A Story of Numbers | CBSE Class 8 Maths Notes

This note covers counting, one-to-one mapping, early number systems, Roman and Egyptian numerals, landmark numbers, bases, arithmetic, the abacus, place value, Mesopotamian, Mayan and Chinese representations, the Indian number system, and zero.

How can we count without modern number names?

Counting determines how many objects a collection contains. People needed to count food, livestock, goods exchanged in trade and offerings in rituals. They also tracked days to anticipate events such as the new moon, full moon and the arrival of a season.

What does matching objects achieve?

Imagine checking whether all the cows in a herd have returned after grazing. Keep a stick for each cow as it leaves. On its return, match the cow with its stick. The collection of sticks records the size of the herd without modern number names.

Definition: A one-to-one mapping pairs each object in one collection with its own object in another collection, without assigning the same partner to two objects.

This matching idea also helps compare two herds. Match their sticks in pairs. Unmatched sticks show which collection is larger, and the collection of unmatched sticks represents the difference. A comparison is possible before giving the quantities familiar written names.

A number system supplies a standard sequence of objects, names or written symbols in a fixed order. Counting matches the objects being counted to that sequence, following its order. The fixed order makes successive counts consistent.

Why must a counting sequence continue?

The sounds of the English letters from a to z could name successive quantities. In this particular arrangement, a names 1, b names 2 and z names 26. It cannot count collections larger than 26 unless the sequence is extended.

Sticks can provide an unending sequence, but a large collection needs equally many sticks. A sequence of names is convenient to say, but must have a rule for continuing. An effective number system must address both convenience and continuation.

Numerals are the symbols representing numbers in a written number system. A quantity and its written representation are different ideas: different number systems can represent the same quantity using different numerals.

What do body counting, tally marks and counting in twos show?

Some groups have used their hands and other body parts as a counting sequence. A group in Papua New Guinea used, and still uses, body parts in this way. The essential requirement is an agreed order for matching objects to the sequence.

How do tally marks record a collection?

Tally marks are marks made for counted objects, including notches cut into bone or another surface. One mark corresponds to each object. The completed collection of marks therefore records the total, much as a collection of sticks does.

Ancient bones have markings that are thought to represent numbers. The Ishango bone dates back 20,000 to 35,000 years; its notches possibly indicate calendrical systems, meaning systems for keeping track of dates or time. The Lebombo bone may have served as a tally stick or lunar calendar.

A lunar calendar follows the Moon's cycle. These possible uses should remain possibilities: markings alone do not establish a single definite purpose. The Lebombo bone has 29 notches and is estimated to be around 44,000 years old.

How are the Gumulgal number names formed?

The Gumulgal, an indigenous group in Australia, used urapon for 1 and ukasar for 2. Their larger number names combine these words by counting in twos. The following names retain the repeated groups rather than introducing a separate word for every quantity.

Any number greater than 6 was called ras. Thus the named sequence did not distinguish every larger quantity. Counting in groups nevertheless introduces a useful idea: a group can become a unit for describing a larger collection.

Most humans find it difficult to count groups having 5 or more objects in a single glance. This limit could have prompted the replacement of a group of tally marks with another symbol. It is a possible explanation, not a certain account of how every system developed.

How do Roman numerals use landmark numbers?

Landmark numbers are recognisable reference quantities used to build representations of other numbers. In the Roman system, selected landmark numbers have their own basic symbols. Other numbers can be formed by combining these symbols rather than making one mark per object.

How does grouping give a Roman representation?

For 27, take as many tens as possible, then as many fives as possible, and finally ones. The result is two tens, one five and two ones. Here + means addition, and = means that the expressions on its two sides have equal value.

Worked example 1. Write 27 using the Roman symbols X for 10, V for 5 and I for 1.

Answer: 27 = 10 + 10 + 5 + 1 + 1. Replacing each group by its symbol gives XXVII. The repeated X symbols account for the tens; V and the two I symbols account for the rest.

Roman representations also use subtraction in cases such as IV, meaning 1 less than 5, and XL, meaning 10 less than 50. Historical practice was not always consistent: 40 was sometimes written XXXX.

What are the advantages and limitations?

Using several landmark numbers makes a representation more compact than using tally marks. However, the successive Roman landmarks are not obtained by repeatedly multiplying by one fixed number. Regrouping therefore requires attention to which landmark comes next.

Roman numerals do not lend themselves to easy arithmetic, particularly multiplication and division. People using them used the abacus, a calculating device, for arithmetic. Only specially trained people used that tool for calculation.

Note: The order in which these systems are studied is an order of mathematical ideas. It should not be interpreted as a history in which each civilisation improved the immediately preceding system.

How does the Egyptian system introduce a base?

The Egyptian written number system developed around 3000 BCE, meaning Before Common Era. It used separate symbols for landmark numbers. Its distinctive feature was the sequence of those landmarks: begin with 1 and repeatedly make a group ten times as large.

Property: Successive landmarks have a fixed multiplier

Ten collections of size 1 give 10. Ten collections of size 10 give 100. Repeating this process produces successive powers of 10. A power records repeated multiplication: 10² means 10 multiplied by 10, while 10³ means 10 multiplied by 10 multiplied by 10.

The superscript is the exponent, which gives the number of repeated factors in these positive powers. The multiplication sign × means “multiplied by”. The initial landmark is written 10⁰ = 1.

Definition: In a base-n number system, n names the fixed multiplier between successive landmark numbers, starting with 1. Its landmark numbers are powers of n. A base-10 system is also called a decimal system.

The letter n stands for the base under discussion. Thus n² means n × n, and n³ means n × n × n. In the Egyptian system, n is 10. A base describes the relationship between landmarks, not the shape of their symbols.

Worked example 2. Group 324 into Egyptian landmark values of 100, 10 and 1.

Answer: 324 = 100 + 100 + 100 + 10 + 10 + 4. Its representation needs three hundred symbols, two ten symbols and four one symbols. Counting each kind separately checks that every part of the number has been included.

Why is this more systematic?

The same grouping rule works at every step. Ten ones become one ten, and ten tens become one hundred. This repeated relationship makes regrouping predictable. It also prepares the way for simpler arithmetic with landmark numbers.

Having a base does not yet mean having place value, where a symbol's position determines its associated landmark. Egyptian symbols directly identify their landmark values. The separate idea of using position comes later in this discussion.

How can a base other than ten represent numbers?

The grouping process need not use ten. A base-5 system begins with 1 and repeatedly groups five collections of the previous size. It therefore has landmarks 1, 5, 25, 125, 625 and 3125, continuing by the same rule.

How do powers describe the landmarks?

The first landmarks are 5⁰ = 1, 5¹ = 5, 5² = 25 and 5³ = 125. The superscript 1 indicates one factor of 5. Each further multiplication by 5 gives the next landmark, just as multiplication by 10 does in a decimal system.

Worked example 3. Group 143 using the base-5 landmarks 125, 25, 5 and 1.

Answer: Start with 125, the largest available landmark below 143. The grouping is 143 = 125 + 5 + 5 + 5 + 1 + 1 + 1. It uses one group of 125, no group of 25, three groups of 5 and three ones.

In this representation, a separate symbol may be assigned to each landmark. An absent landmark simply contributes no symbol. This is a system with a base, but it does not yet use the later method of assigning a fixed place to every landmark.

How should a grouping be checked?

1. Begin with the largest landmark that fits into the number.
2. Take as many groups of that size as possible.
3. Continue with smaller landmark sizes for the remaining quantity.
4. Check that adding all the groups gives the original number.

Regrouping means replacing several smaller groups by an equivalent larger group. In base 5, five equal landmark groups can be replaced by one group of the next landmark. A completed grouping therefore does not need five copies of the same landmark.

The base determines the regrouping rule. It does not change the quantity being represented. The number 143 is the same quantity whether its groups are based on tens or fives; the chosen system changes how those groups are recorded.

Why does a fixed base simplify arithmetic?

Arithmetic operations include addition, subtraction, multiplication and division. A system with a base makes regrouping predictable during these operations. For addition, combine groups of the same landmark size, then exchange enough of them for the next larger landmark.

How does carrying follow from grouping?

Carrying records a larger group produced during addition. In decimal addition, ten ones become one ten. The same idea applies when combining Egyptian symbols: ten copies of a landmark symbol can be exchanged for the next landmark symbol.

Worked example 4. Add 47 and 56 by grouping ones and tens in base 10.

Answer: The ones give 7 + 6 = 13, which is one ten and three ones. The tens give 4 + 5 + 1 = 10 tens after including the carried ten. Ten tens make one hundred, so 47 + 56 = 103.

This explains the written carrying process through quantities rather than a rule to memorise. Carrying preserves the total while changing its grouping. The ones, tens and hundreds are related by the same multiplier.

Property: Multiplying landmark numbers gives a landmark number

In a system whose landmarks are powers of its base, the product, or result of multiplication, of two landmarks is another landmark. Multiplying an Egyptian landmark by 10 moves to the next power of 10; multiplying by 10² moves two powers further.

For a number containing several landmark groups, multiplication can be applied to the parts and the results added. This is the distributive property: multiplication distributes over an addition. It links the multiplication of a whole number to the multiplication of its component groups.

The corresponding process is less straightforward with Roman landmarks because they are not successive powers of one fixed base. The advantage of a base therefore concerns both compact representation and calculation. It provides a repeated structure that can be used throughout the number system.

How does a decimal abacus represent numbers?

A decimal abacus represents quantities with counters placed on lines for powers of 10. Starting from the line for 1, successive lines stand for 10, 100, 1000 and further powers. A counter's location gives the landmark with which it is associated.

How are counters read?

In the board arrangement considered here, a counter above a line represents five units of that line's value. A counter on the line represents one such unit. This permits six ones to be shown with one counter above the ones line and one on it.

The illustration shows horizontal lines labelled 1000, 100, 10 and 1. There are three counters on the 1000 line, four on the 100 line and two on the 10 line. A counter above the 1 line and another on it show six ones.

See Fig. 3.1 in your NCERT textbook

Worked example 5. Read an abacus showing three thousands, four hundreds, two tens and six ones.

Answer: The groups combine as 3426 = 1000 + 1000 + 1000 + 100 + 100 + 100 + 100 + 10 + 10 + 1 + 1 + 1 + 1 + 1 + 1. The represented number is 3426.

How are numbers added?

Place the two numbers on opposite sides of a vertical partition, then bring counters for the same line together. Regroup when a complete group of ten is formed. The procedure expresses the same decimal relationships used in written addition.

Worked example 6. Add 2907 and 43 using decimal groups.

Answer: Seven ones and three ones make ten ones, exchanged for one ten. The four tens from 43 and the new ten make five tens. The nine hundreds and two thousands remain. The result is 2950.

The device helps perform calculations, but it does not by itself remove every limitation of a written system. The Egyptian method still needed new symbols for higher and higher powers of 10. A finite supply of written landmark symbols could not meet that continuing need.

How did Mesopotamian place value reduce the need for new symbols?

The later Mesopotamian system used base 60, also called the sexagesimal system. Its landmarks were powers of 60. It is also called the Babylonian number system. Its influence remains visible in the relationships of 60 minutes to an hour and 60 seconds to a minute.

What does a position represent?

A positional number system, or place value system, uses the position of a symbol to identify its associated landmark. In the Mesopotamian representation, the rightmost group shows ones, the group to its left shows sixties, and the next group shows groups of 3600.

Worked example 7. Express 640 in groups of 60 and 1.

Answer: 640 = 10 × 60 + 40. A place value reading therefore gives ten sixties and forty ones. Separate written symbols for the landmark 60 need not be repeated ten times when position already identifies that landmark.

Worked example 8. Express 7530 using the landmarks 3600, 60 and 1.

Answer: 7530 = 2 × 3600 + 5 × 60 + 30. Read the successive groups as two groups of 3600, five groups of 60 and thirty ones. The groups record how many times each landmark occurs.

When grouping into powers of 60, sixty occurrences of one landmark can be exchanged for the next. This keeps the number of occurrences below 60 after regrouping. The positions replace the need for an unending sequence of new landmark symbols.

What historical qualifications matter?

Different theories explain why base 60 was chosen. They include connections with time periods, the ease of representing fractions and changes to earlier landmark sequences. These are possible explanations rather than a single established reason.

It does not seem that the Mesopotamians reached place value through the same reconstruction used here. Some scholars suggest that similarities between earlier symbols, and their accidental use, might have helped bring about the idea. A mathematical explanation of a system is not necessarily its actual history.

Why does a place value system need a placeholder?

A placeholder marks a position whose landmark makes no contribution to a number. It keeps the remaining symbols in their intended positions. Without a clear marker, an empty position may be confused with ordinary spacing or overlooked altogether.

Why were blank spaces ambiguous?

Mesopotamian numerals initially left a blank when a power of 60 did not occur. Different manuscripts did not maintain consistent spacing. A reader could therefore be unsure which symbols belonged to which powers of 60, or how many empty positions a gap represented.

The problem concerns ambiguity, meaning that the same written representation can be read in more than one way. Place value makes writing compact, but the positions must remain identifiable. A blank of uncertain width does not reliably do that job.

Property: Zero preserves an empty position

Later Mesopotamians assigned a symbol to a blank position. This was like using zero as a placeholder. It made missing groups explicit, helping readers distinguish a position with no contribution from the neighbouring positions that contained symbols.

The improvement did not remove every ambiguity. Their placeholder was primarily used in the middle of numbers, rather than at the end. Consequently, the system was not a fully developed place value system even after this important innovation.

Definition: Zero as a positional digit records that a particular place contains no units of its landmark value. Zero as a number can also take part in arithmetic.

A digit is an individual symbol used in writing a numeral. In the Indian system, zero is treated on par with the other digits. Its written presence records an empty place clearly instead of asking the reader to infer one from a gap.

Keep the two roles distinct. Being a placeholder explains how zero supports unambiguous notation. Being a number explains why zero also has arithmetic properties. The Indian system brings these roles together.

How do the Mayan and Chinese systems use position?

The Mayan civilisation in Central America made great intellectual and cultural progress between the 3rd and 10th centuries CE, meaning Common Era. Its place value notation developed independently of those in Asia and included a seashell-like placeholder for zero.

Why is the Mayan system not an actual base-20 system?

The Mayan landmarks begin with 1, 20 and 360, rather than 1, 20 and 400. Later listed landmarks include 7200 and 144000. Since the multiplier is not consistently 20 at every step, it is not an actual base-20 system.

A dot represents 1 and a bar represents 5. Symbols are arranged vertically, with ones at the bottom, twenties above them and groups of 360 above those. Position therefore contributes to the interpretation even though the landmarks are not all powers of 20.

Some scholars feel that the third landmark, 360, might have something to do with Mayan calendars. This remains a suggested connection. The system's use of position and a zero placeholder was an important advance, despite its computational limitations.

How do Chinese rod numerals work?

The Chinese used a written system for recording quantities and a rod system for computations. Rod numerals are the numerals of that rod-based system. The rod system is decimal and alternates two symbol arrangements called zongs and hengs.

Zongs represent units, hundreds and tens of thousands. Hengs represent tens, thousands and hundreds of thousands. Alternating these arrangements distinguishes adjacent places. A blank indicates a skipped place.

Worked example 9. Read rod numeral groups representing 2 thousands, 6 hundreds, 3 tens and 4 ones.

Answer: 2 × 10³ + 6 × 10² + 3 × 10 + 4 × 1 = 2634. Each group contributes its value multiplied by the landmark assigned to its position.

The slightly more uniform sizes of the Chinese symbols made blanks easier to locate than in the Mesopotamian system. With a symbol for zero, this Chinese system would be a fully developed place value system.

What makes the Indian number system efficient?

The Indian number system, also called the Hindu or Hindu-Arabic number system, uses place value with usually 10 digits, including zero. In its familiar decimal form, these digits are 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.

How does a digit combine with its place?

Each position has a landmark value. Moving left from the ones place gives tens, hundreds, thousands and successive powers of ten. The digit records how many of that landmark occur. One digit in each position, together with zero, removes ambiguity in reading and writing.

Worked example 10. Read the decimal numeral 375 using its place values.

Answer: 375 = 3 × 10² + 7 × 10 + 5 × 1. The 3 records three hundreds, the 7 records seven tens and the 5 records five ones. The digit and its position must be read together.

Draw three columns labelled hundreds, tens and ones. Put 3, 7 and 5 in them respectively. Underneath, write 3 × 10² + 7 × 10 + 5 × 1 = 375.

Can the same place value idea use another base?

Yes. Place value can be built from powers of another base. The quantity stays the same while its written numeral changes. Always identify the base explicitly when comparing such numerals, because the familiar-looking digits acquire different place values.

Worked example 11. Write the decimal number 25 in base 8 and base 5.

Answer: In base 8, 25 = 3 × 8 + 1, so its numeral is 31. In base 5, 25 = 1 × 5² + 0 × 5 + 0 × 1, so its numeral is 100. The two zeros preserve the empty fives and ones places.

Ancient Indians likely used base 10 because humans have 10 fingers and can use them for counting. The word likely matters: it presents an explanation with a qualification, rather than a definite record of a single decision.

How did Indian numerals and zero become widely used?

The modern structure of oral and written numbers originated in India thousands of years ago. Ancient Indian texts named powers of ten. The Yajurveda Samhita includes names such as eka for one, dasha for ten, shata for hundred and sahasra for thousand.

What happened as the notation spread?

The written system using digits from 0 to 9 developed in India around 2000 years ago. The Bakhshali manuscript, dated approximately to the 3rd century CE, uses ten digits with zero represented by a dot. Aryabhata explained and used the Indian system for scientific computations.

The system reached the Arab world by around 800 CE. Al-Khwārizmī and Al-Kindi helped popularise it through works on Hindu numerals. From the Arab world it reached Europe and parts of Africa by around 1100 CE.

Fibonacci argued for its adoption in Europe around 1200. Widespread acceptance took several more centuries because Roman numerals were deeply established. European scholars called the numerals Arabic because they learnt them from the Arab world; Arab scholars called them Hindu numerals.

Here Hindu refers to a geography and people, rather than a religion. The terms Indian numerals, Hindu numerals and Hindu-Arabic numerals help identify the system's origin and transmission.

How did zero's arithmetic role matter?

Indian mathematics treated zero as a number in its own right. Aryabhata used its arithmetic properties in 499 CE: adding zero leaves a number unchanged, and multiplying a number by zero gives zero. Brahmagupta codified arithmetic with zero in 628 CE.

Introducing zero along with negative numbers supported a collection closed under addition, subtraction and multiplication: applying any of these operations to two members gives another member of the collection. This supported later developments in mathematics.

The central ideas can be traced from counting groups, to using landmarks, to choosing powers of a base, to assigning place values, and finally to using zero as a positional digit and number. This sequence summarises mathematical ideas, not a simple chronological chain between civilisations.

Glossary

• One-to-one mapping — Pairing each object with its own partner, without assigning the same partner to two objects.
• Number system — A standard sequence of objects, names or written symbols in a fixed order for representing numbers.
• Numeral — A symbol or written representation used to represent a number within a number system.
• Tally marks — Marks that record counted objects, with one mark corresponding to each object in the collection.
• Landmark numbers — Recognisable reference quantities used to group, represent and work with other numbers in a system.
• Base — The fixed multiplier connecting successive landmark numbers in a system whose landmarks begin with one.
• Decimal system — A base-10 number system whose landmark numbers are successive powers of ten, starting with one.
• Regrouping — Replacing groups of one size with equivalent groups of another size while preserving the total quantity.
• Abacus — A calculating device that represents quantities using counters arranged at positions associated with landmark values.
• Sexagesimal system — A base-60 number system with landmark numbers given by successive powers of sixty, starting with one.
• Place value — The use of a symbol's position to identify the landmark number associated with that symbol.
• Placeholder — A symbol marking an empty position so that other symbols retain their intended place values.
• Rod numerals — Numerals used in the Chinese rod-based decimal system for representing quantities and carrying out computations.
• Digit — An individual symbol used to write numerals, such as any of the ten decimal symbols.

Common errors and misconceptions

• Misconception: Counting requires modern written digits. Correct: Objects can be counted by matching them with a fixed sequence of sticks, body parts, names or other symbols. One-to-one matching is the central idea.
• Misconception: Every system with landmarks has a base. Correct: A base requires successive landmarks to have one fixed multiplier, starting from 1. Roman landmarks do not have this repeated relationship.
• Misconception: Having a base automatically means having place value. Correct: The Egyptian system has base 10 but uses separate landmark symbols. Place value additionally uses position to identify the associated landmark.
• Misconception: The Mayan landmarks are all powers of 20. Correct: They begin 1, 20 and 360. The third landmark is 360 rather than 400, so it is not an actual base-20 system.
• Misconception: A blank is as clear as a zero. Correct: Variable spacing can make empty positions hard to identify. A placeholder explicitly records an empty position and helps prevent confusion.
• Misconception: Zero serves only as a placeholder. Correct: In Indian mathematics, zero is also a number with arithmetic properties. Adding zero leaves a number unchanged; multiplication by zero gives zero.
• Misconception: The sequence of systems proves that each developed from the previous one. Correct: It illustrates mathematical ideas. Their actual histories are more complex and many times not clearly known.

Exam-style questions with model answers

Q1. A herder keeps one stick for each cow leaving to graze. Explain how one-to-one mapping helps check the returning herd without using number names. [2 marks]

1. Pair each returning cow with its own stick, ensuring that no stick is assigned to two cows.
2. If some sticks remain unmatched after all returning cows are paired, they represent cows that have not returned.

Q2. Using X = 10, V = 5 and I = 1, write 27 as a sum of landmark numbers and then in Roman numerals. [2 marks]

1. Grouping into tens, fives and ones gives 27 = 10 + 10 + 5 + 1 + 1.
2. Two tens give XX, five gives V and two ones give II. Therefore, the Roman numeral is XXVII.

Q3. A base-5 system has landmarks 1, 5, 25 and 125. Group 143 using these landmarks and explain why five copies of a landmark are unnecessary in a completed grouping. [3 marks]

1. Begin with 125, the largest listed landmark that fits into 143. Only one group of this size is needed, and no group of 25 is needed.
2. The complete grouping is 143 = 125 + 5 + 5 + 5 + 1 + 1 + 1.
3. Five copies of any landmark can be regrouped into one copy of the next landmark, because successive landmarks differ by a multiplier of five.

Q4. A decimal abacus groups counters into ones, tens, hundreds and thousands, with ten units of each place exchanged for one of the next. Use this rule to add 2907 and 43. [4 marks]

1. The ones from the two numbers combine as seven ones and three ones, giving ten ones altogether.
2. Exchange these ten ones for one ten, leaving no ungrouped ones in the final representation.
3. Add that new ten to the four tens in 43. This gives five tens, while the nine hundreds and two thousands remain unchanged.
4. Reading the resulting thousands, hundreds, tens and ones gives 2950. Thus the required sum is 2907 + 43 = 2950.

Q5. A base-60 place value representation has positions for 3600, 60 and 1, read from left to right. Decompose 7530 into these positions and explain the role of position. [3 marks]

1. The decomposition is 7530 = 2 × 3600 + 5 × 60 + 30, using the supplied landmark values.
2. The groups therefore record two groups of 3600, five groups of 60 and thirty ones, in that order.
3. Position identifies the landmark associated with each group. There is no need to repeat a separate symbol for that landmark for every occurrence, so the representation becomes more compact.

Q6. Explain five stages in the mathematical development of efficient number representation: counting in groups, landmark numbers, a base, place value and zero. [5 marks]

1. Counting in groups lets a repeated collection act as a unit. For example, Gumulgal number names use groups of two instead of a separate tally for every object.
2. Landmark numbers provide several recognisable reference quantities. Roman numerals combine symbols for these landmarks to describe larger quantities more compactly.
3. A base organises landmarks as powers of one fixed number, beginning with one. This creates a repeated regrouping rule that helps arithmetic.
4. Place value lets position identify the associated landmark. It avoids having to invent a new written symbol for every larger landmark.
5. Zero explicitly records an empty position and also functions as a number in arithmetic. These are stages in mathematical ideas, not proof of a direct historical succession between civilisations.

Q7. The Mayan landmarks begin 1, 20 and 360, and its symbols are arranged vertically with ones at the bottom. Explain why its notation uses position but is not an actual base-20 system. [3 marks]

1. The bottom position represents ones, the next represents twenties and the next represents groups of 360. A symbol's location therefore affects its interpretation.
2. In an actual base-20 system, each landmark would be twenty times the previous one, beginning with one.
3. Twenty times twenty gives 400, whereas the given third Mayan landmark is 360. The landmarks therefore do not follow powers of twenty throughout.

Q8. Write the decimal number 25 in base 8 and base 5. Use place values 8 and 1 for base 8, and 25, 5 and 1 for base 5. Explain the zeros in the second representation. [3 marks]

1. In base 8, the grouping is 25 = 3 × 8 + 1. Three eights and one unit give the numeral 31.
2. In base 5, the grouping is 25 = 1 × 25 + 0 × 5 + 0 × 1. This gives the numeral 100.
3. The two zeros explicitly preserve the empty fives and ones positions. They show that the first digit belongs to the twenty-fives position.

Key takeaways

• Counting uses one-to-one matching with a standard sequence of objects, names or symbols arranged in a fixed order.
• Landmark numbers make larger quantities easier to represent by grouping them into recognisable reference quantities.
• A base connects successive landmark numbers by a fixed multiplier, giving predictable regrouping rules for arithmetic.
• The Egyptian system illustrates a decimal base, while its need for further landmark symbols reveals a limitation.
• Place value lets a symbol's position identify its associated landmark, reducing the need for new symbols.
• The Mayan and Chinese systems show important uses of position, with different landmark structures and ways of indicating empty places.
• Zero supports unambiguous place value notation and, in Indian mathematics, also has the status of a number.
• The sequence of mathematical ideas should not be mistaken for a simple chronological history of civilisations improving one another's systems.

Test yourself

What distinguishes a numeral from the quantity it represents?

A numeral is a written representation. The same quantity can be represented using different numerals in different number systems.

What does ukasar-ukasar-urapon represent if ukasar means two and urapon means one?

It represents five, formed by combining two, two and one in the Gumulgal naming system.

Why do Roman landmarks not form a fixed-base sequence?

The multiplier between successive landmarks is not constant. Their sequence is not formed by repeatedly multiplying one by the same base.

Why can five copies of a base-5 landmark be regrouped?

Each next landmark is five times the previous one, so five equal groups form one group of the next size.

How does 640 split into sixties and ones?

It splits into ten sixties and forty ones: 640 = 10 × 60 + 40.

What limitation remained in the Mesopotamian use of a placeholder?

The placeholder was primarily used in the middle of numbers rather than at the end, so some ambiguities remained.

Why is a Mayan third landmark of 360 significant?

An actual base-20 sequence would have 400 as its third landmark. The Mayan sequence therefore is not consistently base 20.

What are zero's two roles in the Indian system?

Zero marks empty positions in written numerals and is also a number with arithmetic properties of its own.