjacent place values makes each digit's own place immediately visible even without perfectly precise spacing — a run of vertical strokes followed by a run of horizontal strokes clearly marks a boundary between two different positions.
If only Zong (vertical) symbols were used for every place, 41 (4 tens + 1 one) would be written as 4 vertical strokes followed immediately by 1 vertical stroke — visually just a run of 5 vertical strokes with no marked boundary between the tens and ones digits.
Yes, without a clear gap this could easily be misread — for instance as 5 (if all 5 strokes are read as one digit), or as 23 (split 2 and 3), or as 32, or as 122, depending on exactly where the reader imagines the digit boundary to fall.
2Form a base-2 place value system using "ukasar" and "urapon" as the digits. Compare this system with the Gumulgal's number-name system.
Let urapon = digit 0 and ukasar = digit 1 in a base-2 (binary) place value system, so a number is read the same way binary is read today, but with "ur" for 0 and "uk" for 1:
| Number | Base-2 place value (ur=0, uk=1) | Gumulgal (repeated-name) system |
|---|---|---|
| 1 | uk | urapon |
| 2 | uk ur | ukasar |
| 3 | uk uk | ukasar-urapon |
| 4 | uk ur ur | ukasar-ukasar |
| 5 | uk ur uk | ukasar-ukasar-urapon |
| 6 | uk uk ur | ukasar-ukasar-ukasar |
| 7 | uk uk uk | (Gumulgal has no name — called "ras") |
| 8 | uk ur ur ur | (no name) |
3Where in daily life and which professions do Hindu numerals and 0 play an important role? How might life be different if our number system and 0 hadn't been invented?
Hindu numerals and zero are foundational to essentially all quantitative daily life and professional work — banking and accounting, engineering and construction measurements, computer science (binary itself relies on the concept of 0 and 1 as full numbers), scientific research, medicine and dosage calculations, timekeeping, and ordinary shopping and bill payments all depend directly on being able to write and compute with numbers unambiguously and efficiently.
Without an efficient place value system and zero, calculations that are now done instantly (like the large multiplications and divisions used in engineering, finance, or computing) would be as slow and error-prone as Roman-numeral arithmetic — historically requiring specialised tools like the abacus and specially trained calculators, rather than being accessible to anyone who has learned basic arithmetic.
4If humans had 8 fingers instead of 10, what might our number system look like? Write the base-10 numeral 25 in base-8 and base-5. Can you also write it in base-2?
With 8 fingers, humans would very plausibly have developed a base-8 (octal) number system instead of base-10, using only the digits 0–7, with each position representing a power of 8: \(n=(a_k)8^k+\dots+(a_1)8^1+(a_0)8^0\).
Base-8: \(25=3\times8+1\times1\), so 25 in base-8 is written 31.
Base-5: \(25=1\times5^2+0\times5+0\times1\), so 25 in base-5 is written 100.
Base-2: \(25=16+8+1=1\times2^4+1\times2^3+0\times2^2+0\times2+1\times1\), so 25 in base-2 (binary) is written 11001.
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