Class 9 · Mathematics · English medium · Part 1
Ganita Manjari Part 1: Class 9 Maths companion
Textbook of Mathematics for Grade 9
An interactive companion to Ganita Manjari (Grade 9 mathematics, Part 1): original explanations, a live coordinate plane, circle and probability explorers, worked examples and practice.
Hindi medium: गणित मंजरी भाग 1
An original companion to Ganita Manjari, NCERT Class 9 Mathematics (2026–27 edition). Official textbook ↗ ncert.nic.in
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Chapters
- ENहिंHinglish
Chapter 1: Orienting Yourself: The Use of Coordinates
Ganita Manjari (Class 9) Chapter 1 introduces coordinates. - ENHinglish
Chapter 2: Introduction to Linear Polynomials
The chapter moves from algebraic expressions to polynomials in one variable (terms, coefficients, constant term, degree) and then studies li - ENHinglish
Chapter 3: The World of Numbers (Number Systems)
From counting notches to the real number line - ENHinglish
Chapter 4: Exploring Algebraic Identities
Squares, strips, tiles and cubes that explain algebra - ENHinglish
Chapter 5: I’m Up and Down, and Round and Round (Circles)
Circles, chords, arcs and the angles they make - ENHinglish
Chapter 6: Measuring Space: Perimeter and Area
Running tracks, the hunt for π, and the art of cutting and rearranging - ENHinglish
Chapter 7: The Mathematics of Maybe: Introduction to Probability
This Class 9 chapter introduces probability as a number from 0 (impossible) to 1 (certain) that measures how likely an event is. - ENHinglish
Chapter 8: Predicting What Comes Next: Exploring Sequences and Progressions
The chapter studies sequences (ordered lists of numbers) and how to predict terms.
Also in हिन्दी / Hinglish
- Chapter 1: अपनी स्थिति पहचानना: निर्देशांकों का उपयोग (हिन्दी) · Apni Jagah Pehchano: Coordinates ka Use (Hinglish)
- Chapter 2: Linear Polynomials se Pehchaan (Hinglish)
- Chapter 3: Numbers ki Duniya (Hinglish)
- Chapter 4: Algebraic Identities ko Explore Karo (Hinglish)
- Chapter 5: Main Upar-Neeche, aur Gol-Gol Ghoomta Hoon (Hinglish)
- Chapter 6: Jagah ko Naapna: Perimeter aur Area (Hinglish)
- Chapter 7: Shayad ka Ganit: Probability ka Introduction (Hinglish)
- Chapter 8: Aage Kya Aayega? Sequences aur Progressions ki Khoj (Hinglish)
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absolute valueनिरपेक्ष मानThe distance of a number from 0 on the number line, written |x|; it is never negative.Ch 3 · The World of Numbersadditive inverseयोज्य प्रतिलोमThe number -a that gives 0 when added to a.Ch 3 · The World of Numbersalgebra tilesSquares and strips of areas x², x and 1 used to model products and factors.Ch 4 · Exploring Algebraic IdentitiesarcचापThe part of a circle traced when moving along it from one point to another. The shorter of the two arcs is the minor arc, the longer the major arc.Ch 5 · I’m Up and Down, and Round and RoundarcचापA piece of a circle between two of its points. An arc subtending θ ° has length 2 π r × θ 360.Ch 6 · Measuring Space: Perimeter and Areaareaक्षेत्रफलThe amount of flat surface a shape covers, measured in square units.Ch 6 · Measuring Space: Perimeter and Areaarithmetic progressionसमांतर श्रेढ़ीA sequence that adds the same number d at every step: a, a + d, a + 2d, …Ch 8 · Predicting What Comes Next: Exploring Sequences and ProgressionsC
Cartesian planeकार्तीय तलThe flat surface on which the coordinate axes are drawn; every point on it has coordinates.Ch 1 · Orienting Yourself: The Use of CoordinateschordजीवाA segment whose two ends lie on the circle.Ch 5 · I’m Up and Down, and Round and Roundcircleवृत्तA round curve whose points are all exactly r from one point inside it, the centre; r is the radius.Ch 5 · I’m Up and Down, and Round and Roundcircumcentreपरिकेंद्रThe centre of a circumcircle: where the perpendicular bisectors of the sides of a triangle meet.Ch 5 · I’m Up and Down, and Round and Roundcircumcircleपरिवृत्तThe one circle that passes through all three vertices of a triangle.Ch 5 · I’m Up and Down, and Round and RoundcircumferenceपरिधिThe perimeter of a circle: C = π d = 2 π r.Ch 6 · Measuring Space: Perimeter and AreaclosedसंवृतA collection of numbers is closed under an operation when doing that operation on any two of its members always gives a member of the collection.Ch 3 · The World of NumberscoefficientगुणांकThe number that multiplies a variable or a power of it. In -x² the coefficient is -1.Ch 2 · Introduction to Linear PolynomialscoefficientगुणांकThe number multiplying the letters in a term; in 7x² it is 7.Ch 4 · Exploring Algebraic IdentitiescollinearसंरेखLying on one straight line. A, B, C are collinear with B between them when AB + BC = AC.Ch 1 · Orienting Yourself: The Use of CoordinatescollinearसंरेखLying on one straight line.Ch 5 · I’m Up and Down, and Round and Roundcommon differenceसार्व अंतरThe fixed number d added at each step of an AP; d = t2 - t1.Ch 8 · Predicting What Comes Next: Exploring Sequences and Progressionscommon ratioसार्व अनुपातThe fixed multiplier r of a GP; r = t2 ÷ t1.Ch 8 · Predicting What Comes Next: Exploring Sequences and Progressionsconcyclicएकवृत्तीयPoints are concyclic when one circle passes through all of them.Ch 5 · I’m Up and Down, and Round and Roundconstant termअचर पदA term with no variable in it, so its value never changes, like -5 in 2x³ - x² + 7x - 5.Ch 2 · Introduction to Linear Polynomialscoordinate systemनिर्देशांक पद्धतिA fixed starting point and reference lines that let us name every location with numbers.Ch 1 · Orienting Yourself: The Use of Coordinatescoordinatesनिर्देशांकThe ordered pair (x, y) that fixes a point: its signed distances from the y-axis and from the x-axis.Ch 1 · Orienting Yourself: The Use of Coordinatescorollaryउपप्रमेयA result that follows almost at once from a theorem just proved.Ch 5 · I’m Up and Down, and Round and Roundcyclic 4-gonचक्रीय चतुर्भुजA quadrilateral whose four vertices lie on one circle.Ch 6 · Measuring Space: Perimeter and Areacyclic numberचक्रीय संख्याA number such as 142857 whose multiples by 1, 2, 3, … up to a limit only rotate its digits.Ch 3 · The World of Numberscyclic quadrilateralचक्रीय चतुर्भुजA quadrilateral whose four vertices lie on one circle.Ch 5 · I’m Up and Down, and Round and RoundD
degreeघातIn a polynomial, the largest power of the variable that appears. 2x³ - x² + 7x - 5 has degree 3.Ch 2 · Introduction to Linear Polynomialsdiameterव्यासA chord through the centre; its length is 2r and no chord is longer.Ch 5 · I’m Up and Down, and Round and Rounddifference of two squaresदो वर्गों का अंतरAn expression a² - b², which factorises as (a + b)(a - b).Ch 4 · Exploring Algebraic Identitiesdistance formulaदूरी सूत्रThe distance between (x1, y1) and (x2, y2) is √ (x2 - x1)² + (y2 - y1)².Ch 1 · Orienting Yourself: The Use of CoordinatesE
equally likelyसमप्रायिकHaving the same chance of happening, because nothing favours one outcome over another.Ch 7 · The Mathematics of Maybe: Introduction to ProbabilityequationसमीकरणA statement that two expressions are equal. An ordinary equation such as 3x - 5 = 16 holds only for particular values, found by solving it; an identity holds for all values.Ch 4 · Exploring Algebraic IdentitieseventघटनाA collection of outcomes, that is, a subset of the sample space.Ch 7 · The Mathematics of Maybe: Introduction to Probabilityexperimental probabilityप्रायोगिक प्रायिकताThe fraction of trials in which an event actually happened.Ch 7 · The Mathematics of Maybe: Introduction to Probabilityexplicit ruleA formula giving the n th term straight from n, such as tn = 7n + 3.Ch 8 · Predicting What Comes Next: Exploring Sequences and ProgressionsF
factorगुणनखंडAn expression that divides another exactly; x + 2 is a factor of x² + 5x + 6.Ch 4 · Exploring Algebraic IdentitiesfactorisationगुणनखंडनWriting an expression as a product of simpler expressions.Ch 4 · Exploring Algebraic Identitiesfinite sequenceपरिमित अनुक्रमA sequence that stops after some number of terms. One that never stops is an infinite sequence.Ch 8 · Predicting What Comes Next: Exploring Sequences and ProgressionsfractalA shape made of smaller and smaller copies of itself, such as the Sierpiński triangle.Ch 8 · Predicting What Comes Next: Exploring Sequences and ProgressionsfunctionफलनA rule that gives exactly one output for each input, such as p(x) = 3x - 4.Ch 2 · Introduction to Linear PolynomialsG
gambler's fallacyThe mistaken belief that a result becomes more likely because it has not happened for a while.Ch 7 · The Mathematics of Maybe: Introduction to Probabilitygeneralisationव्यापकीकरणA result that includes another as a special case, as Brahmagupta’s formula includes Heron’s.Ch 6 · Measuring Space: Perimeter and Areageometric progressionगुणोत्तर श्रेढ़ीA sequence that multiplies by the same number r at every step: a, ar, ar², …Ch 8 · Predicting What Comes Next: Exploring Sequences and ProgressionsI
identityसर्वसमिकाAn equality of two expressions that holds for every value of its letters, such as (a + b)² = a² + 2ab + b².Ch 4 · Exploring Algebraic Identitiesintegerपूर्णांकA natural number, zero, or the negative of a natural number. The collection is written Z.Ch 3 · The World of Numbersirrational numberअपरिमेय संख्याA real number that is not a ratio p q of integers with q ≠ 0, such as √ 2 or π; its decimal neither stops nor repeats.Ch 3 · The World of Numbersirrational numberअपरिमेय संख्याA number that cannot be written as a fraction of two integers; its decimal never ends or repeats.Ch 6 · Measuring Space: Perimeter and AreaL
law of large numbersWith a very large number of trials, the experimental probability of an event settles near its theoretical probability.Ch 7 · The Mathematics of Maybe: Introduction to Probabilitylinear decayरैखिक ह्रासDecline by equal subtractions: in every time step of the same length, the quantity loses the same fixed amount.Ch 2 · Introduction to Linear Polynomialslinear equationरैखिक समीकरणAn equation made by setting a linear polynomial equal to a number, such as 25 + 2p = 85.Ch 2 · Introduction to Linear Polynomialslinear growthरैखिक वृद्धिGrowth by equal additions: in every time step of the same length, the quantity gains the same fixed amount.Ch 2 · Introduction to Linear Polynomialslinear patternरैखिक प्रतिरूपA list of numbers in which the same number is added each time to get the next term.Ch 2 · Introduction to Linear Polynomialslinear polynomialरैखिक बहुपदA polynomial of degree 1, which can be written as ax + b with a ≠ 0.Ch 2 · Introduction to Linear Polynomialslinear relationshipरैखिक संबंधA link between two changing quantities of the form y = ax + b; its graph is a straight line.Ch 2 · Introduction to Linear PolynomialslocusबिंदुपथThe set of exactly those points that obey a given rule, and no others.Ch 5 · I’m Up and Down, and Round and Roundlowest termsसरलतम रूपA fraction p q is in lowest terms when 1 is the only positive whole number dividing both p and q, and q > 0.Ch 3 · The World of NumbersM
medianमाध्यिकाThe segment joining a corner of a triangle to the middle point of the side facing it. It halves the area.Ch 6 · Measuring Space: Perimeter and Areamidpointमध्य-बिंदुThe point halfway along a segment; its coordinates are the averages of the ends' coordinates.Ch 1 · Orienting Yourself: The Use of CoordinatesN
natural numberप्राकृत संख्याA counting number: 1, 2, 3, … The collection is written N.Ch 3 · The World of NumbersO
one-to-one correspondenceएकैकी संगतताPairing each object of one collection with exactly one object of another, so the two collections can be compared without counting.Ch 3 · The World of Numbersordered pairक्रमित युग्मA pair of numbers in which order matters: (a, b) = (b, a) only when a = b.Ch 1 · Orienting Yourself: The Use of CoordinatesoriginमूलबिंदुThe point O(0, 0) where the two axes cross.Ch 1 · Orienting Yourself: The Use of CoordinatesoutcomeपरिणामOne possible result of a random experiment.Ch 7 · The Mathematics of Maybe: Introduction to ProbabilityP
perfect-square trinomialपूर्ण वर्गA three-term expression a² ± 2ab + b², equal to (a ± b)².Ch 4 · Exploring Algebraic IdentitiesperimeterपरिमापThe total length of the boundary of a closed shape.Ch 6 · Measuring Space: Perimeter and Areaperpendicular bisectorलंब समद्विभाजकThe line through the midpoint of a segment at right angles to it; its points are exactly those equally far from the two ends.Ch 5 · I’m Up and Down, and Round and RoundpolynomialबहुपदA sum of terms, each a number times a whole-number power of a variable, such as 2x³ - x² + 7x - 5.Ch 2 · Introduction to Linear Polynomialsprobabilityप्रायिकताA number from 0 to 1 that measures how likely an event is to happen.Ch 7 · The Mathematics of Maybe: Introduction to Probabilityproof by contradictionविरोधाभास द्वारा उपपत्तिA proof that assumes the opposite of the statement and reasons correctly to something impossible, so the assumption must be false.Ch 3 · The World of NumbersQ
quadrantचतुर्थांशOne of the four regions made by the axes, numbered I to IV anticlockwise from the top right.Ch 1 · Orienting Yourself: The Use of CoordinatesR
radiusत्रिज्याA segment from the centre to a point of the circle, or its length r.Ch 5 · I’m Up and Down, and Round and Roundrandom experimentयादृच्छिक प्रयोगA repeatable action whose possible results are known but whose next result is not, such as rolling a die.Ch 7 · The Mathematics of Maybe: Introduction to Probabilityrational expressionपरिमेय व्यंजकA fraction whose numerator and denominator are polynomials, with a non-zero denominator.Ch 4 · Exploring Algebraic Identitiesrational numberपरिमेय संख्याA number that can be written as p q with p, q integers and q ≠ 0. The collection is written Q.Ch 3 · The World of Numbersreal numberवास्तविक संख्याAny rational or irrational number; each point of the number line. The collection is written R.Ch 3 · The World of Numbersrecursive ruleपुनरावर्ती नियमA rule giving each term from earlier terms, plus the starting term(s), such as t1 = 6, tn = tn-1 + 5.Ch 8 · Predicting What Comes Next: Exploring Sequences and Progressionsreflectionपरावर्तनThe mirror image of a point or shape in a line. In the y-axis, (x, y) goes to (-x, y).Ch 1 · Orienting Yourself: The Use of Coordinatesrelative frequencyसापेक्ष बारंबारताHow often something happened divided by the total number of observations; the same as experimental probability.Ch 7 · The Mathematics of Maybe: Introduction to Probabilityrepeating decimalअसांत आवर्ती दशमलवA decimal that never stops but in which a block of digits repeats for ever, such as 0. 185.Ch 3 · The World of NumbersS
sampleप्रतिदर्शA smaller group chosen from a population in order to study it. A good sample is large and chosen at random.Ch 7 · The Mathematics of Maybe: Introduction to Probabilitysample spaceप्रतिदर्श समष्टिThe set S containing every outcome that a random experiment can produce.Ch 7 · The Mathematics of Maybe: Introduction to Probabilitysectorत्रिज्यखंडThe region of a disc between two radii and their arc; area π r² × θ 360.Ch 6 · Measuring Space: Perimeter and Areasegmentवृत्तखंडThe region of a disc between a chord and its arc.Ch 6 · Measuring Space: Perimeter and Areasegment of a circleवृत्तखंडThe region between a chord and one of the two arcs it cuts off.Ch 5 · I’m Up and Down, and Round and Roundsemi-perimeterअर्धपरिमापHalf the perimeter, s = 1 2 (a + b + c) for a triangle.Ch 6 · Measuring Space: Perimeter and Areasequenceअनुक्रमA list of numbers in a definite order, such as 3, 10, 17, 24, 31.Ch 8 · Predicting What Comes Next: Exploring Sequences and ProgressionsslopeढालThe number a in y = ax + b: how much y changes when x goes up by 1.Ch 2 · Introduction to Linear Polynomialsspecial caseविशेष स्थितिA result obtained from a more general one by adding an extra condition, such as b = a.Ch 6 · Measuring Space: Perimeter and Areasplitting the middle termFactorising ax² + bx + c by writing bx as two terms whose coefficients multiply to ac.Ch 4 · Exploring Algebraic IdentitiesstaggerThe distance between the starting marks of neighbouring lanes on a running track; for one lap it is 2 π w for lanes w wide.Ch 6 · Measuring Space: Perimeter and Areasubtendअंतरित करनाA chord or arc AB subtends ∠ APB at a point P: the angle formed by joining P to A and to B.Ch 5 · I’m Up and Down, and Round and RoundT
termपदOne of the parts of an expression that are added together, such as 4f, 6s and 11 in 4f + 6s + 11.Ch 2 · Introduction to Linear PolynomialstermपदOne number in a sequence. The term in position n is written tn.Ch 8 · Predicting What Comes Next: Exploring Sequences and Progressionsterminating decimalसांत दशमलवA decimal that stops after a finite number of digits, such as 0.4375.Ch 3 · The World of Numberstheoretical probabilityसैद्धांतिक प्रायिकताFavourable outcomes divided by all possible outcomes, when every outcome is equally likely.Ch 7 · The Mathematics of Maybe: Introduction to Probabilitytree diagramवृक्ष आरेखA branching diagram that lists the outcomes of a multi-step experiment; each path is one outcome.Ch 7 · The Mathematics of Maybe: Introduction to Probabilitytrialअभिप्रयोगOne performance of a random experiment, such as a single roll of a die.Ch 7 · The Mathematics of Maybe: Introduction to Probabilitytriangular numberत्रिभुजीय संख्याA number of the form 1 + 2 + ⋯ + n = n(n + 1) 2: 1, 3, 6, 10, 15, …Ch 8 · Predicting What Comes Next: Exploring Sequences and ProgressionsU
unbiasedअनभिनतOf a coin, die or spinner: made so that every outcome has the same chance.Ch 7 · The Mathematics of Maybe: Introduction to ProbabilityV
variableचरA letter that stands for a number that can change, such as x or t.Ch 2 · Introduction to Linear PolynomialsVirahānka–Fibonacci sequenceEach term is the sum of the two before it, starting 1, 2: so 1, 2, 3, 5, 8, 13, …Ch 8 · Predicting What Comes Next: Exploring Sequences and ProgressionsX
x-axisx-अक्षThe horizontal number line of the coordinate plane. Its points have the form (x, 0).Ch 1 · Orienting Yourself: The Use of Coordinatesx-coordinateभुजThe first number of (x, y): how far the point lies to the right (+) or left (−) of the y-axis.Ch 1 · Orienting Yourself: The Use of CoordinatesY
y-axisy-अक्षThe vertical number line of the coordinate plane. Its points have the form (0, y).Ch 1 · Orienting Yourself: The Use of Coordinatesy-coordinateकोटिThe second number of (x, y): how far the point lies above (+) or below (−) the x-axis.Ch 1 · Orienting Yourself: The Use of Coordinatesy-intercepty-अंतःखंडThe number b in y = ax + b; the line meets the y -axis at (0, b).Ch 2 · Introduction to Linear PolynomialsΠ
π (pi)पाईThe fixed ratio circumference ÷ diameter of every circle, about 3.14159. It is irrational.Ch 6 · Measuring Space: Perimeter and Areancert.ai is an independent learning companion aligned to the NCERT curriculum. It is not affiliated with or endorsed by NCERT.