Unit II – Number Systems and Units of Measurement

 Unit II – Number Systems and Units of Measurement

Dr. Alok Pawar

 

2.1     Number Systems in India – Historical Evidence

India is one of the oldest civilizations to develop scientific, logical, and symbolic number systems.

Historical Evidence

1.     Harappan Civilization (Indus Valley): • Use of linear scales, standardized weights, decimals in trade. Binary-like weight ratios (1:2:4:8…)

2.     Vedic Period: Large numbers mentioned in the Vedas – śata (100), sahasra (1000), ayuta, niyuta. Use of powers of 10.

3.     Panini (5th century BCE): Used sophisticated numeric classification in his grammar.

4.     Jain Mathematics: Highly advanced classification of infinity (अनन्त / ananta). Use of extremely large numbers.

5.     Aryabhata, Brahmagupta, Bhāskarācārya: Establishment of zero (śūnya), decimal place value system, algebra.

6.     Epigraphical Evidence: Numerals found on ancient copper plates, stone inscriptions, coins.

 

2.2     Salient Features of the Indian Numeral System

1.     Use of Ten Symbols (0–9) : India introduced a simple set of ten digits used to form all numbers.

2.     Concept of Zero (शून्य / śūnya): Zero was used both as a placeholder and as an independent number, a unique contribution of India.

3.     Place Value System (स्थानमूल्य पद्धति): The value of a digit depends on its position (units, tens, hundreds…). This makes representation efficient.

4.     Decimal System (दशमान पद्धति): Base-10 system used in calculations, commerce, astronomy, weights, and measures.

5.     Ability to Represent Very Large Numbers: Indian texts used names for extremely large numbers (lakṣa, koṭi, ayuta, niyuta).

6.     Efficient Arithmetic Operations: Addition, subtraction, multiplication, division become simpler due to positional notation.

7.     Universality & Adaptability: The system evolved into the modern Hindu–Arabic numeral system, now used worldwide.

 

2.2.1 Concept of Zero and Its Importance (शून्य / Śūnya)

Zero was first used in India as:

1.     A placeholder – to indicate absence of a value in a position (e.g., 205).

2.     A number with its own value – capable of being used in arithmetic.

Mathematicians like Aryabhata, Brahmagupta, and later Bhāskara developed rules for zero.

Importance of Zero

1.     Foundation of the Place-Value System: Zero allows correct positioning of digits (10, 100, 1000).

2.     Enables the Decimal System: Without zero, decimal notation and modern calculations are impossible.

3.     Simplifies Arithmetic: Operations like addition, subtraction, and multiplication become efficient.

4.     Allows Representation of Large Numbers Easily: Complex numbers can be written using a few digits.

5.     Essential in Algebra & Calculus: Zero led to concepts like equations, limits, and advanced mathematics.

6.     Global Impact: Zero transformed mathematics worldwide, forming the basis for modern science, computing, and technology.

 

2.2.2 Large Numbers and Their Representation

Ancient India developed a highly structured system to name and represent very large numbers. This system was used in mathematics, astronomy, commerce, and literature.

1. Use of Named Large Numbers

Indian texts provide names for systematically increasing powers of ten:

·         Daśa (10)

·         Śata (100)

·         Sahasra (1,000)

·         Ayuta (10,000)

·         Lakṣa (1,00,000)

·         Koṭi (1,00,00,000)

Even larger units like arbuda, abja, kharva, nikharva are also mentioned.

2. Efficient Place-Value System

Large numbers could be written easily using digits (0–9) because of the place value (स्थानमूल्य) system. Example: 56,78,90,123 uses only 9 digits instead of long words.

3. Use of Zero (śūnya)

Zero enabled representation of extremely large numbers with clarity.

4. Use in Astronomy & Mathematics

Indian mathematicians like Āryabhata and Brahmagupta used large numbers for:

·         planetary distances

·         time cycles (yugas)

·         calculations involving millions and billions

5. Bhūta-saṅkhyā (word-number representation)

Large numbers were also expressed using words symbolizing fixed numerical values in poetry and literature.

 

2.2.3 Place Value of Numerals (स्थानमूल्य प्रणाली)

The place value system means that the value of a digit depends on its position in a number. This system was first fully developed in ancient India and later spread worldwide.

1.     Value depends on position

o    The same digit has different values based on where it appears.
Example: In 252, the digit 2 represents

§  200 (hundreds place)

§  2 (units place)

2.     Use of Base-10 (दशमान पद्धति)

o    Each place is 10 times bigger than the previous one:
Units → Tens → Hundreds → Thousands …

3.     Role of Zero (शून्य / śūnya)

o    Zero is used as a placeholder, making place value possible.
Example: 207 (0 shows nothing in the tens place).

4.     Efficient representation of numbers

o    Very large numbers can be written using few symbols.

5.     Foundation for modern arithmetic

o    All operations (addition, subtraction, multiplication, division) become easier due to positional notation.

 

2.2.4 Decimal System (दशमान पद्धति)

The decimal system is a base-10 (दशमान) number system developed in ancient India. It uses ten digits (0–9) to represent all numbers.

Key Features

1.     Base-10 Structure:

o    Each place represents a power of 10:
Units (10⁰), Tens (10¹), Hundreds (10²), Thousands (10³)…

2.     Use of Zero (शून्य / śūnya)

o    Zero acts as a placeholder and also a number, making the system efficient.

3.     Place-Value System (स्थानमूल्य)

o    Meaning of a digit depends on its position.
Example: 5 in 50 = 5 × 10¹.

4.     Compact Representation of Large Numbers

o    Huge numbers can be expressed with few digits.

5.     Simplifies Arithmetic

o    Addition, subtraction, multiplication, and division become easy and systematic.

6.     Globally Adopted

o    This Indian innovation evolved into the modern Hindu–Arabic numeral system used worldwide.

 

2.3     Unique Approaches to Represent Numbers

Ancient India developed several innovative and creative methods to express numbers, especially useful in mathematics, poetry, astronomy, and coding of texts.

 

2.3.1 Bhūta-Saṃkhyā System (भूतसंख्या पद्धति)

A symbolic method where common objects or concepts represent fixed numbers.
Used in poetry (especially astronomy texts) to encode numerical values.

Examples:

·         Moon (चंद्र) = 1

·         Eyes (नेत्र) = 2

·         Vedas (वेद) = 4

·         Senses (इंद्रिय) = 5

·         Mountains (पर्वत) = 7

·         Directions (दिशा) = 10

This allowed authors to write numbers in verse form without disturbing the meter.

2.3.2  Śūnyabindu System (शून्यबिन्दु पद्धति)

• A system where a dot (bindu) was used to represent zero (śūnya).
This helped in maintaining place value and is considered an early form of zero notation.

Uses:

·         Representing large numbers concisely

·         Ensuring accurate positional values

This system laid the foundation for modern decimal notation.

 

2.3.3 Piṅgala and the Binary System (पिङ्गल छन्दः – द्विमान पद्धति)

भारतीय छंदशास्त्रातील महान विद्वान पिंगल यांनी इ.स.पू. 2ऱ्या शतकात द्वि-आधारी संख्या पद्धती (Binary System) चे पहिले तत्त्व मांडले. त्यांनी लघु (॰) आणि गुरु (–) या दोन मात्रांच्या साहाय्याने छंदातील क्रम मांडले. हाच दोन चिन्हांवर आधारित क्रम म्हणजे बायनरी.

 

पिंगलाच्या बायनरी पद्धतीचा मूलभूत सिद्धांत

• लघु = 0
• गुरु = 1
  हे 0 आणि 1 यांचे विविध संयोजन म्हणजेच आजचे ‘Binary Numbers’.

 

उदाहरण 1 : तीन मात्रांचा क्रम (३-bit binary)

तीन स्थानांवर लघु (0) आणि गुरु (1) ठेवून ० ते ७ पर्यंत संख्यांचा क्रम:

क्रमांक	पिंगलाचा स्वरूप	आधुनिक बायनरी	दहावी संख्या
1	लघु-लघु-लघु	000	0
2	लघु-लघु-गुरु	001	1
3	लघु-गुरु-लघु	010	2
4	लघु-गुरु-गुरु	011	3
5	गुरु-लघु-लघु	100	4
6	गुरु-लघु-गुरु	101	5
7	गुरु-गुरु-लघु	110	6
8	गुरु-गुरु-गुरु	111	7

 

उदाहरण 2 : “गुरु-लघु-गुरु” (1 0 1)

• पिंगलानुसार = गुरु (1), लघु (0), गुरु (1)
• आधुनिक बायनरी = 101
• दहावी संख्या =

= 1×2²+0×2¹+1×2⁰

= 4+0+1

= 5

 

हे महत्त्वाचे का?

• पिंगलाने दिलेली ही पद्धती आधुनिक कॉम्प्युटर सायन्समधील बायनरी कोडची प्रारंभिक मूळ रुप मानली जाते.
• फक्त “दोन प्रकार” वापरून मोठे संख्यात्मक प्रणाली मांडण्याचा हा सर्वात जुना पुरावा आहे.

 

2.4     Measurements for Time, Distance and Weight in Ancient India

Indian measurement systems were systematic, scientific, and used widely in rituals, astronomy, and trade.

A. Time Measurement (कालमापन)

Ancient units:

Unit	Sanskrit Name	Approx. Value
Nimeṣa (निमेष)	Blink of an eye	~ 0.214 sec
Kāṣṭhā (काष्ठा)	18 nimeṣa	~ 4 seconds
Kalā (कला)	30 kāṣṭhā	~ 2 minutes
Muhūrta (मुहूर्त)	30 kalā	48 minutes
Day/Night (अहोरात्र)	30 muhūrta	24 hours
Tithi (तिथि)	Lunar day	~1 day
Nakṣatra	Star-based unit	—

Astronomy texts like Surya Siddhanta mention extremely small time units:
Truti, Tatpara, Nimesha etc.

 

B. Distance Measurement (अंतर मापन / मिती)

Unit	Approx. Value
Aṅgula (अंगुल)	~1.9 cm
Vitasti (वितस्ति)	12 aṅgula
Hasta (हस्त)	24 aṅgula
Dhanus (धनुः)	96 aṅgula
Krośa (क्रोश)	~3 km
Yojana (योजन)	~12–15 km

Used in architecture, pilgrimage, astronomy.

 

C. Weight Measurement (भार मापन)

Unit	Sanskrit Name	Value
Ratti (रत्ती)	0.121g
Masha (माष)	8 ratti
Karsha (कर्ष)	12 masha
Tola (तोळा)	11.6g
Pal (पल)	4 karsha
Tula (तुला)	100 pal

Used for metals, medicines, trade.

 

2-Mark Questions

1.     Define the Indian numeral system in brief.

2.     What is śūnya (zero) and why is it important?

3.     Give two examples of Bhūta-saṃkhyā system.

4.     Write the place value of 5 in the number 352.

5.     What is decimal system (दशमान पद्धति)?

6.     Name any two large numbers used in ancient India.

7.     What does Piṅgala’s binary system represent using लघु and गुरु?

8.     Give the approximate value of 1 aṅgula in modern measurement.

9.     What is muhūrta in minutes?

10. Name any two units of weight used in ancient India.

 

4-Mark Questions

1.     Explain the salient features of the Indian numeral system.

2.     Discuss the concept of zero and its importance in mathematics.

3.     Describe large numbers and their representation in ancient India with examples.

4.     Explain the place value system (स्थानमूल्य प्रणाली) with an example.

5.     Describe the decimal system and its key features.

6.     Explain unique approaches to represent numbers: Bhūta-saṃkhyā, Śūnyabindu, and Piṅgala’s binary system.

7.     Give an account of time, distance, and weight measurements in ancient India with examples.

8.     Explain the historical evidence of number systems in India.

9.     Convert “गुरु-लघु-गुरु” (1 0 1) in Piṅgala’s binary system to decimal.

10. Discuss the importance of place-value system and zero in representing large numbers efficiently.