This is the National Council of Educational Research and Training (NCERT) mathematics textbook for Class IX. It is an English medium (Roman script) book — the complete text, theorems, worked examples, exercises and answers are in English, and there is no Hindi content in it. It is prescribed for CBSE schools and for state boards that follow the NCERT curriculum. Class 9 is where mathematics becomes a subject of proof rather than procedure. The book opens with number systems — rational and irrational numbers, real numbers on the number line, laws of exponents — then moves to polynomials, the remainder and factor theorems and algebraic identities. Coordinate geometry introduces the cartesian plane, and linear equations in two variables introduces their graphical solution. The geometry section is the heart of the year: Euclid's definitions, axioms and postulates set out what a proof is, and the chapters on lines and angles, triangles and their congruence, quadrilaterals and circles then prove theorem after theorem from those foundations. The book closes with Heron's formula, surface areas and volumes of solids, and statistics. The reason to work from the prescribed text here is that the theorem proofs asked in the board paper are the proofs as written in this book, in this order and with this notation. A guide's condensed version will not do — the examiner is looking for the given, to prove, construction and proof structure the text establishes. The exercises are graded, with the later questions in each set requiring the student to choose which theorem applies rather than apply a named one. Beyond the board examination, Class 9 mathematics is the foundation year for NTSE, olympiad and early JEE preparation. Polynomials, coordinate geometry and circles at this level are the direct groundwork for what Class 10 and 11 build on, and those programmes work from the NCERT exercises rather than around them. Answers are provided at the end for independent checking. Key Features: English medium book — complete content written in English (Roman script), no Hindi content Published by NCERT (National Council of Educational Research and Training) Mathematics — Textbook for Class IX (Class 9) Prescribed for CBSE schools and state boards following the NCERT curriculum Covers Number Systems — rational and irrational numbers, real numbers, exponents Covers Polynomials — remainder theorem, factor theorem and algebraic identities Covers Coordinate Geometry and Linear Equations in Two Variables Covers Introduction to Euclid's Geometry — definitions, axioms and postulates Covers Lines and Angles, Triangles and congruence criteria Covers Quadrilaterals and Circles with full theorem proofs Covers Heron's Formula Covers Surface Areas and Volumes Covers Statistics Theorem proofs set out in the structure and notation the board expects Worked examples followed by graded exercise sets Answers provided at the end for independent checking Base textbook for the CBSE Class 9 mathematics examination Foundation text for NTSE, olympiad and early JEE preparation Standard NCERT text — the base book before any guide or question bank यह पुस्तक राष्ट्रीय शैक्षिक अनुसंधान और प्रशिक्षण परिषद (NCERT) द्वारा कक्षा IX (कक्षा 9) के लिए निर्धारित गणित की पाठ्यपुस्तक है। ध्यान दें: यह पुस्तक पूर्णतः अंग्रेज़ी भाषा एवं रोमन लिपि में है — इसमें हिन्दी भाषा का प्रयोग नहीं है। इसमें संख्या पद्धति, बहुपद (शेषफल एवं गुणनखंड प्रमेय, बीजीय सर्वसमिकाएँ), निर्देशांक ज्यामिति, दो चरों वाले रैखिक समीकरण, यूक्लिड की ज्यामिति का परिचय, रेखाएँ एवं कोण, त्रिभुज, चतुर्भुज, वृत्त, हीरोन का सूत्र, पृष्ठीय क्षेत्रफल एवं आयतन तथा सांख्यिकी — ये अध्याय सम्मिलित हैं। इसी वर्ष गणित प्रक्रिया से प्रमाण की ओर बढ़ता है — बोर्ड परीक्षा में प्रमेयों की सिद्धि उसी क्रम एवं संकेतन में अपेक्षित होती है जैसी इस पुस्तक में दी गई है, अतः गाइड के संक्षिप्त रूप के स्थान पर मूल पुस्तक से पढ़ना आवश्यक है। हल किए गए उदाहरण, क्रमबद्ध अभ्यास-प्रश्नावली तथा अंत में उत्तर दिए गए हैं। NTSE, ओलंपियाड एवं JEE फाउंडेशन की तैयारी भी इन्हीं अभ्यासों पर आधारित होती है।