Real numbers form the foundation of Class 10 mathematics. From solving equations to calculating measurements, understanding these numbers is essential for both exams and practical applications. Real numbers include rational numbers, which can be expressed as fractions, and irrational numbers, which cannot. Mastering their formulas simplifies problem-solving, reduces errors, and builds confidence in algebra, HCF-LCM calculations, and more.

This guide covers all important real number formulas, explains their applications with clear examples, and shares proven tips to memorize them quickly.

CBSE Class 10 Maths Real Numbers Formulas

1. Euclid’s Division Lemma

For any two positive integers aaa and bbb, there exist unique integers qqq (quotient) and rrr (remainder) such that:

a=bq+r,0≤r<ba = bq + r, \quad 0 \le r < ba=bq+r,0≤r<b

2. HCF and LCM Formulas

• Relation between HCF and LCM: HCF(a,b)×LCM(a,b)=a×bHCF(a, b) \times LCM(a, b) = a \times bHCF(a,b)×LCM(a,b)=a×b
• Prime factorization method:
  • HCF: Multiply the lowest powers of common prime factors.
  • LCM: Multiply the highest powers of all prime factors.

Example:
For 12 and 18:

• Prime factors of 12 → 22×32^2 \times 322×3
• Prime factors of 18 → 2×322 \times 3^22×32
• HCF → 21×31=62^1 \times 3^1 = 621×31=6
• LCM → 22×32=362^2 \times 3^2 = 3622×32=36

3. Fundamental Theorem of Arithmetic

• Every composite number can be expressed as a product of prime numbers, and this factorization is unique (order does not matter).
• Example: 36 = 22×322^2 \times 3^222×32

4. Rational and Irrational Numbers

• Rational Numbers: Can be expressed as pq\frac{p}{q}qp​, where ppp and qqq are integers and q≠0q \ne 0q=0.
  • Example: 34,−72\frac{3}{4}, \frac{-7}{2}43​,2−7​
    
• Irrational Numbers: Cannot be expressed as a fraction; their decimal expansions are non-repeating and non-terminating.
  • Example: 2,π\sqrt{2}, \pi2​,π

5. Irrationality Proofs

Example: Prove that 2\sqrt{2}2​ is irrational.

• Assume 2=mn\sqrt{2} = \frac{m}{n}2​=nm​ in simplest form.
• Then 2n2=m22n^2 = m^22n2=m2, implying m is even.
• Substitute m=2km = 2km=2k: 2n2=4k22n^2 = 4k^22n2=4k2 → n2=2k2n^2 = 2k^2n2=2k2 → n is even.
• Contradiction: m and n cannot both be even if fraction is simplest.
• Conclusion: 2\sqrt{2}2​ is irrational.
  

6. Special Theorems

• Prime Divides Square Theorem: If a prime number xxx divides a2a^2a2, then xxx also divides aaa.

Useful in proofs of irrationality and HCF calculations.

7. Number Type Reference

Applications of Real Numbers Formulas – Class 10

• Arithmetic Problems: HCF and LCM formulas help in dividing items into equal groups and solving problems involving fractions, ratios, and multiples.
• Algebra Simplification: Properties of real numbers (commutative, associative, distributive) are used to simplify expressions and solve equations efficiently.
• Number Theory & Proofs: Euclid’s Division Lemma and prime factorization formulas help calculate HCF/LCM of large numbers and prove rationality or irrationality of numbers.
• Exam Calculations: Formulas reduce errors and save time in complex numerical problems, including problems on divisibility, factorization, and algebraic manipulation.

Tips to Master Real Numbers Formulas

• Understand the concept behind each formula instead of rote memorization.
• Use flashcards or formula sheets to revise regularly.
• Group formulas by type—arithmetic, algebra, or geometry—for easier recall.
• Apply formulas in practical scenarios, like dividing items equally or calculating distances.
• Create mnemonics or memory aids for algebraic identities and HCF/LCM rules.
• Practice multiple examples from CBSE textbooks and sample papers to reinforce understanding.

Real Numbers Formulas for Class 10 – Examples

Example 1: Prove that √3 is an irrational number

Solution:

1. Assume √3 is rational. Then it can be written as a fraction pq\frac{p}{q}qp​, where p and q are coprime integers.
2. Squaring both sides:
   3=p2q2  ⟹  p2=3q23 = \frac{p^2}{q^2} \implies p^2 = 3q^23=q2p2​⟹p2=3q2
3. This shows that p² is divisible by 3, so p must also be divisible by 3. Let p=3kp = 3kp=3k.
4. Substitute back:
   (3k)2=3q2  ⟹  9k2=3q2  ⟹  q2=3k2(3k)^2 = 3q^2 \implies 9k^2 = 3q^2 \implies q^2 = 3k^2(3k)2=3q2⟹9k2=3q2⟹q2=3k2
5. This implies q is divisible by 3.
6. Contradiction: p and q were supposed to have no common factors.

Conclusion: √3 cannot be expressed as a fraction; hence, it is irrational.

Example 2: Organizing items into equal groups

Problem: A shop has 540 red pencils and 360 blue pencils. How many pencils can be placed in each equal stack?

Solution:

1. Find the HCF of 540 and 360:
   • Prime factorization:
     • 540 = 2² × 3³ × 5
     • 360 = 2³ × 3² × 5
   • HCF = 2² × 3² × 5 = 180
2. Divide the pencils into stacks of 180 each.

Answer: Each stack will contain 180 pencils.