Problem 97 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Let \(f(x)=1\), if \(x\) is a rational number \(=0\), if \(x\) is a irrational number. Find:- i. \(f\left(\frac{1}{3}\right) ;\\{\) Ans. 1\(\\}\) ii. \(f(\sqrt{7}) ;\\{\) Ans. 0\(\\}\) iii. \(f\left(\frac{22}{7}\right) ;\\{\) Ans. 1\(\\}\) iv. \(f(\pi) ;\\{\) Ans. 0\(\\}\) v. \(f(e) ;\\{\) Ans. 0\(\\}\) vi. \(f(f(1.4327)) ;\\{\) Ans. 1\(\\}\) \text { vii. } f(f(\sqrt{3})) \text { . }

Step-by-Step Solution

Answer

Short Answer: i. \(f\left(\frac{1}{3}\right) = 1\) ii. \(f(\sqrt{7}) = 0\) iii. \(f\left(\frac{22}{7}\right) = 1\) iv. \(f(\pi) = 0\) v. \(f(e) = 0\) vi. \(f(f(1.4327)) = 1\) vii. \(f(f(\sqrt{3})) = 1\)

Step 1: i. Evaluate \(f\left(\frac{1}{3}\right)\)

Since \(\frac{1}{3}\) is a rational number, we can apply the function definition and determine that \(f\left(\frac{1}{3}\right) = 1\).

Step 2: ii. Evaluate \(f(\sqrt{7})\)

Since \(\sqrt{7}\) is an irrational number, we can apply the function definition and determine that \(f(\sqrt{7}) = 0\).

Step 3: iii. Evaluate \(f\left(\frac{22}{7}\right)\)

Since \(\frac{22}{7}\) is a rational number, we can apply the function definition and determine that \(f\left(\frac{22}{7}\right) = 1\).

Step 4: iv. Evaluate \(f(\pi)\)

Since \(\pi\) is an irrational number, we can apply the function definition and determine that \(f(\pi) = 0\).

Step 5: v. Evaluate \(f(e)\)

Since \(e\) (Euler's number) is an irrational number, we can apply the function definition and determine that \(f(e) = 0\).

Step 6: vi. Evaluate \(f(f(1.4327))\)

First, we need to determine the value of \(f(1.4327)\). Since 1.4327 is a rational number, we can apply the function definition and find that \(f(1.4327)=1\). Now, we can evaluate \(f(f(1.4327)) = f(1)\). Since 1 is a rational number, we can apply the function definition again and find that \(f(f(1.4327)) = f(1) = 1\).

Step 7: vii. Evaluate \(f(f(\sqrt{3}))\)

First, we need to determine the value of \(f(\sqrt{3})\). Since \(\sqrt{3}\) is an irrational number, we can apply the function definition and find that \(f(\sqrt{3}) = 0\). Now, we can evaluate \(f(f(\sqrt{3})) = f(0)\). Since 0 is a rational number, we can apply the function definition again and find that \(f(f(\sqrt{3})) = f(0) = 1\).

Key Concepts

Rational Numbers

Rational numbers are those numbers that can be expressed as a fraction of two integers, where the denominator is not zero. They have a distinct \/ repeating decimal form. Examples include \( \frac{1}{2} \), \( -3 \), and \( 0.75 \). This set of numbers includes integers, fractions, and finite decimals. When we talk about a number like \( \frac{1}{3} \), it's a rational number because it can be written as a simple fraction.
Rational numbers are crucial to understand because they help us see patterns and relationships between quantities in mathematical problems.
In the Dirichlet function, rational numbers lead to an output of 1, demonstrating their unique role in function evaluation.

Irrational Numbers

Irrational numbers are numbers that cannot be expressed as a simple fraction. These numbers have a non-terminating, non-repeating decimal form. Examples include \( \pi \) and \( \sqrt{2} \).
In the exercise above, numbers like \( \pi \), \( \sqrt{7} \), and \( e \) demonstrate their standing as irrational numbers. In the Dirichlet function's context, an irrational number will lead to an output of 0.
This characteristic highlights the distinct gap between rational numbers and their irrational counterparts, which plays a crucial role in real number analysis and mathematical logic.

Function Evaluation

Function evaluation in mathematics refers to the process of substituting a value into a function to determine the output. For the Dirichlet function in the exercise, we substitute a number \( x \) to find \( f(x) \).
Depending on whether \( x \) is rational or irrational, the Dirichlet function will yield a result of 1 or 0.

• If \( x \) is rational, \( f(x)=1 \).
• If \( x \) is irrational, \( f(x)=0 \).

This method of evaluation allows us to apply the rules of the function effectively, ensuring that we derive the correct solution for any input provided.

Mathematical Logic

Mathematical logic is a field of mathematics that examines formal systems in relation to the way they influence mathematical reasoning. In the context of the Dirichlet function, the logic involves determining the rationality or irrationality of a number.
By applying mathematical logic, we can evaluate function consistency, make comparisons, and draw conclusions. For example, each evaluated term in the exercise uses logical reasoning to deduce whether a number is rational or irrational. This step-by-step deduction emphasizes the understanding of different number properties and allows accurate function evaluation. Mathematical logic also enables us to track consistency systematically, particularly when nested functions are present, ensuring accurate outcomes.