Question

Every point on a number line represents :

• A. An Irrational number
• B. A Rational number
• C. A Unique real number
• D. A Natural number

Hint:

The number line is a complete representation of all {real numbers}. Real numbers include:
Rational numbers (integers, fractions, terminating/repeating decimals)
Irrational numbers (non-terminating, non-repeating decimals like \(\pi, \sqrt{2}\)) Each point on the line maps to one specific real number, and vice versa.

Answer 1

The Correct Option is C

Concept: The number line is a visual representation of numbers. The types of numbers that can be represented on it are important.

Step 1: Understanding the Number Line A number line is a straight line with points that are used to represent numbers. Typically, zero is placed at the center, positive numbers extend to the right, and negative numbers extend to the left.

Step 2: Types of Numbers 
Natural Numbers: \(1, 2, 3, \ldots\) 
Whole Numbers: \(0, 1, 2, 3, \ldots\) 
Integers: \(\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots\) 
Rational Numbers: Numbers that can be expressed as a fraction \(p/q\), where \(p\) and \(q\) are integers and \(q \neq 0\). This includes integers, terminating decimals, and repeating decimals. Examples: \(1/2, -3/4, 5, 0.25\). 
Irrational Numbers: Numbers that cannot be expressed as a simple fraction \(p/q\). Their decimal representations are non-terminating and non-repeating. Examples: \(\pi, \sqrt{2}, e\). 
Real Numbers: The set of all rational numbers and all irrational numbers combined. Real numbers fill the entire number line.

Step 3: What each point on the number line represents There is a one-to-one correspondence between the set of real numbers and the points on a number line. This means: % Option (A) Every real number corresponds to exactly one unique point on the number line. % Option (B) Every point on the number line corresponds to exactly one unique real number. The number line includes points for integers, fractions (rational numbers), and numbers like \(\sqrt{2}\) or \(\pi\) (irrational numbers).

Step 4: Analyzing the options 
(1) An Irrational number: Incorrect. Points like 2 or 0.5 are on the number line but are rational. 
(2) A Rational number: Incorrect. Points like \(\sqrt{2}\) or \(\pi\) are on the number line but are irrational. 
(3) A Unique real number: Correct. Every point corresponds to one specific real number (which could be rational or irrational), and this real number is unique to that point. 
(4) A Natural number: Incorrect. Points like -1, 0.5, or \(\sqrt{2}\) are on the number line but are not natural numbers. Therefore, every point on a number line represents a unique real number.