Rational Or Irrational Numbers

Step 1: Define the sets and .

The even numbers less than 10 are . Thus:

The prime numbers less than 10 are . Thus:

Step 2: Find the intersection .

The intersection contains all elements that are common to both and . Comparing the elements of and :

The only common element is . Thus:

Step 3: Find .

The number of elements in is:

Final Answer: The value of is , which corresponds to option .

Step 1: Recall the definition of co-prime numbers.

Two numbers are co-prime if their only common factor is 1. In other words, their HCF is always 1.

Step 2: Match the definition to the options.

• (1) : Incorrect, as the HCF is not equal to .
• (2) : Incorrect, as the HCF is not equal to .
• (3) : Incorrect, as the HCF is not the product of and .
• (4) 1: Correct, as co-prime numbers have an HCF of 1.

Final Answer: The HCF of and is , which corresponds to option .

Step 1: Recall the associative property of addition.

The associative property states that for any real numbers , , and :

Step 2: Match the property to the options.

• (1) : This represents the commutative property of addition, not the associative property.
• (2) : This is the associative property of addition.
• (3) : This represents the distributive property, not the associative property.
• (4) None of the above: Incorrect, as option (2) matches the associative property.

Final Answer: The correct option is .

Step 1: Recall the condition for a rational number to have a terminating decimal expansion.

A rational number (in its simplest form) has a terminating decimal expansion if and only if the denominator has no prime factors other than 2 or 5.

Step 2: Analyze each option.

• (1) : The denominator is , which includes the prime factor 3. Hence, this does not have a terminating decimal expansion.
• (2) : The denominator is , which contains only the prime factors 2 and 5. Hence, this has a terminating decimal expansion.
• (3) : The denominator is , which includes the prime factor 7. Hence, this does not have a terminating decimal expansion.
• (4) : The denominator is , which includes the prime factor 3. Hence, this does not have a terminating decimal expansion.

Final Answer: The rational number with a terminating decimal expansion is , which corresponds to option .

Step 1: Recall the definition of an irrational number.

An irrational number is a number that cannot be expressed as a fraction , where and are integers and . Irrational numbers have non-terminating and non-repeating decimal expansions.

Step 2: Analyze each option.

• (1) 0.2: This is a terminating decimal, so it is a rational number.
• (2) : This is a mixed fraction, which can be written as . It is a rational number.
• (3) 1212121……: This decimal repeats the pattern "12" indefinitely. Repeating decimals are rational numbers.
• (4) is an irrational number.

We are asked to identify which of the following numbers is not an irrational number.

• is an irrational number.
• is an irrational number.
• is a rational number because it equals .
• is an irrational number.

Thus, the number which is not an irrational number is , which is a rational number.

The correct option is (C):

Let's analyze each statement:

Statement (1): The sum of a rational number and an irrational number is an irrational number.

This is correct. Proof: If rational + irrational = rational, then irrational = rational - rational = rational, which is a contradiction.

Statement (2): The sum of two irrational numbers need not be an irrational number.

This is correct. Example: , which is rational.

Statement (3): The product of a non-zero rational number and an irrational number is an irrational number.

This is correct. Proof: If rational × irrational = rational, then irrational = rational ÷ rational = rational, which is a contradiction.

Statement (4): The product of two irrational numbers is always an irrational number.

This is not correct. Counterexample: , which is rational.

Conclusion

The statement that is not correct is (4) The product of two irrational numbers is always an irrational number.

Step 1: Understand the concept of irrational numbers
An irrational number is a number that cannot be expressed as a ratio of two integers, meaning its decimal form is non-terminating and non-repeating. Examples include , etc. A rational number is a number that can be expressed as the ratio of two integers.
Step 2: Analyze each option
Option (1): is an irrational number because is irrational, and subtracting an irrational number from a rational number results in an irrational number.
Option (2): , which is a rational number. Hence, this is not irrational.
Option (3): is irrational, as the sum of two irrational numbers is usually irrational.
Option (4): is also irrational, as the difference of two irrational numbers is typically irrational.
Step 3: Conclusion
The only rational number is , so the correct answer is option .