Number Systems - Irrational Numbers

Definition of Irrational Numbers: An irrational number is a number that cannot be expressed in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. These numbers form a set that, together with rational numbers, makes up the Real Number system.

Decimal Expansion: Unlike rational numbers, the decimal expansion of an irrational number is non-terminating and non-recurring. For example, $\pi \approx 3.14159265...$ and $\sqrt{2} \approx 1.41421356...$ continue infinitely without any repeating pattern of digits.

Square Roots of Non-Perfect Squares: If $n$ is a positive integer that is not a perfect square, then $\sqrt{n}$ is always an irrational number. For instance, $\sqrt{2}, \sqrt{3}, \sqrt{5}, \text{ and } \sqrt{7}$ are all irrational because their radicands are not squares of integers.

Representing $\sqrt{n}$ on the Number Line: We can locate irrational numbers on a number line using the Pythagoras Theorem. To represent $\sqrt{2}$, visualize a right-angled triangle with a base of $1$ unit and a height of $1$ unit. The hypotenuse, according to the theorem, is $\sqrt{1^2 + 1^2} = \sqrt{2}$. By placing a compass at the origin and drawing an arc with the hypotenuse as the radius, the point where the arc meets the number line is the exact location of $\sqrt{2}$.

Properties of Operations: The sum or difference of a rational number and an irrational number is always irrational. Similarly, the product or quotient of a non-zero rational number and an irrational number is irrational. However, the sum, difference, product, or quotient of two irrational numbers may or may not be irrational.

Rationalizing the Denominator: This is the process of removing a radical (square root) from the denominator of a fraction. To rationalize $\frac{1}{\sqrt{a}}$, we multiply both the numerator and denominator by $\sqrt{a}$. For expressions like $\frac{1}{a + \sqrt{b}}$, we multiply by the conjugate $a - \sqrt{b}$.

The Square Root Spiral: This is a geometric construction where right-angled triangles are built consecutively. Starting with a triangle of legs $1$ and $1$ (hypotenuse $\sqrt{2}$), a second triangle is built using the $\sqrt{2}$ side as a base and a new perpendicular side of $1$ unit. This creates a hypotenuse of $\sqrt{3}$. Continuing this creates a visual spiral of lengths $\sqrt{2}, \sqrt{3}, \sqrt{4}, \sqrt{5}, \dots$ representing the square roots of consecutive natural numbers.