14. Assertion (A): V2 is an irrational number. Reason (R) : Decimal expansion of an irrational number is non-recurring and non terminating???? a) Both (A) and (R) are true and Reason (R) is the correct explanation of Assertion (A) b) Both (A) and ( R) are true but Reason (R) is not a correct explanation of (A) c) Assertion (A) is true and Reason (R) is false d) Assertion (A) is false and Reason (R) is true
Answers
Answer:
Integers are the complete numbers, i.e. they are free of any decimal parts.
e.g. ...,−2,−1,0,1,2,...
Rational numbers are the numbers which can be expressed as a ratio of two integers. i.e. they can be written in the form of pq,(p,q)∈Z,q≠0.
Irrational numbers are the numbers which cannot be expressed as a ratio of two integers.
e.g. 2–√=1.4142135...,π=3.141592...,e=2.71828...,log5=0.69897...
Complete step by step answer:
Let us examine both the statements one by one:
Assertion: 2 is a rational number.: CORRECT, because we can write 2 as 21, where both 2 and 1 are integers.
Reason: The square roots of all positive integers are irrationals.: INCORRECT, because square roots of perfect square positive integers are themselves integers, not irrational. e.g. 4–√=2.
Since the reason is incorrect, the question of it being a correct explanation does not arise at all.
The correct answer is C. Assertion is correct but Reason is incorrect.
Note: Rational numbers are either terminating or non-terminating and repeating in decimal form.
e.g. 3.4, 5.7777... = 5.7¯, 8.13 etc.
Irrational numbers are neither terminating nor repeating in decimal form.
e.g. 2–√=1.4142135...,π=3.141592...,e=2.71828...,log5=0.69897...
An even root of a negative number is a complex number. The complex unit is defined as
Both (A) and (R) are true and Reason (R) is the correct explanation of Assertion (A).
Assertion: Yes, √2 is an irrational number.
Reason: The decimal expansion of √2 is 1.41421356237 which is a non-recurring and non-terminating number.