Real Numbers: Introduction and Properties

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What is the significance of Euclid’s division algorithm?

It helps in computing the HCF of two positive integers and has applications related to the divisibility properties of integers.

What is the essence of the Fundamental Theorem of Arithmetic?

It states that every composite number can be expressed as a product of primes in a unique way.

What properties of positive integers are discussed in Sections 1.2 and 1.3?

Euclid’s division algorithm and the Fundamental Theorem of Arithmetic.

What are the main applications of the Fundamental Theorem of Arithmetic?

It has deep and significant applications in the field of mathematics. Signup and view all the answers

What did you learn about real numbers in Class IX?

Encountered irrational numbers. Signup and view all the answers

What are the two main applications of the Fundamental Theorem of Arithmetic?

Proving the irrationality of numbers and exploring the nature of the decimal expansion of rational numbers Signup and view all the answers

Can any natural number be obtained by multiplying prime numbers?

Yes Signup and view all the answers

What is the prime factorization of 32760?

2^3 × 3^2 × 5 × 7 × 13 Signup and view all the answers

What is the prime factorization of 123456789?

3^2 × 3803 × 3607 Signup and view all the answers

Are there infinitely many primes?

Yes Signup and view all the answers

What process maximizes the number of successful variations?

Sexual reproduction Signup and view all the answers

What is the mechanism studied in this chapter?

The mechanism by which variations are created and inherited Signup and view all the answers

In Figure 8.1, what does each of the four individuals in the bottom row represent?

Different individuals in the next generation Signup and view all the answers

What provides both a common basic body design and subtle changes for the next generation?

Inheritance from the previous generation Signup and view all the answers

What gives rise to new individuals that are similar but subtly different?

Reproductive processes Signup and view all the answers

In a population of an asexually reproducing species, if a trait A exists in 10% of the population and a trait B exists in 60% of the same population, which trait is likely to have arisen earlier?

Trait B is likely to have arisen earlier. Signup and view all the answers

How does the creation of variations in a species promote survival?

The creation of variations in a species promotes survival by increasing the likelihood of some individuals having advantages that are beneficial for their survival in specific environmental conditions. Signup and view all the answers

What is the significance of Mendel's contributions to the rules for inheritance of traits?

Mendel's contributions provided foundational rules for the inheritance of traits, establishing the basis for understanding genetic inheritance and variability in offspring. Signup and view all the answers

How do environmental factors contribute to the selection of variants in a species?

Environmental factors contribute to the selection of variants in a species by favoring individuals with traits that are advantageous in specific environmental conditions, leading to the survival and proliferation of those traits. Signup and view all the answers

How do the rules of heredity affect the generation of individuals with variations?

The rules of heredity govern the transmission of genetic traits from parents to offspring, leading to the generation of individuals with variations that contribute to the overall genetic diversity within a population. Signup and view all the answers