1 Real Number introduction for class 10th both CBSE and state syllabus students can go through this
- 1. STD- X CHAPTER- 1 REAL NUMBERS
- 2. PREVIOUS KNOWLEDGE • Natural numbers : 1,2,3,4,5………………… • Whole numbers : 0,1,2,3,4,……………….. • Integers : -3,-2,-1,0,1,2,3………….. • Rational numbers - Numbers of the form where p and q are integers and q ≠ 0 • Irrational Numbers -Numbers that cannot be written in the form • Real numbers - Rational numbers and Irrational numbers together form set of real numbers • Decimal expansion of rational numbers are either terminating or non- terminating recurring • Decimal expansion of irrational numbers are non-terminating and non recurring • Co-prime numbers - numbers having no common factor other than one • Dividend = Divisor X Quotient + ramainder
- 3. NEW TERMS USED • Algorithm - A series of well defined steps which gives a procedure for solving a type of problem • Lemma : A proven statement used for proving another statement
- 4. CONTENT
- 5. EUCLID ‘S DIVISION LEMMA • Given positive integers a and b, there exist unique integers q and r satisfying • a = bq+ r where 0 ≤ r < b.
- 6. E UCLID’S DIVISION ALGORITHM •This is based on Euclid’s division lemma. According to this, the HCF of any two positive integers c and d, with c > d is obtained as follows: • Step 1: Apply the division lemma to candd.wecan find wholenumbers q and r suchthat c = d q + r , 0 ≤ r < d. • Step 2 : If r = 0, the HCF is d. If r ≠ 0, apply Euclid’s lemma to d and r. • Step 3 : Continue the process till the remainder is zero. The divisor at this stage will be HCF ( c,d).
- 7. FUNDAMENTAL THEOREM OF ARITHMETIC • Every composite number can be expressed as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur.
- 8. HOW TO FIND HCF AND LCM OF NUMBERS HCF • 1)Write the prime factorisation of the numbers • 2)List the common prime factors of the numbers • 3)Write the smallest power of each common prime factor • 4)Find the product of these LCM • 1)Write the prime factorisation of the numbers • 2)List all the factors involved in the numbers • 3)Write the greatest power of the prime factors • 4)Find the product of these
- 9. If a and b are two positive integers then HCF(a,b)X LCM(a,b) =a X b •If a and b are two co- prime positive integers then •HCF(a,b) = 1
- 10. SHOW THAT √ 2 IS IRRATIONAL. Let us assume the contrary that√2 is rational. • Then, √2 = a/b • where a and b are co-prime integersandbnot zero. ……………………… (1) • Or √2b = a or 2b² = a² It means 2 divides a² •and hence 2 divides a. ……………………….(2) •Let, a= 2c for some integer c. •then, 2b² = (2c)² = 4c² or b² = 2c² It means 2 divides b² • and hence 2 divides b. ……………………………(3) • From statements (2) and (3), it is clear that 2 is a common factor of a and b both. •But, from statement (1), p and q are co-prime which means that they cannot have any common • factor other than 1. • This indicates that our supposition that √2 is rational, is wrong. •Hence, √2 is irrational.
- 11. CONDITION FOR A RATIONAL NUMBER TO HAVE TERMINATING OR NON-TERMINATING REPEATING DECIMAL EXPANSION. • Terminating: A rational number p/q, where p and q are co- prime integers will have a terminating decimal expansion if the denominator is of the form 2ⁿ × 5m, where n and m are non- negative integers. •Non- Terminating: A rational number p/q, where p and q are co- prime integers will havea non- terminating repeating decimal expansion if the denominator is not of the form 2ⁿ × 5m, where n and m are non- negative integers.
- 12. THANK YOU