Q1(ii):

State whether the following statements are true or false. Justify your answers. (ii) Every point on the number line is of the form $\sqrt{m}$, where $m$ is a natural number.

Step 1: Defining the Mathematical Sets

To evaluate the statement, we must first define the mathematical entities involved:

• The Number Line: Represents the set of all real numbers ($\mathbb{R}$). This continuous line includes positive numbers ($\mathbb{R}^+$), negative numbers ($\mathbb{R}^-$), and zero ($0$).
• Natural Numbers ($\mathbb{N}$): The set of positive integers, defined as $\mathbb{N} = \{1, 2, 3, 4, \dots\}$.

Step 2: Analyzing the Properties of $\sqrt{m}$

The expression $\sqrt{m}$ denotes the principal (non-negative) square root of a natural number $m$. [By the definition of the principal square root function over real numbers], the output of $\sqrt{m}$ is strictly a positive real number for any $m \in \mathbb{N}$.

Mathematically, if $m \in \mathbb{N}$, then $\sqrt{m} > 0$.

Step 3: Evaluating the Composition of the Real Number Line

The real number line extends infinitely in both the positive and negative directions. Therefore, it contains infinitely many negative real numbers (e.g., $-1, -2, -3.5, -\pi$).

Step 4: Establishing the Logical Contradiction

The given statement asserts that every point on the number line can be mapped to the form $\sqrt{m}$. To falsify a universal statement, we only need to provide a single counter-example [Per the rules of formal logic and proof by contradiction].

Consider the point $-2$ on the number line. Because the principal square root of any natural number is always positive, there exists no natural number $m$ such that:

$\sqrt{m} = -2$

Furthermore, the square of any real number is non-negative, meaning the square root of a positive integer can never yield a negative value in the real number system. Therefore, no negative number on the number line can be expressed in the form $\sqrt{m}$.

Note: Additionally, there are infinitely many positive real numbers that cannot be expressed as the square root of a natural number. For example, the point $1.5$ is on the number line, but $1.5 = \sqrt{2.25}$, and $2.25 \notin \mathbb{N}$.

Step 5: Visual Proof via the Real Number Line

The following diagram maps the exact coordinates of natural number roots ($\sqrt{1}, \sqrt{2}, \sqrt{3}, \sqrt{4}$) and highlights the negative domain, which entirely escapes the form $\sqrt{m}$.

Final Solution: False. The number line contains negative numbers, and no negative number can be expressed as the principal square root of a natural number ($\sqrt{m}$).