These notes revise Chapter 6, “Mathematics in India,” of the NCERT Class 11 Knowledge Traditions and Practices of India (KTPI) textbook — pages 99 to 116 of the rationalised book — and they follow the 2026-27 session. The chapter traces Indian mathematics from Mohenjodaro (3000 B.C.)
to the seventeenth century: the invention of decimal place value, the discovery of the zero symbol, board arithmetic, Sulbasūtra geometry, algebra and trigonometry.
Use this page for a complete fast revision: anchor yourself with the timeline, learn the eight fundamental operations and their Sanskrit names, then run through the original board-method worked examples and the common-slips table. The chapter’s five exercise questions are mapped out at the end.
The official NCERT PDF of this chapter is available directly from NCERT, and the full rationalised book sits on the NCERT textbook portal. For the rest of the subject, see the KTPI Class 11 notes hub; the companion chapter on visual culture has its own Indian Art and Architecture notes. You can also browse the full CBSE notes library.
The Arc of the Chapter: From Mohenjodaro to Mādhava
The chapter’s sweep is one claim: decimal place value was invented and first used in India (NCERT, p. 99), and every major branch of mathematics grew continuously from the earliest times to the seventeenth century. Keep this timeline as your skeleton.
| Period | Milestone |
|---|---|
| 3000 B.C. | Mohenjodaro: the inhabitants of the land of the Sindhu lived a highly organised life (p. 99) |
| 2000 B.C. | Brāhmaṇa literature — partly ritualistic, partly philosophical (p. 99) |
| 800 B.C. | Sulbasūtras composed, recording much older knowledge (p. 111) |
| 300 B.C. | King Asoka’s inscriptions — common use of numerical symbols; Brāhmī numerals (p. 100) |
| A.D. 499 | Āryabhaṭa I and the Āryabhaṭīya (p. 101) |
| A.D. 500–1200 | Golden (Siddhāntic) period of Indian mathematics (p. 100) |
| A.D. 628 | Brahmagupta’s Brahmasphuṣasiddhānta (p. 101) |
| A.D. 1114 | Bhāskara II, author of Līlāvatī and Bījagaṇita (p. 101) |
| A.D. 1500 | Mādhava and Nīlakaṇṭha — the great \( \pi \) approximations (p. 113) |
Memory hook for the Golden period: the names run Āryabhaṭa I → Varāhamihira → Bhāskara I → Brahmagupta → Mahāvīra → Śrīdhara → Śrīpati → Bhāskara II. Their initials — A, V, B, B, M, S, S, B — make the sentence “A Very Big Book Makes Students Sharper, Brilliantly.”
The trap is the two Bhāskaras: Bhāskara I (A.D. 600) was Āryabhaṭa’s commentator, while Bhāskara II (A.D. 1114) closed the period with Līlāvatī and Bījagaṇita. The chapter also prints portraits of Āryabhaṭa and Brahmagupta; only the Āryabhaṭa portrait is credited to NCERT’s Mathematics Class XI textbook.
Numerals, Brāhmī Signs and the Birth of Zero
India counted in base ten from the earliest times, and long lists of number names survive: the Yajurveda Saṃhitā refers to numerical denominations as large as \( 10^{12} \) (NCERT, p. 100). The archaeological trail of the symbols:
- 300 B.C. — Asoka’s inscriptions: numerical symbols were in common use across his dominions.
- Brāhmī numerals: a purely Indian invention, known from Asoka’s time; 1, 2 and 3 were written as one, two and three horizontal lines placed one below the other (p. 100).
- A.D. 1st–2nd century: inscriptions containing numerals found in a cave in the Nasik district of the then Bombay presidency.
The zero story: the symbol for zero was discovered by Āryabhaṭa I (born A.D. 496) in connection with the decimal expression of numbers — “the vacant places should be filled up with a circle,” which looks like śūnya, as illustrated by his commentator Bhāskara I (A.D. 600) (NCERT, p. 101).
This brought a revolution in computation: the whole technique of expressing numbers was simplified to nine numerical symbols plus zero, replacing all earlier notation.
Pāṅgaṇita: Board Arithmetic and Its Eight Operations
Pāṅgaṇita is a compound of pāṅ (“board”) and gaṇita (“science of calculation”) — arithmetic done on a writing board. Calculations were also called dhūli-karma (“dust-work”), because figures were written on dust spread on a board or on the ground (NCERT, p. 101).
Brahmagupta’s frame for pāṅgaṇita is twenty operations and eight determinations; the eight fundamental operations are the chapter’s backbone (NCERT, p. 101):
- addition
- subtraction
- multiplication
- division
- square
- square-root
- cube
- cube-root
Who gave the root rules: Āryabhaṭa I gave the rules for square and cube-roots, while Brahmagupta gave the cube-root rule only (NCERT, p. 102). Bhāskara I saw the deep structure: every operation is a variation of increase or decrease — addition is increase, subtraction is decrease, multiplication and evolution are repeated addition, division and involution repeated subtraction.
The treatises are famously compact: known formulae and results were stated so briefly that they are sometimes hardly understandable without a commentary — the exposition in the Āryabhaṭīya is more compact than in the later works (p. 101).
Working the Pāṭi: Addition, Subtraction, Multiplication, Division
All four basic operations were performed on the pāṭi with the same two styles — direct and inverse. The comparison table below is the single best revision tool for this part of the chapter.
| Operation | Direct process | Inverse process | How partial results are corrected |
|---|---|---|---|
| Addition (saṅkalita) | Sum of the units (rightmost) column is written first, then tens, then hundreds | Sum of the extreme-left (last) column is written first | Direct: the new column sum is added to the partial sum’s figure in that place, which is then substituted. Inverse: the carried figure is added to the digit already written; the old digit is rubbed out |
| Subtraction (vyutkalita) | Subtract place by place from the right; borrow a ten from the next place when a figure cannot be subtracted | Begins at the last (leftmost) place of the minuend | Inverse: previously obtained partial differences are rubbed out and corrected as later places are worked |
| Multiplication (guṇana) | Begins with the rightmost digit of the multiplicand | Begins with the leftmost digit of the multiplicand (the general fashion) | Each used digit of the multiplicand is rubbed out and replaced by its product; carried figures are added to the left neighbour, which is then substituted |
Addition — saṅkalita
Aryabhata II (A.D. 950) defined addition as “the making into one of several numbers.” The ancient name is saṅkalita (“made together”), with synonyms saṅkalana, miśraṇa, sammelana, prakṣepaṇa, saṃyojana, ekīkaraṇa, yoga and abhyāsa (NCERT, p. 102).
Bhāskara II’s rule in the Līlāvatī: “Add the figures in the same places in the direct or the inverse order.”
Subtraction — vyutkalita
Aryabhata II: “The taking out (of some number) from the sarvadhana (total) is subtraction; what remains is called śeṣa (remainder)” (NCERT, p. 103). The minuend is the sarvadhana (or viyojya), the subtrahend the viyojaka, and the remainder the śeṣa (or antara). Other names for subtraction: vyutkalana, śodhana (“clearing”), patana (“causing to fall”), viyoga.
The borrowing logic, as in 1000 − 360: six cannot be taken from the zero in the tens place, so take ten, subtract six, and place the remainder four below; the borrowed ten is then deducted from the next place, and so on until it is exhausted at the last place (p. 103).
The inverse process is identical but begins at the last (leftmost) place of the minuend — ideal for a board where figures can be easily rubbed out and corrected (p. 104).
Multiplication — guṇana
Guṇana is the oldest name for multiplication, found in Vedic literature. The names hanana, vadha, kṣaya — all meaning “killing” or “destroying” — came into use with the decimal place-value method, because the figures of the multiplicand were successively rubbed out and replaced by the figures of the product (NCERT, p. 104).
The multiplicand is the guṇya, the multiplier the guṇaka, and the product the guṇana-phala or pratyutpanna (“reproduced”).
Why the names make sense (word-origin device):
- hanana, vadha and kṣaya = “killing”: the scribe physically destroyed each digit of the multiplicand as he worked, writing the product over it.
- pratyutpanna = “reproduced”: the product is a new number born from the killed figures — the book says 12 and 135 “have been killed and a new number 1620 is born.”
- dhūli-karma = “dust-work”: figures on dust are easy to erase, which is exactly what the method needs.
- labdhi = “what is obtained”: the quotient is what you gain from division.
Brahmagupta lists four multiplication methods — gomutrikā, khanda, bheda, iṣṭa — and Śrīpati (A.D. 1039) named the plan of placing multiplicand below multiplier “as in the junction of two doors”: kapāṇa-sandhi (NCERT, p. 105).
Division — bhāgāhara
The division names — bhāgāhara, bhājana, harāṇa, chedana — literally mean “to break into parts”; harana, “to take away,” shows division’s link to subtraction (NCERT, p. 106). The dividend is the bhājya, the divisor the bhājaka (or bhāgahara, hara), and the quotient the labdhi — “what is obtained” (p. 107).
The board method of long division begins at the extreme left of the dividend, places each quotient digit in a separate line of quotients, and rubs out figures as each product is subtracted.
Written out one step below another instead of obliterated, it becomes the modern method of long division; the method seems to have been invented in India about the fourth century A.D. (NCERT, p. 107).
Fractions, Squares and Square-Roots: Varga, Mūla and the Bhāga Names
Fractions
The oldest fractions on record are ardha (half) and tri-pāda (three-fourths, literally “three feet”) in the Ṛgveda — probably the oldest record of a composite fraction known to us (NCERT, p. 108). The Maitrāyaṇi Saṃhitā adds kala (\( \frac{1}{16} \)), kustha (\( \frac{1}{12} \)), sapha (\( \frac{1}{8} \)) and pāda (\( \frac{1}{4} \)).
Unit fractions were named with a cardinal or ordinal plus bhāga/aṃśa (“part”), and composites combined the number with the ordinal:
| Fraction | Sanskrit name | How it is built |
|---|---|---|
| \( \frac{1}{2} \) | ardha | half (Ṛgveda) |
| \( \frac{3}{4} \) | tri-pāda | three feet of a quadruped |
| \( \frac{1}{16} \) | kala | one-sixteenth (Maitrāyaṇi Saṃhitā) |
| \( \frac{1}{12} \) | kustha | one-twelfth |
| \( \frac{1}{8} \) | sapha | one-eighth |
| \( \frac{1}{4} \) | pāda | one-fourth |
| \( \frac{1}{15} \) | pañca-daśa-bhāga | cardinal fifteen + bhāga |
| \( \frac{1}{5} \) | pañcama-bhāga | ordinal fifth + bhāga |
| \( \frac{3}{8} \) | tri-aṣṭama | three-eighths |
| \( \frac{2}{7} \) | dvi-saptama | two-sevenths |
Square — varga
The Sanskrit term is varga (“rows” or “troops”) or kṛti (“doing, making”). Āryabhaṭa I’s definition: “A square figure of four equal sides and the (number representing its) area are called varga. The product of two equal quantities is also varga” (NCERT, p. 108).
On the pāṭi, squaring begins from the last (leftmost) place — the book works out \( 125^2 \) on p. 109; an original example is below.
Square-root — varga-mūla
Mūla and pāda both mean “root”: the foot, the lowest part, the basis, the cause or origin of anything. So varga-mūla is literally the “cause or origin of the square” — or the side of the square figure (NCERT, p. 110).
The Āryabhaṭīya rule, in one line: “Always divide the even place by twice the square-root (upto the preceding odd place); after having subtracted from the odd place the square (of the quotient), the quotient put down at the next place (in the line of the root) gives the root.”
How to read it: mark the places of the number alternately odd and even from the left. Subtract the largest possible square from the first odd place; divide the even place by twice the running root; subtract the square of the new quotient from the next odd place; repeat.
The book’s worked example is \( \sqrt{54756} = 234 \) (pp. 110–111); a full original example is in the worked-examples section.
Sulbasūtra Geometry: Fire Altars, Triangles and the “Pythagoras” Result
The Sulbasūtras of Baudhāyana, Apastamba and Kātyāyana are the oldest known mathematics texts, composed around 800 B.C. while recording far older knowledge — their authors say they are stating facts already known to the composers of the Brāhmaṇas and Saṃhitās (NCERT, p. 111).
Their purpose was practical: designing and constructing Vedic fire altars with rich symbolic shapes — a falcon in flight with curved wings, a chariot-wheel complete with spokes, a tortoise with extended head and legs — all needing accurate geometry.
The Sulbasūtras name three triangle types (p. 111):
| Sanskrit name | Type |
|---|---|
| sama | equilateral |
| dvisama | isosceles |
| viṣamatribhūja | scalene |
Two pillars of plane geometry stand in the Sulbasūtras: (i) the result now known as the Pythagoras theorem, stated and applied to construct a square equal in area to the sum or the difference of two squares, or to a rectangle; and (ii) the properties of similar figures (NCERT, p. 112).
Later, Āryabhaṭa I gave correct area and perimeter formulae for the common geometrical figures, and Brahmagupta gave the area of a triangle, the area of a cyclic quadrilateral when its sides are known, and the diagonals of a cyclic quadrilateral (p. 112). The chapter also prints a portrait of Baudhāyana.
Bījagaṇita: The Science of Calculation with Unknowns
Bījagānita = bīja (“element” or “analysis”) + gānita (“science of calculation”) — the science of analytical calculation, what we call algebra.
Brahmagupta (A.D. 628) called it kuṇaka-gānita or simply kuṇaka, “the pulveriser,” from its branch on indeterminate equations of the first degree — so important that the whole science was named after it in the beginning of the seventh century (NCERT, p. 112).
Algebra is avyakta-gānita (calculation with unknowns), in contrast to vyakta-gānita (calculation with knowns — arithmetic including geometry and mensuration).
Its origin lies in the Sulba (800–500 B.C.) and Brāhmaṇa (2000 B.C.) periods, but it was then mostly geometrical (NCERT, p. 113):
- The transformation of a square into a rectangle of given side is equivalent to the linear equation \( ax = c^{2n} \).
- The altar problem — making a figure similar to a given one but differing in area by a specified amount — has its counterpart in the quadratic equation; the usual method increased the unit of measure of the linear dimensions.
- Āryabhaṭa I contributed to the binomial expansion, used for operations on indices — square, square-root, cube, cube-root.
Brahmagupta’s opening verse makes the Indian attitude clear: “Since questions can scarcely be known (i.e., solved) without algebra, therefore, I shall speak of algebra with examples.” Bhāskara II went further, defining algebra as the science that treats numbers expressed by means of symbols, with scope and primary need for intelligent artifices and ingenious devices (p. 113).
Trigonometry: R-Sines, Cosines and Indian Values of Pi
The Greeks founded trigonometry but their progress halted for lack of adequate algebraic machinery and notation. Indian mathematicians invented the sine and cosine functions, discovered most of the standard formulae including \( \sin(A \pm B) \), and constructed fairly accurate sine tables (NCERT, p. 113).
Āryabhaṭa I defined, for the first time, a right triangle within a quarter-circle: the perpendicular, jyā or jīva, is \( R\sin\theta \), and the base, ko-jyā, is \( R\cos\theta \), satisfying the fundamental identity (NCERT, p. 114):
\[ (R\sin\theta)^2 + (R\cos\theta)^2 = R^2 \]
where R is the radius of the circle and θ the angle. He gave the R-sine values the chapter prints in a table:
| θ | R Sin θ (as printed in the chapter) |
|---|---|
| \( 0^\circ \) | \( 0 \) |
| \( 30^\circ \) | \( R/\sqrt{3} \) * |
| \( 45^\circ \) | \( R/\sqrt{2} \) |
| \( 60^\circ \) | \( R\sqrt{3}/2 \) |
| \( 90^\circ \) | \( R \) |
* The 30° entry needs a correction — see Common Slips.
Āryabhaṭa I noted that sine values increase while cosine values decrease as the angle grows from \( 0^\circ \) to \( 90^\circ \) (for example, \( R\cos 0^\circ = R \) and \( R\cos 90^\circ = 0 \)). For intermediate angles, Brahmagupta (A.D. 628) and Govindaswami (A.D. 880) gave interpolation formulae — special cases of the Newton-Stirling and Newton-Gauss formulae for second-order difference (p. 113).
The chapter’s famous approximations of \( \pi \):
- \( 3.1416 \)} — Āryabhaṭa I (A.D. 499)
- \( 3.14159265359 \)} — Mādhava (A.D. 1500)
- \( \frac{355}{113} \)} — Nīlakaṇṭha (A.D. 1500)
Sanskrit Terms You Must Know
One sentence before the table: these are the terms the chapter’s exercise asks for by name, so learn the meaning alongside the spelling.
| Term | Meaning | Where it appears / example |
|---|---|---|
| gaṇita | mathematics; the science of calculation | valued by Jainas and Buddhists (pp. 99–100) |
| gaṇita anuyoga | the exposition of the principles of mathematics | Jaina religious literature (pp. 99, 116) |
| saṅkhyāna | the science of numbers — arithmetic and astronomy | a principal accomplishment of the Jaina priest (pp. 99, 116) |
| pāṅgaṇita | calculation on the board (pāṅ = board + gaṇita) | arithmetic forms the major part of pāṅgaṇita (p. 101) |
| dhūli-karma | “dust-work” — figures written on dust spread on a board or the ground | another name for carrying out calculations (p. 101) |
| pāṭi | the writing board | the board long-division method was worked on the pāṭi (p. 104) |
| śūnya | “the vacant places should be filled up with a circle” — the zero symbol | Āryabhaṭa I’s discovery, illustrated by Bhāskara I (p. 101) |
| saṅkalita | addition — “made together” | synonyms: saṅkalana, miśraṇa, sammelana, prakṣepaṇa, saṃyojana, ekīkaraṇa, yoga (p. 102) |
| vyutkalita | subtraction — “made apart” | synonyms: vyutkalana, śodhana, patana, viyoga (p. 103) |
| śeṣa | the remainder (also antara = difference) | what remains after taking from the sarvadhana (p. 103) |
| sarvadhana | the total or minuend (also viyojya) | the subtrahend is viyojaka (p. 103) |
| guṇana | multiplication — the oldest Sanskrit name, from Vedic literature | hanana, vadha, kṣaya (“killing”) are synonyms (p. 104) |
| guṇya / guṇaka | multiplicand / multiplier | in \( 124 \times 23 \), guṇya = 124, guṇaka = 23 (p. 104) |
| pratyutpanna | the product — “reproduced” (also guṇana-phala) | the new number born from the killed figures (p. 104) |
| kapāṇa-sandhi | “junction of two doors” — a method of multiplying numbers | named by Śrīpati (A.D. 1039) (p. 105) |
| bhāgāhara | division — “to break into parts” | synonyms: bhājana, harāṇa, chedana (p. 106) |
| labdhi | the quotient — “what is obtained” | dividend = bhājya, divisor = bhājaka (p. 107) |
| varga | square — “rows” or “troops”; also kṛti | the product of two equal quantities is varga (p. 108) |
| mūla | root — “cause or origin” (also pāda) | varga-mūla = square-root (p. 110) |
| tri-pāda | three-fourths — “three feet” | oldest record of a composite fraction, in the Ṛgveda (p. 108) |
| bhāga / aṃśa | part — used with cardinals and ordinals to name unit fractions | pañca-daśa-bhāga = \( \frac{1}{15} \); pañcama-bhāga = \( \frac{1}{5} \) (p. 108) |
| bījagānita | algebra — “science of calculation with elements” | also avyakta-gānita, calculation with unknowns (p. 112) |
| kuṇaka | “pulveriser” — Brahmagupta’s name for algebra | branch on indeterminate equations of the first degree (p. 112) |
| vyakta / avyakta | known / unknown | vyakta-gānita = arithmetic, geometry, mensuration; avyakta-gānita = algebra (p. 112) |
| jyā / jīva | the perpendicular — \( R\sin\theta \) | ko-jyā is the base — \( R\cos\theta \) (p. 114) |
| Sulbasūtra | from the practice of surveyors marking straight lines with cords joined by pegs | texts of Baudhāyana, Apastamba, Kātyāyana, ~800 B.C. (pp. 111, 116) |
| Brāhmī | the purely Indian invention of numerals; 1, 2, 3 written as horizontal lines | known from the time of King Asoka (300 B.C.) (p. 100) |
Key Results and Formulae at a Glance
Everything numeric worth recalling in one box. In all entries, R is the radius of the circle, θ the angle, and the R-sine values are lengths (radius × ratio).
| Result | Origin and meaning |
|---|---|
| \( ax = c^{2n} \) | The Sulba transformation of a square into a rectangle of a given side — a linear equation in one unknown (p. 113) |
| \( (R\sin\theta)^2 + (R\cos\theta)^2 = R^2 \) | Āryabhaṭa I’s quarter-circle triangle: jyā = \( R\sin\theta \), ko-jyā = \( R\cos\theta \) (p. 114) |
| \( \sin(A \pm B) \) | Standard formula discovered and used by Indian mathematicians (p. 113) |
| \( \pi \approx 3.1416 \) | Āryabhaṭa I (A.D. 499) |
| \( \pi \approx 3.14159265359 \) | Mādhava (A.D. 1500) |
| \( \pi \approx \frac{355}{113} \) | Nīlakaṇṭha (A.D. 1500) |
| Newton-Stirling and Newton-Gauss second-order difference formulae | Special cases behind the interpolation formulae of Brahmagupta (A.D. 628) and Govindaswami (A.D. 880) |
R-sine values (standard values — the 30° correction is in Common Slips): \( R\sin 0^\circ = 0 \), \( R\sin 30^\circ = R/2 \), \( R\sin 45^\circ = R/\sqrt{2} \), \( R\sin 60^\circ = R\sqrt{3}/2 \), \( R\sin 90^\circ = R \).
Vocabulary: in multiplication, guṇya (multiplicand) \( \times \) guṇaka (multiplier) = pratyutpanna (product); in division, bhājya (dividend) \( \div \) bhājaka (divisor) = labdhi (quotient). The eight fundamental operations anchor pāṅgaṇita: addition, subtraction, multiplication, division, square, square-root, cube, cube-root.
Pāṭi Methods, Step by Step: Original Worked Examples
Six examples with fresh numbers, each following the board logic of the chapter. Work through them once on paper and the terminology will stick.
Example 1 — Addition by the direct process: 47 + 38
Step 1: Write 47 above 38 and draw a line.
Add the units column: \( 7 + 8 = 15 \).
Write 15 below the line — 5 in the units place, 1 in the tens place of the partial sum.
Step 2: Add the tens column: \( 4 + 3 = 7 \).
Add this to the partial sum’s tens figure: \( 7 + 1 = 8 \).
Rub out the 1 and write 8 in its place.
| Step | Board on the pāṭi | What you do |
|---|---|---|
| 1 | 47 / +38 | write the numbers and draw the line |
| 2 | 47 / +38 / 15 | units sum 15 written below the line |
| 3 | 47 / +38 / 85 | \( 4 + 3 = 7 \); \( 7 + 1 = 8 \) replaces the 1 |
Final answer: \( 47 + 38 = 85 \). Modern check: \( 40 + 30 = 70 \), \( 7 + 8 = 15 \), \( 70 + 15 = 85 \).
Example 2 — Kapāṇa-sandhi multiplication (inverse process): 124 × 23
Step 1: Write the multiplier 23 above the multiplicand 124, aligned to the right.
Begin with the leftmost digit of the multiplicand.
Step 2: Digit 1: \( 1 \times 3 = 3 \) replaces the 1; \( 1 \times 2 = 2 \) is written to its left.
The digit 1 has been “killed” and 2 is “born.”
Step 3: Move the multiplier one place right.
Digit 2: \( 2 \times 3 = 6 \) replaces it; \( 2 \times 2 = 4 \) is added to the left neighbour 3, giving \( 3 + 4 = 7 \), which replaces the 3.
Step 4: Move the multiplier one place right.
Digit 4: \( 4 \times 3 = 12 \) — write 2, carry 1; \( 4 \times 2 = 8 \), plus the carry 1 gives 9, plus the left neighbour 6 gives \( 9 + 6 = 15 \) — write 5, carry 1; the carry 1 added to 7 gives 8.
| Step | Board on the pāṭi | What you do |
|---|---|---|
| 1 | 23 / 124 | multiplier above multiplicand, aligned right |
| 2 | 23 / 2324 | 1 killed → 3; 2 born to the left |
| 3 | 23 / 2764 | 2 killed → 6; \( 3 + 4 = 7 \) |
| 4 | 23 / 2852 | 4 killed → 2 (carry 1); \( 8 + 1 + 6 = 15 \) → 5 (carry 1); \( 1 + 7 = 8 \) |
Final answer: \( 124 \times 23 = 2852 \). The figures 23 and 124 have been killed; 2852 is the pratyutpanna — the “reproduced” product. Modern check: \( 124 \times 20 = 2480 \), \( 124 \times 3 = 372 \), \( 2480 + 372 = 2852 \).
Example 3 — Long division on the board: 2852 ÷ 23
Step 1: Write the dividend 2852 above and the divisor 23 below.
Begin from the extreme left.
Step 2: \( 28 \div 23 = 1 \), remainder 5.
Write 1 in the line of quotients; rub out 28 and write 5 in its place.
Step 3: Move the divisor one place right.
\( 55 \div 23 = 2 \) (since \( 23 \times 2 = 46 \)), remainder 9.
Write 2 in the line of quotients; rub out 55 and write 9.
Step 4: Move the divisor one place right.
\( 92 \div 23 = 4 \) exactly, remainder 0.
Write 4 in the line of quotients; rub out 92.
| Step | Board on the pāṭi | Line of quotients | What you do |
|---|---|---|---|
| 1 | 2852 / 23 | — | write dividend above divisor |
| 2 | 552 / 23 | 1 | \( 28 \div 23 = 1 \), remainder 5 replaces 28 |
| 3 | 92 / 23 | 12 | \( 55 \div 23 = 2 \), remainder 9 replaces 55 |
| 4 | — / 23 | 124 | \( 92 \div 23 = 4 \), no remainder |
Final answer: \( 2852 \div 23 = 124 \). Note: written one line below another instead of rubbed out, these steps are exactly the modern long-division layout (NCERT, p. 107). Check: \( 23 \times 124 = 2852 \).
Example 4 — Square of 21 by the pāṭi algorithm
Step 1: Write 21.
The process begins from the last (leftmost) place.
Step 2: Square the leftmost digit: \( 2^2 = 4 \), placed over the 2.
Step 3: Twice the leftmost digit: \( 2 \times 2 = 4 \), placed below the remaining digit 1; the 2 is rubbed out.
Step 4: Multiply the remaining digit by 4: \( 1 \times 4 = 4 \), placed over the 1; the lower 4 is removed.
Step 5: Move the remaining digit 1 one place to the right.
Step 6: Square the last digit: \( 1^2 = 1 \), placed over itself — it now sits past the right end of the top row, so the 1 becomes a new rightmost digit.
Step 7: Twice the last digit: \( 2 \times 1 = 2 \) — no digits remain to multiply; the 1 is rubbed out and the work ends.
| Step | Board on the pāṭi | What you do |
|---|---|---|
| 1 | 21 | write the number |
| 2 | 4 / 21 | \( 2^2 = 4 \) over the 2 |
| 3 | 4 / 1 / 4 | twice 2 = 4 below the 1; the 2 is rubbed out |
| 4 | 44 / 1 | \( 1 \times 4 = 4 \) over the 1; the lower 4 is removed |
| 5 | 44 / 1 → | the 1 moves one place right |
| 6 | 441 | \( 1^2 = 1 \) appended as the new rightmost digit |
| 7 | 441 | work ends; the 1 is rubbed out |
Final answer: \( 21^2 = 441 \). Check: \( 21 \times 21 = 441 \).
Example 5 — Square-root of 28224 by Āryabhaṭa’s odd-and-even-place rule
Setup: Mark the places of 28224 alternately from the left: 2 (odd), 8 (even), 2 (odd), 2 (even), 4 (odd).
Step 1 (odd place): First odd place is 2.
The largest square not exceeding 2 is \( 1^2 = 1 \).
\( 2 – 1 = 1 \). First root digit 1 — root = 1 (I).
Step 2 (even place): Bring down the even-place digit 8, giving 18.
Divide by twice the root (\( 2 \times 1 = 2 \)): \( 18 \div 2 \).
The quotient must let the next subtraction work, so choose 6: \( 2 \times 6 = 12 \), remainder 6.
(Quotient 9 leaves remainder 0, but the next odd-place value would be 2, smaller than \( 9^2 = 81 \) — impossible.)
Root digit 6 — root = 16 (II).
Step 3 (odd place): Remainder 6, bring down the odd-place digit 2, giving 62.
Subtract \( 6^2 = 36 \): \( 62 – 36 = 26 \).
Step 4 (even place): Bring down the even-place digit 2, giving 262.
Divide by twice the root (\( 2 \times 16 = 32 \)): \( 32 \times 8 = 256 \), remainder 6.
Root digit 8 — root = 168 (III).
Step 5 (odd place): Remainder 6, bring down the last digit 4, giving 64.
Subtract \( 8^2 = 64 \): \( 64 – 64 = 0 \).
The process ends.
| Step | Working on the pāṭi | Root line |
|---|---|---|
| 1 | First odd place 2: \( 2 – 1^2 = 1 \) | 1 (I) |
| 2 | Even place 18: \( 18 \div (2 \times 1) \); \( 2 \times 6 = 12 \), remainder 6 | 16 (II) |
| 3 | Odd place 62: \( 62 – 6^2 = 26 \) | — |
| 4 | Even place 262: \( 262 \div (2 \times 16) \); \( 32 \times 8 = 256 \), remainder 6 | 168 (III) |
| 5 | Odd place 64: \( 64 – 8^2 = 0 \) | process ends |
Final answer: \( \sqrt{28224} = 168 \). Check: \( 168 \times 168 = 28224 \).
Example 6 — Naming a fraction in Sanskrit, and verifying the identity at 45°
Step 1: \( \frac{3}{7} \) follows the composite pattern of the chapter: tri (three) + saptama (seventh) = tri-saptama, exactly as \( \frac{3}{8} \) is tri-aṣṭama and \( \frac{2}{7} \) is dvi-saptama (NCERT, p. 108).
Step 2: From the chapter’s table, \( R\sin 45^\circ = R/\sqrt{2} \) and \( R\cos 45^\circ = R/\sqrt{2} \) (p. 114).
Substitute into the identity:
\[ (R\sin 45^\circ)^2 + (R\cos 45^\circ)^2 = \left(\frac{R}{\sqrt{2}}\right)^2 + \left(\frac{R}{\sqrt{2}}\right)^2 = \frac{R^2}{2} + \frac{R^2}{2} = R^2 \]
Final answer: \( \frac{3}{7} \) is tri-saptama, and the identity \( (R\sin\theta)^2 + (R\cos\theta)^2 = R^2 \) is verified for \( \theta = 45^\circ \).
Common Slips Students Make in This Chapter
This chapter is a name-and-date trap. The pairs below are the mix-ups that cost accuracy — including one printing slip in the textbook’s own table.
| Students write… | Correct is… | How to check your answer |
|---|---|---|
| “Bhāskara I wrote Līlāvatī and Bījagaṇita” | Bhāskara I (A.D. 600) was Āryabhaṭa’s commentator; Bhāskara II (A.D. 1114) wrote Līlāvatī and Bījagaṇita | Attach the date to the work: Līlāvatī belongs to the twelfth century, so it must be Bhāskara II |
| “Āryabhaṭa I and Aryabhata II are the same person” | Āryabhaṭa I (A.D. 499) wrote the Āryabhaṭīya; Aryabhata II (A.D. 950) defined addition and subtraction | The definition “the making into one of several numbers is addition” belongs to Aryabhata II |
| “Mahāvīra and Śrīdhara are one mathematician” | Both worked around A.D. 850 but are two separate mathematicians | Count them as two entries in the Golden period list, not one |
| “guṇya is the multiplier” | guṇya = multiplicand (the number repeated); guṇaka = multiplier (how many times) | In \( 124 \times 23 \), 124 is the guṇya and 23 the guṇaka |
| “The inverse method of addition starts from the units column” | The inverse process starts from the extreme left (the last place) | The first partial sum written is for the leftmost column, with corrections rubbed in as you move right |
| “3.1416 is Mādhava’s value of \( \pi \)” | 3.1416 → Āryabhaṭa I (A.D. 499); \( 3.14159265359 \) → Mādhava (A.D. 1500); \( \frac{355}{113} \) → Nīlakaṇṭha (A.D. 1500) | Match the number of digits to the period: the oldest value has the fewest decimals |
| “\( R\sin 30^\circ = R/\sqrt{3} \)” | The chapter’s table prints \( R\sin 30^\circ = R/\sqrt{3} \), but this is a slip — the standard value is \( \sin 30^\circ = \frac{1}{2} \), so \( R\sin 30^\circ = R/2 \) | Quote the standard value \( R/2 \) in your answer and flag the table’s 30° entry as a slip |
| “The Sulbasūtras prove Indians invented the Pythagoras theorem” | The Sulbasūtras contain a statement of the result now called the Pythagoras theorem, with applications in altar construction | Use the precise phrase: “a statement of the theorem with applications” — not an invention claim |
| “Brahmagupta knew twenty-eight operations” | He lists twenty operations and eight determinations — two separate lists | Count them as 20 + 8; never merge into 28 |
How to Answer the Chapter’s Exercise Questions
This is answer-writing guidance for the five questions at the end of the chapter (NCERT, p. 114) — what each complete answer must contain.
Q1 — The fundamental operations
State the number (eight) and list all eight: addition, subtraction, multiplication, division, square, square-root, cube, cube-root. Add who gave the root rules: Āryabhaṭa I gave the square and cube-root rules; Brahmagupta gave the cube-root rule only (p. 102).
Q2 — Mathematicians of geometry and trigonometry, and similarities with modern concepts
Build a table of mathematicians with periods: for geometry, the Sulbasūtra authors (Baudhāyana, Apastamba, Kātyāyana, around 800 B.C.), Āryabhaṭa I (A.D. 499) and Brahmagupta (A.D. 628); for trigonometry, Āryabhaṭa I, Brahmagupta, Govindaswami (A.D. 880), Mādhava (A.D. 1500) and Nīlakaṇṭha (A.D. 1500).
Then give at least three similarities with today’s mathematics: the R-sine notation (jyā = \( R\sin\theta \)) compared with the modern sine function, the identity \( (R\sin\theta)^2 + (R\cos\theta)^2 = R^2 \), the formula \( \sin(A \pm B) \), and the \( \pi \) approximations versus the modern value.
Q3(a) — Old board process vs the present system
Contrast the two settings: calculations were performed on a board (pāṅgaṇita, dhūli-karma) where figures were rubbed out and corrected, versus paper today where the working stays visible. Mention the direct and inverse processes, and that the board method of long division becomes the modern layout once the steps are written one below the other (pp. 107–108).
Q3(b) — Which process is easier, and why
Give a reasoned personal opinion with a concrete reason — for example, the inverse process suits a board because corrections are rubbed in, while the modern paper method suits a written answer because every step remains visible. Any clearly argued answer works; the reason matters more than the choice.
Q4 — Three terms per operation
Write at least three terms with meanings for each operation: addition — saṅkalita, saṅkalana, miśraṇa, sammelana, prakṣepaṇa, saṃyojana, ekīkaraṇa, yoga; subtraction — vyutkalita, vyutkalana, śodhana, patana, viyoga, with śeṣa for the remainder; multiplication — guṇana, hanana, vadha, kṣaya, with pratyutpanna for the product; division — bhāgāhara, bhājana, harāṇa, chedana, harana, with labdhi for the quotient (pp. 102–107).
Q5 — Other concepts from the literature
Answer as a short report on one or two concepts found in the chapter’s bibliography (p. 115) — for example, M.D. Srinivas’s lecture on Piṅgala’s Chandahasastra (the metrical combinatorics of Piṅgala) or the Subramanian–Srinivas study of the development of calculus in India. Name the source you used.
Two general habits: keep dates consistent (always A.D. or B.C., never mixed silently), and write Sanskrit terms with diacritics plus a one-word translation the first time you use them. All chapters of the rationalised book are on the official NCERT textbook portal.
Last-Minute Revision Sheet
Ten minutes before you close the book: three tables and three takeaways.
Table 1 — Who did what, when:
| Mathematician | Period | Major work or contribution |
|---|---|---|
| Āryabhaṭa I | A.D. 499 | Āryabhaṭīya; symbol for zero; square and cube-root rules; R-sines; \( \pi \approx 3.1416 \) |
| Varāhamihira | A.D. 505 | Golden-period astronomer-mathematician |
| Bhāskara I | A.D. 600 | Commentator on Āryabhaṭa; all operations are increase or decrease |
| Brahmagupta | A.D. 628 | Brahmasphuṣasiddhānta; twenty operations, eight determinations; cube-root rule; kuṇaka (pulveriser); area formulae |
| Mahāvīra | A.D. 850 | Worked on the pāṭi squaring method |
| Śrīdhara | A.D. 850 | Worked on the pāṭi squaring method |
| Aryabhata II | A.D. 950 | Defined addition and subtraction |
| Śrīpati | A.D. 1039 | Named the kapāṇa-sandhi multiplication method |
| Bhāskara II | A.D. 1114 | Līlāvatī and Bījagaṇita |
| Mādhava | A.D. 1500 | \( \pi \approx 3.14159265359 \) |
| Nīlakaṇṭha | A.D. 1500 | \( \pi \approx \frac{355}{113} \) |
Table 2 — The operations and their Sanskrit names:
| Operation | Sanskrit name | Key terms |
|---|---|---|
| Addition | saṅkalita | saṅkalana, miśraṇa, sammelana, prakṣepaṇa, saṃyojana, ekīkaraṇa, yoga |
| Subtraction | vyutkalita | vyutkalana, śodhana, patana, viyoga; śeṣa (remainder); sarvadhana (minuend); viyojaka (subtrahend) |
| Multiplication | guṇana | hanana, vadha, kṣaya (“killing”); guṇya, guṇaka; pratyutpanna (product) |
| Division | bhāgāhara | bhājana, harāṇa, chedana, harana; bhājya, bhājaka, labdhi |
| Square | varga | kṛti |
| Square-root | varga-mūla | mūla, pāda |
Table 3 — Field to modern counterpart:
| Field | Indian contribution | Modern counterpart |
|---|---|---|
| Numeral system | Decimal place value; nine symbols plus zero | Place-value notation used worldwide |
| Arithmetic | Board operations (pāṅgaṇita) with the eight fundamental operations | The four basic operations plus roots |
| Geometry | Sulbasūtra statement and applications of the Pythagoras result; area and perimeter formulae | Pythagoras theorem applications; mensuration |
| Algebra | Bījagānita; kuṇaka (pulveriser); linear and quadratic equations from geometry | Algebra with unknowns |
| Trigonometry | Sine and cosine functions; the identity; \( \sin(A \pm B) \); \( \pi \) approximations | Modern trigonometry |
Three one-line takeaways:
- Decimal place value and the symbol for zero are India’s foundational contribution to mathematics.
- All of pāṅgaṇita rests on eight fundamental operations: the four basics plus square, square-root, cube, cube-root.
- The Golden (Siddhāntic) period runs A.D. 500–1200, from Āryabhaṭa I to Bhāskara II.
For every Class 11 subject in one place, see the Class 11 CBSE notes index.
Quick Doubts, Answered
Who discovered zero, and where does the chapter say the symbol came from?
The chapter says the symbol for zero was discovered by Āryabhaṭa I (born A.D. 496) in connection with the decimal expression of numbers: “the vacant places should be filled up with a circle” — the śūnya — as illustrated by his commentator Bhāskara I (A.D. 600) (NCERT, p. 101). It revolutionised computation: nine numerical symbols plus zero replaced all earlier notation.
What is the Golden (Siddhāntic) period of Indian mathematics, and who brackets it?
The Golden (Siddhāntic) period runs A.D. 500–1200. It opens with Āryabhaṭa I (born A.D. 496) and closes with Bhāskara II (born A.D. 1114); between them come Varāhamihira (A.D. 505), Bhāskara I (A.D. 600), Brahmagupta (A.D. 628), Mahāvīra (A.D. 850), Śrīdhara (A.D. 850) and Śrīpati (A.D. 1039) (NCERT, p. 100).
What are the eight fundamental operations of ancient gaṇita?
Addition, subtraction, multiplication, division, square, square-root, cube and cube-root (NCERT, p. 102). Āryabhaṭa I gave the rules for square and cube-roots; Brahmagupta gave the cube-root rule only.
What does kapāṇa-sandhi mean, and which mathematician named it?
Kapāṇa-sandhi means “junction of two doors” — the multiplicand is placed below the multiplier as two doors meeting at a junction, and the multiplication proceeds in the direct or inverse order. Śrīpati (A.D. 1039) gave the name (NCERT, p. 105); the chapter’s glossary spells it “kapaya-sandhi.”
Does the Sulbasūtra really contain the Pythagoras theorem?
It contains a statement of the result now known as the Pythagoras theorem along with applications — constructing a square equal in area to the sum or the difference of two squares, or to a rectangle (NCERT, p. 112). So the precise phrasing is “statement and applications,” not “invented by Indians.”
Which Indian mathematicians gave values of pi, and what approximations did they give?
Āryabhaṭa I (A.D. 499) gave \( 3.1416 \); Mādhava (A.D. 1500) gave \( 3.14159265359 \); Nīlakaṇṭha (A.D. 1500) gave \( \frac{355}{113} \) (NCERT, p. 113).
Reference: NCERT Class 11 Knowledge Traditions and Practices of India textbook, chapter “Mathematics in India”.
Explore Class 11 Knowledge Traditions Practices Of India Notes
- Class 11 Knowledge Traditions Practices Of India Notes
- Class 11 CBSE Notes
- CBSE Notes for Classes 1 to 12
- Previous: Indian Art and Architecture
Class 11 Knowledge Traditions Practices Of India on LearnCBSE:
Official source: download the NCERT textbook free from ncert.nic.in.