CBSE Class 8 Maths Revision Notes Chapter 3: A Story of Numbers

Humans used objects, sounds, body parts and marks for counting before modern numerals were developed. The Hindu number system uses ten digits, place value and zero to represent numbers clearly and perform calculations efficiently.

Numbers are used for counting objects, measuring time, recording trade and solving scientific problems. However, the symbols 0, 1, 2, 3 and the other digits that we use today developed slowly over thousands of years.

These CBSE Class 8 Maths Revision Notes Chapter 3 explain the evolution of number systems from sticks and tally marks to Roman, Egyptian, Mesopotamian, Mayan, Chinese and Hindu systems. The chapter also covers one-to-one mapping, landmark numbers, the base-n number system, place value and the importance of zero.

Key Takeaways

• Number system: A fixed sequence of objects, names or symbols used for counting.
• Landmark numbers: Reference numbers used to represent larger quantities.
• Base-n system: A system whose landmark numbers are powers of n.
• Zero: It acts as both a placeholder and an independent number.

Access Class 8 Maths Chapter 3 A Story of Numbers Notes in 30 Minutes

Spend the first 10 minutes revising early counting methods, one-to-one mapping, tally marks and Roman numerals. Use the next 10 minutes for landmark numbers, the Egyptian system and the idea of a base.

Use the final 10 minutes to revise place value systems, Mesopotamian, Mayan and Chinese numerals, the Hindu number system and zero.

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History of Numbers in Class 8 Maths Chapter 3 Notes

Humans needed to count even during the Stone Age. They counted food, animals, traded goods, ritual offerings, passing days, seasons and phases of the Moon.

Early people did not use the numerals that we use now. Different communities developed physical objects, spoken names and written symbols to represent quantities.

The modern spoken and written number system developed in India. Ancient Indian texts used number names based on powers of 10, while the written digits 0 to 9 gradually developed into their present forms.

How Did Early Humans Count?

Before standard numerals existed, people matched each object with another object, mark, sound or body part. This allowed them to compare quantities without saying or writing modern numbers.

For example, a person could keep one stick for every cow. When the herd returned, each cow could be matched with one stick to check whether any cow was missing.

One-to-One Mapping

One-to-one mapping means pairing every object in one collection with exactly one object in another collection.

For example:

1 cow ↔ 1 stick
2 cows ↔ 2 sticks
3 cows ↔ 3 sticks

No two cows should be matched with the same stick. This method helps compare the sizes of two collections and identify missing objects.

Uses of One-to-One Mapping

One-to-one mapping can answer questions such as:

• Have all the animals returned?
• Which group has more objects?
• Do two groups contain the same number of objects?
• How many objects must be added to make the groups equal?

The idea continues to be important in mathematics.

What Is a Number System?

A number system is a standard sequence of objects, names or written symbols arranged in a fixed order and used for counting.

A useful number system should be able to:

• Continue without ending
• Represent small and large quantities
• Follow a clear order
• Support calculations
• Avoid confusion

A collection of sticks can represent any quantity, but it becomes inconvenient for large numbers. A limited sequence of letters is easy to use, but it cannot continue forever without additional rules.

What Is a Numeral?

A numeral is a written symbol or group of symbols used to represent a number.

Examples in the Hindu number system include:

0, 5, 19, 236 and 1729

A number is the mathematical idea of quantity, while a numeral is its written representation.

The same number can be written in different ways:

5 in Hindu numerals
V in Roman numerals
Five in words

Early Number Systems

Different communities created number systems according to their needs, available materials and languages. Some used body parts, while others used sticks, marks, sounds or written symbols.

These early systems gradually introduced important ideas such as grouping, landmark numbers, bases and place value.

Counting with Body Parts

Several groups across the world used fingers, hands, arms and other body parts as a counting sequence. Each body part represented a particular position in the sequence.

This system was convenient because people always carried the counting tool with them. However, representing large quantities using body parts could become difficult.

The ten fingers on human hands may be one reason why base 10 became common in many number systems.

Tally Marks

Tally marks are marks made on a surface to record quantities. One mark is made for each object counted.

For example:

1 = |
2 = ||
3 = |||
4 = ||||
5 = |||||

This method is similar to using one stick for every object. It is simple for small numbers but becomes long and difficult to read for larger quantities.

Ancient Tally Bones

Archaeologists have found bones with notches that may have been used to record quantities or time.

The Ishango bone was discovered in the Democratic Republic of Congo and is believed to be between 20,000 and 35,000 years old. The Lebombo bone from South Africa has 29 notches and may be around 44,000 years old.

These artefacts suggest that people recorded quantities long before modern written numerals appeared.

Counting in Groups

Grouping numbers makes representation more efficient than using one mark for every object. Different societies counted in groups of 2, 5, 10 or 20.

The Gumulgal people of Australia used number names mainly based on groups of two.

Their system followed patterns such as:

3 = 2 + 1
4 = 2 + 2
5 = 2 + 2 + 1
6 = 2 + 2 + 2

Grouping reduced the number of separate words or marks required. However, using only one group size could still make large numbers difficult to represent.

Roman Numerals in A Story of Numbers Class 8 Notes

The Roman number system used special symbols for certain important quantities.

These important reference numbers are called landmark numbers.

Writing Numbers in Roman Numerals

Numbers are represented using combinations of landmark symbols.

For example:

27 = 10 + 10 + 5 + 1 + 1

Therefore:

27 = XXVII

Another example:

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

Therefore:

2367 = MMCCCLXVII

Subtraction Rule in Roman Numerals

A smaller numeral may be written before a larger numeral to show subtraction.

IV = 5 − 1 = 4
IX = 10 − 1 = 9
XL = 50 − 10 = 40
XC = 100 − 10 = 90

Examples

302 = CCCII

715 = DCCXV

1222 = MCCXXII

2999 = MMCMXCIX

Limitations of Roman Numerals

Roman numerals are more efficient than tally marks, but arithmetic operations are difficult to perform with them.

Addition requires careful regrouping of symbols. Multiplication and division are harder because the landmark numbers do not follow one regular multiplication pattern.

The system also does not use place value or zero. People using Roman numerals often depended on tools such as the abacus for calculations.

Landmark Numbers

Landmark numbers are easily recognised reference numbers used to represent and understand other numbers.

In the Roman system, landmark numbers include:

1, 5, 10, 50, 100, 500 and 1000

In the decimal system, they include:

1, 10, 100, 1000, 10,000 and so on

When landmark numbers follow a regular pattern, number representation and calculations become easier.

Egyptian Number System

The ancient Egyptian number system developed around 3000 BCE. It used separate symbols for different powers of 10.

Its landmark numbers were:

10⁰ = 1
10¹ = 10
10² = 100
10³ = 1000
10⁴ = 10,000

Each landmark number was ten times the previous one.

Representing Numbers in the Egyptian System

The number 324 can be expanded as:

324 = 100 + 100 + 100 + 10 + 10 + 1 + 1 + 1 + 1

The Egyptian numeral would use the symbol for 100 three times, the symbol for 10 twice and the symbol for 1 four times.

More examples:

1023 = 1000 + 10 + 10 + 1 + 1 + 1

2660 = 2000 + 600 + 60

784 = 700 + 80 + 4

1111 = 1000 + 100 + 10 + 1

The Idea of a Base

A base-n number system has the following features:

• Its first landmark number is 1.
• Every next landmark number is obtained by multiplying the previous one by n.

Its landmark numbers are therefore:

n⁰ = 1
n¹ = n
n²
n³
n⁴ and so on

Base-10 Number System

In a base-10 system, each landmark number is ten times the previous one.

10⁰ = 1
10¹ = 10
10² = 100
10³ = 1000
10⁴ = 10,000

A base-10 system is also called a decimal system. Both the Egyptian system and the Hindu number system use powers of 10 as landmark numbers.

Base-5 Number System

In a base-5 system, each landmark number is five times the previous one.

5⁰ = 1
5¹ = 5
5² = 25
5³ = 125
5⁴ = 625
5⁵ = 3125

For example:

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

This can also be written as:

143 = 1 × 125 + 3 × 5 + 3 × 1

The landmark numbers in this system are powers of 5.

Base-7 Number System

The landmark numbers in a base-7 system are:

7⁰ = 1
7¹ = 7
7² = 49
7³ = 343
7⁴ = 2401

In general, the landmark numbers of a base-n system are:

n⁰, n¹, n², n³, ...

Advantages of a Base-n Number System

A base-n system follows one fixed rule for moving from one landmark number to the next. This regularity simplifies representation and arithmetic.

For example, in base 10:

10 × 10² = 10³

100 × 1000 = 10² × 10³ = 10⁵

The product of two landmark numbers is another landmark number.

In general:

nᵃ × nᵇ = nᵃ⁺ᵇ

This makes multiplication easier than in the Roman system, where landmark numbers do not follow a single regular pattern.

Abacus and Decimal Grouping

An abacus is a calculating device that uses positions to represent landmark numbers such as 1, 10, 100 and 1000.

For example:

3426 = 3 × 1000 + 4 × 100 + 2 × 10 + 6 × 1

During addition, ten counters in one position can be regrouped as one counter in the next position.

For example:

7 ones + 3 ones = 10 ones

10 ones = 1 ten

This is similar to carrying a digit in modern addition.

Limitation of the Egyptian Number System

The Egyptian system used powers of 10 and was useful for calculations. However, it required a new symbol for every higher power of 10.

Representing larger numbers would therefore require an endless sequence of new symbols. Place value systems solved this problem by using the position of a symbol to show its value.

What Is a Place Value System?

A place value system is a number system in which the value of a symbol depends on its position.

The rightmost position represents the smallest landmark number. Each position to the left represents a higher power of the base.

For example:

375 = 3 × 100 + 7 × 10 + 5 × 1

375 = 3 × 10² + 7 × 10¹ + 5 × 10⁰

The digit 3 represents 300 because it is in the hundreds place. The digit 7 represents 70 because it is in the tens place.

Mesopotamian Number System

The later Mesopotamian or Babylonian number system used base 60. It is also called the sexagesimal system.

Its landmark numbers were:

60⁰ = 1
60¹ = 60
60² = 3600
60³ = 216,000

The influence of base 60 can still be seen in time measurement:

1 hour = 60 minutes

1 minute = 60 seconds

Representing Numbers in Base 60

For example:

640 = 10 × 60 + 40

Another example:

7530 = 2 × 3600 + 5 × 60 + 30

The position of each group of symbols showed whether it represented ones, sixties, 3600s or larger powers of 60.

Problem of Empty Positions

The Mesopotamians sometimes left a blank space when a particular power of 60 was missing. This created confusion because the number of blank places was not always clear.

Later, they introduced a placeholder symbol for an empty position. This was similar to zero as a placeholder, though their system did not use zero as completely as the Hindu number system.

Mayan Number System

The Mayan civilisation developed a place value system independently in Central America. It used a dot for 1, a bar for 5 and a shell-like symbol for zero.

The symbols were arranged vertically. The lowest position represented ones, the next represented twenties and higher positions represented larger landmark numbers.

Examples of expansion include:

77 = 3 × 20 + 17 × 1

100 = 5 × 20 + 0 × 1

361 = 1 × 360 + 0 × 20 + 1 × 1

721 = 2 × 360 + 0 × 20 + 1 × 1

The system was related to base 20, although one of its landmark numbers was 360 instead of 400.

Chinese Number System

The Chinese used rod numerals for calculations. Their rod-based system followed base 10 and used alternating vertical and horizontal forms to distinguish neighbouring positions.

For example:

2634 = 2 × 1000 + 6 × 100 + 3 × 10 + 4 × 1

2634 = 2 × 10³ + 6 × 10² + 3 × 10¹ + 4 × 10⁰

The change in symbol direction reduced confusion between digits in adjacent places.

Blank spaces were used for missing positions. The system would become fully developed as a place value system when a symbol for zero was included.

Hindu Number System

The Hindu number system is the decimal place value system used throughout the world today.

It uses ten digits:

0, 1, 2, 3, 4, 5, 6, 7, 8 and 9

Every whole number can be written using only these ten symbols.

Place Value Example

375 = 3 × 10² + 7 × 10¹ + 5 × 10⁰

375 = 3 × 100 + 7 × 10 + 5 × 1

375 = 300 + 70 + 5

The same digit can have different values depending on its position.

For example, in 505:

The first 5 represents 500.

The second 5 represents 5.

Importance of Zero

Zero has two major roles in the Hindu number system:

• It acts as a placeholder.
• It is treated as an independent number.

Zero as a Placeholder

Compare:

52
502
5002

In 502, zero shows that there are no tens.

502 = 5 × 100 + 0 × 10 + 2 × 1

Without zero, it would be difficult to distinguish 52 from 502 or 5002.

Zero as a Number

Zero has its own arithmetic properties.

a + 0 = a

a − 0 = a

a × 0 = 0

0 × a = 0

These properties make zero essential in arithmetic, algebra, science and computing.

Origin and Spread of Hindu Numerals

The modern system of ten digits developed in India around 2000 years ago. An early form of zero written as a dot appears in the Bakhshali manuscript.

Aryabhata used the Indian place value system for detailed scientific calculations. Brahmagupta later described arithmetic rules involving zero and negative numbers.

The system reached the Arab world by around 800 CE. Al-Khwarizmi and Al-Kindi helped explain and popularise Hindu numerals.

It later reached Europe, where Fibonacci strongly supported its use. Since Europeans learned the system through Arab scholars, they often called the symbols Arabic numerals.

More accurate names include:

• Hindu numerals
• Indian numerals
• Hindu-Arabic numerals

Why Is the Hindu Number System Efficient?

The Hindu number system combines several important ideas:

• Base 10
• Ten digits
• Place value
• Zero as a placeholder
• Zero as a number
• Clear representation
• Easy calculations

Unlike Roman numerals, it does not require new symbols for every larger value. The same ten digits can represent every whole number.

Evolution of Number Systems

The evolution of number systems can be understood through the following stages:

1. Matching each object with a stick, mark or body part.
2. Counting in groups such as 2, 5, 10 or 20.
3. Using landmark numbers.
4. Choosing powers of a number as landmark numbers.
5. Using position to show the value of symbols.
6. Introducing zero as a placeholder.
7. Treating zero as an independent number.

Each stage made number representation and calculation more efficient.

Comparison of Important Number Systems

Class 8 Maths Chapter 3 Summary Notes

Important Terms from A Story of Numbers

Number system: A fixed sequence of objects, names or symbols used for counting and representing numbers.

Numeral: A written symbol or group of symbols representing a number.

One-to-one mapping: Pairing every member of one collection with exactly one member of another.

Tally mark: A mark made for each object counted.

Landmark number: An important reference number used to represent other quantities.

Base-n number system: A system whose landmark numbers are powers of n.

Decimal system: A base-10 number system.

Place value system: A system in which the value of a digit depends on its position.

Placeholder: A symbol used to show an empty place in a numeral.

Zero: A digit that acts as a placeholder and an independent number.

Sexagesimal system: A base-60 number system.

Hindu number system: The decimal place value system that uses the digits 0 to 9.

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