Rational and Irrational Numbers - Surds and rationalization

Rational Numbers are numbers that can be expressed in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. Visually, these include all integers, terminating decimals (like $0.5$), and repeating decimals (like $0.333...$) placed precisely on a number line.

Irrational Numbers are numbers that cannot be expressed as a simple fraction $\frac{p}{q}$. Their decimal expansions are non-terminating and non-recurring. On a number line, they fill the gaps between rational numbers, such as $\sqrt{2}$ or $\pi$. Geometrically, $\sqrt{2}$ represents the diagonal of a square with side length 1.

A Surd is an irrational root of a positive rational number. For example, $\sqrt{3}$ is a surd because 3 is rational but its square root is irrational. However, $\sqrt{4}$ is not a surd because it simplifies to the rational number 2. Visualizing a surd $\sqrt{x}$ involves finding a side of a square whose area is exactly $x$ units.

Pure and Mixed Surds: A surd consisting entirely of an irrational number, like $\sqrt{50}$, is a pure surd. A mixed surd has a rational coefficient, like $5\sqrt{2}$. You can convert a pure surd to mixed by factoring out the largest perfect square (e.g., $\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}$).

Like and Unlike Surds: Like surds have the same irrational factor (e.g., $3\sqrt{7}$ and $2\sqrt{7}$). Only like surds can be added or subtracted, similar to how like terms $3x$ and $2x$ are combined in algebra. Unlike surds, such as $\sqrt{2}$ and $\sqrt{3}$, cannot be combined into a single radical term.

Rationalization is the process of eliminating a radical from the denominator of a fraction. If the denominator is a monomial surd like $\sqrt{a}$, we multiply both numerator and denominator by $\sqrt{a}$. If the denominator is a binomial like $a + \sqrt{b}$, we multiply by its conjugate $a - \sqrt{b}$.

The Conjugate of a binomial surd $a + \sqrt{b}$ is $a - \sqrt{b}$. When these two are multiplied, the result is always a rational number ($a^2 - b$). This property is the foundation for rationalizing denominators containing two terms.

Geometric Representation: Irrational numbers like $\sqrt{x}$ can be represented on a number line using the Pythagorean theorem. To represent $\sqrt{2}$, construct a right-angled triangle with base and height of 1 unit. The hypotenuse will be $\sqrt{2}$, which can then be projected onto the number line using a compass arc.