Real Numbers Class 10 In Telugu | Haripriya Mam @VedantuTelugu8910
Vedantu Telugu 8,9 & 10・19 minutes read
The session covers important topics like division, the fundamental theorem of arithmetic, and rational numbers, emphasizing unique prime factorization and decimal expansions of rational numbers. Practical exercises include finding HCF and LCM, understanding co-prime numbers, and distinguishing between terminating and non-terminating decimals based on factors in the denominator.
Insights
- The fundamental theorem of arithmetic asserts that every positive integer has a unique prime factorization, allowing for the factorization of composite numbers into primes regardless of their order.
- Understanding the application of the theorem extends to practical scenarios like calculating the HCF and LCM of numbers, where the LCM involves listing prime factors with the highest powers while the HCF requires listing common prime factors with the least powers.
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Recent questions
What is the fundamental theorem of arithmetic?
Every positive integer has a unique prime factorization.
How are HCF and LCM calculated?
HCF involves common prime factors, LCM takes highest powers.
What are co-prime numbers?
Numbers with HCF of 1 and unique prime factorization.
How are rational numbers distinguished by decimal expansion?
Terminating decimals have factors of 2 and 5.
How are irrational numbers identified?
Through contradiction in assuming rationality.
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Summary
00:00
"NCRT Real Numbers: Division, Prime Factorization, Theorems"
- The session covers the first chapter of NCRT, "Real Numbers," focusing on important questions and topics like division, fundamental theorem of arithmetic, and revisiting irrational numbers.
- ID division involves the formula: dividend = divisor x quotient + remainder, with methods like factor tree and long division for prime factorization.
- The fundamental theorem of arithmetic states that every positive integer has a unique prime factorization, making it a crucial concept.
- The theorem emphasizes that every composite number can be factorized uniquely as a product of primes, regardless of the order in which the prime factors occur.
- A practical example involves finding the prime factorization of 225 through long division, showcasing the application of the fundamental theorem of arithmetic.
- The application of the theorem extends to finding the HCF and LCM of numbers, with specific steps for calculating both and understanding their relationship.
- The LCM is determined by listing prime factors and taking the highest powers, while the HCF involves listing common prime factors and taking the least powers.
- A practical exercise involves finding the minimum distance three persons should walk to cover the same distance in complete steps, using the concepts of HCF and LCM.
- The session also covers the concept of co-prime numbers, where the HCF is 1, and the product of HCF and LCM is equal to the product of two co-prime numbers.
- Additional examples include finding the HCF and LCM of numbers through prime factorization and understanding the decimal expansion of rational numbers, distinguishing between terminating and non-terminating decimals based on the denominator's factors.
32:56
Rational and Irrational Decimal Expansions Explained
- Rational numbers of the form 2 to the power M multiplied by 5 to the power N, where M and N are whole numbers, result in terminating or non-terminating decimal expansions.
- The decimal expansion of 43 by 162 is non-terminating and recurring due to not being of the form 2 to the power M multiplied by 5 to the power N.
- The number of decimal digits after the decimal point is equal to the number of twos or fives, depending on which power is greater in the denominator.
- The decimal expansion of 25 by 24 divided by 75 will have two digits after the decimal point.
- Root 2 and root 3 plus root 5 are proven to be irrational numbers through a contradiction in assuming they are rational.
