Denseness of Rationals in ℝ | Rational & Irrational Numbers | Real Analysis-I | B.Sc 1st Year

Denseness of Rationals in ℝ | Rational & Irrational Numbers | Real Analysis-I | B.Sc 1st Year Welcome to K.Mathematics 🎯 In this lecture of Real Analysis-I, we explore one of the most important and beautiful concepts: 👉 Denseness of Rational Numbers in ℝ This video is specially designed for B.Sc 1st Year Mathematics students and also helpful for CSIR-NET / GATE aspirants. --- 🔥 What you will learn in this video: ✔ Concept of Rational and Irrational Numbers ✔ Proof that √2 is irrational (Theorem-1) ✔ Density Theorem – Between any two real numbers lies a rational number ✔ Denseness of Irrational Numbers in ℝ ✔ Why there are infinitely many rational numbers between two real numbers ✔ Solved problems for better understanding --- ⏱️ Timestamps: ⏱️ 0:00 – Introduction ⏱️ 0:32 – Rational and Irrational Numbers ⏱️ 2:43 – Theorem 1 (√2 is Irrational) ⏱️ 4:59 – Density Theorem (Rationals in ℝ) ⏱️ 9:41 – Solved Problems ⏱️ 12:17 – Outro --- 📚 Why this topic is important? Fundamental concept in Real Analysis-I Frequently asked in university exams Builds base for limits, sequences, and continuity Very useful for competitive exams (CSIR-NET, GATE) --- 👉 Don’t forget to Like 👍, Share 📤 and Subscribe 🔔 to K.Mathematics for more Real Analysis lectures! 📌 Keep learning, keep growing! 🚀 Your Queries: real analysis real analysis 1 denseness of rationals density theorem rational numbers irrational numbers rational and irrational numbers real numbers number system density of rational numbers in R denseness of Q in R infinite rational numbers rational numbers between two real numbers irrational numbers between two numbers sqrt 2 irrational proof proof sqrt 2 irrational theorem rational numbers density theorem proof archimedean property supremum property completeness of real numbers real analysis bsc 1st year bsc mathematics bsc math real analysis real analysis lecture mathematics lecture k mathematics csir net mathematics gate mathematics ugc net maths basic real analysis intro to real analysis properties of real numbers algebraic properties of R order properties of R examples of rational numbers examples of irrational numbers decimal expansion rational numbers non terminating non repeating decimals rational vs irrational real line density between any two real numbers infinite numbers between two points rational numbers proof irrational numbers proof exam important real analysis university maths lecture mathematics concepts pure mathematics analysis for beginners math for bsc students real analysis notes real analysis concepts real numbers theory number theory basics advanced mathematics mathematics youtube lecture density of irrationals infinite irrationals between numbers density of the irrational numbers in the reals denseness of rational numbers irrational numbers are dense in real numbers sum of rational and irrational number irrational numbers are dense in the real line density of rationals and irrationals in R rational numbers are dense in real numbers bsc maths 1st year real analysis density of rational number in bangla density of rationals and irrationals density of the irrationals in the reals 6.4 density of rationals and irrationals #RealAnalysis #DensenessOfRationals #DensityTheorem #RationalNumbers #IrrationalNumbers #RealNumbers #NumberSystem #BScMathematics #BScMaths #RealAnalysis1 #MathLecture #Mathematics #PureMathematics #KMathematics #MathConcepts #MathsForStudents #HigherMathematics #CSIRNETMaths #GATEMaths #UGCNETMaths #MathEducation#kmathematics #education #bscmathematics #nep2020 #utkaluniversity #fms #fmuniversity #LearnMathematics #MathsClass #MathsTeacher #OnlineMaths #MathTutorial #MathsVideo #MathsLecture #ExamPreparation #UniversityMaths #RealAnalysisNotes #MathsRevision #MathsBasics #AdvancedMaths #DensityOfRationals #IrrationalProof #Sqrt2Irrational #InfiniteNumbers #BetweenTwoNumbers #MathsTopic