📘 Concept & Theory A rational number is any number that can be written in the form: ›
where:
- \(p\) and \(q\) are integers
- \(q \ne 0\)
The denominator cannot be zero because division by zero is not defined.
Zero will be called a rational number if it can be represented in the above form.
🗺️ Solution Roadmap Step-by-step Plan ›
- Recall the definition of a rational number.
- Try to express zero in the form \(\frac{p}{q}\).
- Verify that denominator is not equal to zero.
📊 Graph / Figure Graph / Figure ›
✏️ Solution Complete Solution ›
A rational number is any number that can be written in the form:
We know that zero can be written as:
Here:
- \(p = 0\)
- \(q = 1\)
Since \(1 \ne 0\), the condition for a rational number is satisfied.
Similarly, zero can also be written as:
In all these forms, the denominator is not zero.
🎯 Exam Significance Exam Significance ›
- This question builds the foundation of the Number System chapter.
- Very important for understanding rational and irrational numbers in higher classes.
- Frequently asked in school exams, MCQs, viva questions, and Olympiad foundation tests.
- Important for competitive examinations such as NTSE, Olympiads, SSC, Polytechnic Entrance, Railway Exams, NDA foundation level and other aptitude-based exams.
- Helps students avoid the common misconception that zero is neither positive nor negative, so it cannot be rational.
📘 Concept & Theory Rational numbers can always be found between any two rational numbers. ›
To find several rational numbers between two integers, we first convert the integers into fractions having the same denominator.
Then we choose numerators lying between them.
Since we need 6 rational numbers, we take a denominator greater than 6.
Multiplying numerator and denominator by the same non-zero number does not change the value of a rational number.
🗺️ Solution Roadmap Step-by-step Plan ›
- Choose a denominator greater than 6.
- Convert 3 and 4 into equivalent fractions.
- Identify fractions lying between them.
- Write the required six rational numbers.
📊 Graph / Figure Graph / Figure ›
✏️ Solution Complete Solution ›
We need to find 6 rational numbers between \(3\) and \(4\).
Since we need 6 numbers, we take a number greater than 6.
Let us take:
Now convert \(3\) and \(4\) into fractions having denominator \(7\).
We know:
Multiplying numerator and denominator by \(7\):
Similarly,
Therefore, we need six rational numbers between:
Fractions lying between them are obtained by increasing the numerator one by one.
🎯 Exam Significance Exam Significance ›
- This is one of the most important concepts from the chapter Number Systems.
- Questions based on finding rational numbers frequently appear in school exams, MCQs, viva tests, and assignments.
- Important for competitive exams such as NTSE, Olympiads, SSC foundation, Polytechnic Entrance Exams, Railway Exams, NDA foundation level and aptitude tests.
- Helps students understand the property: There are infinitely many rational numbers between any two rational numbers.
- This method is also used later in higher mathematics while studying decimal expansions and real numbers.
📘 Concept & Theory Between any two rational numbers, infinitely many rational numbers can be found. ›
To find a required number of rational numbers between two fractions:
- Convert the fractions into equivalent fractions having a larger common denominator.
- Then select numerators lying between them.
Since we need 5 rational numbers, we choose a multiplier greater than 5.
Multiplying numerator and denominator by the same non-zero number does not change the value of the fraction.
🗺️ Solution Roadmap Step-by-step Plan ›
- Choose a number greater than 5.
- Convert both fractions into equivalent fractions.
- Identify fractions lying between them.
- Write the required five rational numbers.
📊 Graph / Figure Graph / Figure ›
✏️ Solution Complete Solution ›
We need to find 5 rational numbers between:
Since we need 5 rational numbers, we choose a number greater than 5.
Let us take:
Now multiply numerator and denominator of both fractions by \(6\).
For \(\frac{3}{5}\):
For \(\frac{4}{5}\):
Therefore, we need five rational numbers between:
Increment the numerator one by one:
🎯 Exam Significance Exam Significance ›
- This problem develops a strong understanding of equivalent fractions and rational numbers.
- Frequently asked in school examinations, MCQs, viva voce, and worksheet-based assessments.
- Important for competitive examinations such as NTSE, Olympiads, SSC foundation, Polytechnic Entrance Exams, Railways, NDA foundation level and aptitude tests.
- Helps students understand that infinitely many rational numbers exist between any two rational numbers.
- This concept is later used in decimal expansion, real numbers, coordinate geometry, and algebra.
📘 Concept & Theory To solve True/False questions from Number Systems, we must clearly understand the relationship between: ›
- Natural Numbers
- Whole Numbers
- Integers
- Rational Numbers
Their definitions are:
| Type of Numbers | Definition | Examples |
|---|---|---|
| Natural Numbers | Counting numbers starting from 1 | \(1,2,3,4,\dots\) |
| Whole Numbers | Natural numbers together with 0 | \(0,1,2,3,\dots\) |
| Integers | Negative numbers, zero and positive numbers | \(\dots,-2,-1,0,1,2,\dots\) |
| Rational Numbers | Numbers of the form \(\frac{p}{q}\), \(q\ne0\) | \(\frac{1}{2}, -3, 0, \frac{7}{9}\) |
🗺️ Solution Roadmap Step-by-step Plan ›
- Recall the definition of each type of number.
- Check whether the statement satisfies the definition.
- Give a supporting example or counterexample.
📊 Graph / Figure Graph / Figure ›
✏️ Solution Complete Solution ›
-
Statement: Every natural number is a whole number.
Natural numbers are:
\[ 1,2,3,4,\dots \]Whole numbers are:
\[ 0,1,2,3,4,\dots \]Every natural number is included in the set of whole numbers.
Therefore, the statement is True. -
Statement: Every integer is a whole number.
Integers include negative numbers also.
\[ \dots,-3,-2,-1,0,1,2,3,\dots \]Whole numbers do not include negative numbers.
For example:
\[ -1 \]is an integer but not a whole number.
Therefore, the statement is False. -
Statement: Every rational number is a whole number.
Rational numbers are numbers of the form:
\[ \frac{p}{q}, \quad q \ne 0 \]Some rational numbers are fractions and are not whole numbers.
For example:
\[ \frac{1}{2} \]is a rational number but not a whole number.
Therefore, the statement is False.
🎯 Exam Significance Exam Significance ›
- These concepts form the foundation of the entire Number System chapter.
- Very important for school examinations, MCQs, viva questions, and assertion-reason questions.
- Frequently asked in NTSE, Olympiads, SSC foundation, Polytechnic Entrance Exams, Railway Exams and other aptitude-based tests.
- Helps students clearly differentiate between natural numbers, whole numbers, integers, and rational numbers.
- Strong conceptual understanding of number classification is essential for higher mathematics.