Rational and Irrational Numbers - Representation on the number line

Rational Numbers are numbers expressed in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. On a number line, a rational number like $\frac{3}{5}$ is represented by dividing the unit length between $0$ and $1$ into $5$ equal segments and selecting the $3^{rd}$ division point from the origin.

Irrational Numbers are real numbers that cannot be expressed as a simple fraction $\frac{p}{q}$. Their decimal expansions are non-terminating and non-recurring (e.g., $\sqrt{2} \approx 1.414...$ or $\pi \approx 3.1415...$). Visually, these points fill the 'gaps' left by rational numbers on the number line.

The Pythagoras Theorem ($h^2 = b^2 + p^2$) is the fundamental tool for representing square roots of non-perfect squares on the number line. For example, to represent $\sqrt{2}$, we visualize a right-angled triangle with a base of $1$ unit and a height of $1$ unit; the hypotenuse length will be exactly $\sqrt{2}$.

To represent $\sqrt{n}$ on a number line (where $n$ is a positive integer), we first locate $\sqrt{n-1}$. By constructing a perpendicular line segment of unit length ($1$) at the point $\sqrt{n-1}$, the distance from the origin to the top of this perpendicular becomes $\sqrt{n}$ units. This creates a 'Square Root Spiral' effect when repeated.

Geometric Construction of $\sqrt{x}$ for any positive real number $x$: Draw a line segment $AB = x$ units and extend it to $C$ such that $BC = 1$ unit. Find the midpoint $O$ of $AC$ and draw a semi-circle with radius $OA$. A perpendicular line drawn from $B$ to the semi-circle at point $D$ will have a length $BD = \sqrt{x}$. This length can then be marked on the number line using a compass.

Successive Magnification is a visual method to represent terminating or non-terminating recurring decimals. It involves looking through a 'magnifying glass' at the number line. To find $2.665$, we first look between $2$ and $3$, then zoom into the interval $2.6$ and $2.7$, and further zoom into $2.66$ and $2.67$ until $3$ decimal places are reached.

The Real Number Line Property states that every point on the number line represents a unique real number (either rational or irrational), and conversely, every real number can be represented by a unique point on the line. This establishes a one-to-one correspondence between real numbers and points on a geometric line.