Q1
State whether the following statements are true or false. Justify your answers.
- Every irrational number is a real number. True, since collection of real numbers is made up of rational and irrational numbers.
- Every point on the number line is of the form \(\sqrt{m}\) , where m is a natural number. False, no negative number can be the square root of any natural number.
- Every real number is an irrational number. False, for example 2 is real but not irrational.
📘 Concept & Theory ›
- Real Numbers: The collection of all rational and irrational numbers together forms the set of real numbers.
- Rational Numbers: Numbers which can be written in the form \[ \frac{p}{q} \] where \(p\) and \(q\) are integers and \(q \neq 0\).
- Irrational Numbers: Numbers which cannot be written in the form \[ \frac{p}{q} \] are called irrational numbers.
- Examples of irrational numbers: \[ \sqrt{2}, \sqrt{3}, \pi \]
- Every irrational number belongs to the set of real numbers.
- Every real number is not necessarily irrational because rational numbers are also real numbers.
- Square roots of natural numbers are not always integers. Example: \[ \sqrt{2}, \sqrt{5} \] are irrational.
🗺️ Solution Roadmap Step-by-step Plan ›
- Identify the type of numbers mentioned in the statement.
- Recall definitions of rational, irrational, and real numbers.
- Check whether the statement follows the definition correctly.
- If the statement is false, provide a counter-example.
- Write proper mathematical justification step-by-step.
✏️ Solution Complete Solution ›
1. State whether the following statements are true or false. Justify your answers.
-
Statement: Every irrational number is a real number.
Solution:We know that the set of real numbers consists of:
- Rational numbers
- Irrational numbers
Hence, the statement is True. -
Statement:
Every point on the number line is of the form
\[
\sqrt{m}
\]
where \(m\) is a natural number.
Solution:We know that:
- \[ \sqrt{m} \] where \(m\) is a natural number, is always non-negative.
- But the number line also contains negative numbers such as: \[ -1,\ -2,\ -5 \]
No negative number can be written in the form: \[ \sqrt{m} \] where \(m\) is a natural number.Therefore, the given statement is False.Counter Example: \[ -2 \] is on the number line but cannot be expressed as \[ \sqrt{m} \] for any natural number \(m\). -
Statement: Every real number is an irrational number.
Solution:We know that real numbers include:
- Rational numbers
- Irrational numbers
Therefore, all real numbers are not irrational.For example: \[ 2 \] is a real number.Also, \[ 2 = \frac{2}{1} \]Since it can be written in the form \[ \frac{p}{q} \] where \(q \neq 0\), it is a rational number.Hence, the statement is False.
🎯 Exam Significance Exam Significance ›
Importance for Board Exams and Competitive Exams
Board Examination Importance
- Conceptual true/false questions are frequently asked in CBSE exams.
- Definitions of rational and irrational numbers are fundamental.
- Improves mathematical reasoning and justification writing.
- Helpful for proving statements in higher mathematics.
Competitive Exam Importance
- Asked in Olympiads and NTSE foundation examinations.
- Forms the base for algebra and coordinate geometry.
- Improves logical elimination techniques in MCQs.
- Important for future preparation of JEE and other entrance exams.