Maths Teacher

Class 10 Maths – Theorem : Fundamental Theorem of Arithmetic 🧠 What Does the Theorem Say? " Every composite number can be expressed (factorised) as a product of prime numbers, and this factorisation is unique , apart from the order of the prime factors. " ✅ Let’s Break It Down: A composite number is a natural number that is not prime and has more than two factors . According to the theorem: Every composite number can be written as a product of prime numbers . This factorisation is unique , except for the order in which the primes are written. 🔢 Example Take the number 32760 We can write: 32760 = 2 × 2 × 2 × 3 × 3 × 5 × 7 × 13 = 2³ × 3² × 5 × 7 × 13 Now, whether you write it as: 2 × 3 × 5 × 7 × 13 × 2 × 3 × 2 or 3 × 2 × 2 × 5 × 7 × 3 × 2 × 13 👉 It doesn’t matter — it’s still the same set of prime numbers multiplied together. ✅ That’s what “ unique, except for order ” means. 💡 Important Notes A prime factorisation is a way to express a ...

Story: "The Secret Code of Numbers – From Euclid to Gauss" “Behind every number lies a hidden code. One that only the greatest minds in history could fully understand...” 🌿 Chapter 1: The Ancient Whisper – Euclid’s Discovery It all began more than 2,000 years ago in the great library of Alexandria , where a wise Greek mathematician named Euclid was writing his famous book – The Elements . In Book IX, Proposition 14 , Euclid hinted at something amazing: "Every number is built from primes… like bricks building a wall." But he didn’t call it a "theorem." He didn’t even prove it the way we do now. It was more like a quiet whisper from the past – a mystery waiting to be unlocked. 👑 Chapter 2: Enter the Prince of Mathematicians – Carl Friedrich Gauss Fast forward to the year 1801 in Germany. A young genius named Carl Friedrich Gauss , only 24 years old , published his masterpiece: 📘 Disquisitiones Arithmeticae — a powerful book that changed...

  The Fundamental Theorem of Arithmetic – CBSE Class 10  🔍 What You Already Know From earlier classes, you know that: Every natural number can be written as a product of prime numbers . Example: 2 = 2 4 = 2 × 2 253 = 11 × 23 But now, we ask the reverse: Can every natural number be made by multiplying prime numbers? 📌 Understanding with Examples Take a set of primes: 2, 3, 7, 11, 23 Now multiply some or all of them: 7 × 11 × 23 = 1771 2 × 3 × 7 × 11 × 23 = 10626 3 × 7 × 11 × 23 = 5313 2² × 3 × 7 × 11 × 23 = 21252 You can create infinitely many such numbers using combinations and powers of primes. 📘 Important Idea There are infinitely many prime numbers . All composite numbers are made by multiplying primes. But here's the big question : Is it possible that some composite number cannot be expressed as a product of primes? Let’s explore by factorising numbers. 🌳 Factor Tree Method Let’s take a number and break it into its prime f...

Real Numbers – Introduction CBSE Class 10 Maths 🔍 What You Already Know (from Class 9) : Real Numbers = Rational + Irrational numbers Rational Numbers : Can be written as p/q (q ≠ 0) Irrational Numbers : Cannot be written as p/q Examples: √2, √5, π 🧠 What You Will Learn in This Chapter : In Class 10, we go deeper into the properties of real numbers, especially positive integers , through: Euclid’s Division Algorithm Fundamental Theorem of Arithmetic These concepts form the foundation of number theory , with applications in HCF, irrational numbers, and decimal expansions. 📍1. Euclid’s Division Algorithm 📌 Definition : Given two positive integers a and b (a > b), there exist integers q and r such that: a = bq + r {where } 0≤r< b This is basically the long division method you already know! 🔧 Key Use : Helps us find the HCF (Highest Common Factor) of two numbers. Example: To find HCF of 210 and 45, we apply Euclid’s algorithm step by s...