Exercise 1.4: Introduction to Bhartiya System of Numeration
Comprehensive guide and practice questions for CBSE Class 11 Applied Mathematics curriculum (2026-27).
1.4 Introduction to Bhartiya System of Numeration
India pioneered the use of the decimal place value system, which represents numbers using digits one to nine and zero (Śūnya). Mathematical giants like Āryabhaṭa, Brahmagupta, and Bhāskara employed this system to describe basic and advanced arithmetic operations.
In addition to decimal representation, ancient Indian mathematicians invented ingenious alphabetic encoding systems where numbers were encoded as words or syllables, notably the Bhūta-saṅkhyā, Kaṭapayādi, and Āryabhaṭa system (Āryabhaṭīya paddhati). These systems enabled complex astronomical data and tables to be memorized as poetic Sanskrit verses.
Āryabhaṭa's Alphabetic System of Numeration
In Āryabhaṭīyam (composed in 499 CE), Āryabhaṭa assigns numbers to the 33 Sanskrit consonants and uses vowels to denote powers of 10.
Table 1: 33 Sanskrit Consonants (Varga and Avarga)
Class
1
2
3
4
5
Guttural
क (k) = 1
ख (kh) = 2
ग (g) = 3
घ (gh) = 4
ङ (ṅ) = 5
Palatal
च (c) = 6
छ (ch) = 7
ज (j) = 8
झ (jh) = 9
ञ (ñ) = 10
Retroflex
ट (ṭ) = 11
ठ (ṭh) = 12
ड (ḍ) = 13
ढ (ḍh) = 14
ण (ṇ) = 15
Dental
त (t) = 16
थ (th) = 17
द (d) = 18
ध (dh) = 19
न (n) = 20
Labial
प (p) = 21
फ (ph) = 22
ब (b) = 23
भ (bh) = 24
म (m) = 25
Avarga
य (y) = 30
र (r) = 40
ल (l) = 50
व (v) = 60
श (ś) = 70
Avarga (cont.)
ष (ṣ) = 80
स (s) = 90
ह (h) = 100
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Table 2: Nine Sanskrit Vowels as Powers of 10
अ/आ (a)
इ/ई (i)
उ/ऊ (u)
ऋ/ॠ (ṛ)
लृ (ḷ)
ए (e)
ऐ (ai)
ओ (o)
औ (au)
$10^0$
$10^2$
$10^4$
$10^6$
$10^8$
$10^{10}$
$10^{12}$
$10^{14}$
$10^{16}$
Encoding & Decoding Rules:
Multiply the consonant value by the power of 10 indicated by the accompanying vowel.
In a conjunct consonant (e.g. vya or kti), add the consonant values first, then multiply by the vowel power of 10.
Example 1: Decode the word "guṇa" (गुण).
Show Step-by-Step Solution
Split "guṇa" into two sub-units: gu and ṇa.
• $gu = g \times 10^4 = 3 \times 10,000 = 30,000$
• $ṇa = ṇ \times 10^0 = 15 \times 1 = 15$
Add both values: $30,000 + 15 = \mathbf{30,015}$.
Example 2 (Ex 17(i) in Book): Decode "vyakti" (व्यक्ति).
Show Step-by-Step Solution
Split into syllables: $(v + y + a) + (k + t + i)$
• First syllable: $(v + y) \times 10^0 = (60 + 30) \times 1 = 90$
• Second syllable: $(k + t) \times 10^2 = (1 + 16) \times 100 = 17 \times 100 = 1700$
Total: $90 + 1700 = \mathbf{1790}$.
Example 3 (Ex 17(ii) in Book): Decode "khyughṛ" (ख्युघृ) from Āryabhaṭīyam.
Show Step-by-Step Solution
In verse 4 of Āryabhaṭīyam, reckoning revolutions of the sun in a yuga: "yugaravibhagaṇāḥ khyughṛ".
Split: $(kh + y + u) + (gh + ṛ)$
• $khyu = (kh + y) \times 10^4 = (2 + 30) \times 10,000 = 320,000$
• $ghṛ = gh \times 10^6 = 4 \times 1,000,000 = 4,000,000$
Total: $320,000 + 4,000,000 = \mathbf{43,20,000}$ (4.32 million solar revolutions in a Mahāyuga).
CHECK YOUR PROGRESS 1.5
Practice decoding and encoding classical Indian mathematical words into standard decimal numbers using the Āryabhaṭīya paddhati.
Problem 1Decoding Classical Sanskrit Words
Decode the numbers from these words according to the Āryabhaṭa method:
(i) kṛṣṇa (कृष्ण)
(ii) mukti (मुक्ति)
(iii) jyeṣṭha (ज्येष्ठ)
The speed of light is approximately $3 \times 10^8\text{ m/s}$. Express this number using the Āryabhaṭa system.
Show Step-by-Step Solution
Step 1: Identify the digit and multiplier: $3 \times 10^8$. Step 2: From Table 1, the consonant for digit 3 is ग (g). Step 3: From Table 2, the vowel for $10^8$ is लृ (ḷ). Answer: Combining consonant and vowel gives गॢ (gḷ).
Exercise 1.4 Revision Worksheet
Printable questions on historical Indian numeration systems: Āryabhaṭīya paddhati, consonant values, and vowel power multipliers.
Encode the number 4,000,000 using a single Sanskrit syllable in the Āryabhaṭa system.
Show Model Solution
• $4,000,000 = 4 \times 10^6$
• Digit 4 is represented by consonant घ (gh)
• Power $10^6$ is represented by vowel ऋ (ṛ) Answer:घृ (ghṛ).
Section C: Comprehensive Historical Application1 Question • 4 Marks
Q64 Marks
In the 4th verse of Āryabhaṭīyam, the phrase "yugaravibhagaṇāḥ khyughṛ" records the revolutions of the sun in a Mahāyuga. Decode the complete numerical value of khyughṛ showing individual phoneme contributions.
High-Yield Revision Sheet: Bhartiya System of Numeration
Summary tables for Āryabhaṭa's 33 consonants, 9 vowels, and conjunct decoding rules.
Quick Reference: Vowel Multipliers
a (अ): $10^0$ (1)
i (इ): $10^2$ (100)
u (उ): $10^4$ (10K)
ṛ (ऋ): $10^6$ (1M)
ḷ (लृ): $10^8$ (100M)
e (ए): $10^{10}$
ai (ऐ): $10^{12}$
o (ओ): $10^{14}$
au (औ): $10^{16}$
4-Step Decoding Algorithm
Step 1: Partition the word into discrete syllables consisting of consonant(s) + vowel. Step 2: In each syllable, sum the values of all consonants in the conjunct: $(C_1 + C_2 + \dots)$. Step 3: Multiply this sum by the power of 10 designated by the vowel: $(C_1 + C_2) \times 10^{2k}$. Step 4: Add the numerical contributions of all syllables to obtain the final decimal number.
Time: 45 Minutes Max Marks: 20 CBSE Board Blueprint
Assertion (A): Āryabhaṭa's numeration system allowed ancient Indian astronomers to express large astronomical parameters within brief poetic meters. Reason (R): By mapping 33 consonants to digits 1 to 100 and 9 vowels to powers of 10 ($10^0, 10^2, \dots, 10^{16}$), multi-digit numbers could be encoded as compact Sanskrit syllables.
(A) Both A and R are true, and R is the correct explanation of A.
(B) Both A and R are true, but R is not the correct explanation of A.
(C) A is true, but R is false.
(D) A is false, but R is true.
Solution: Both A and R are true, and R correctly explains the mathematical elegance of Āryabhaṭa's alphabetic numeration system.
Section B: Short Answer Type I (Q5 – Q6)2 Questions • 2 Marks Each
Question 52 Marks
Decode the word 'vyakti' (व्यक्ति) into decimal notation showing individual syllable calculations.
Section D: Long Answer Type (Q9)1 Question • 4 Marks
Question 94 Marks
Explain the mathematical difference between Varga and Avarga consonants in Āryabhaṭa's system and demonstrate how conjunct consonants are decoded.
Show Marking Scheme & Solution (4 Marks)
Varga vs Avarga (2 Marks): The 25 Varga consonants represent sequential numbers 1 to 25. The 8 Avarga consonants represent tens from 30 to 100 ($y=30, r=40, \dots, h=100$). Conjunct Rule (2 Marks): When two or more consonants are combined into a conjunct without intervening vowels, their numerical values are summed together first before multiplying by the vowel's power of 10: $(C_1 + C_2 + \dots) \times 10^{2k}$.
Section E: Case-Based / Source-Based Problem (Q10)1 Case Study • 4 Marks
Question 10: Case Study4 Marks Total
Case Background: In ancient Indian astronomy, astronomical revolutions over a Mahāyuga of 4,320,000 solar years were encoded into poetic Sanskrit verses for oral transmission.
(i) [1 Mark]: Identify the vowel multiplier corresponding to $10^6$ in this system.
Show Solution for (i)
[1 Mark]: The vowel is ऋ (ṛ).
(ii) [1 Mark]: If an astronomical constant is represented by $(gh + y) \times 10^4$, calculate its numerical value.
Show Solution for (ii)
[1 Mark]: $gh = 4, y = 30$. Total $= (4 + 30) \times 10,000 = \mathbf{340,000}$.
(iii) [2 Marks]: Decode the term $(kh + v) \times 10^2 + m \times 10^4$.