Number - Real number systems (rational and irrational numbers)

Classification of Real Numbers ($\mathbb{R}$): The real number system is the set of all possible points on a continuous number line. It is divided into two main, mutually exclusive groups: Rational numbers ($\mathbb{Q}$) and Irrational numbers ($\mathbb{I}$). You can visualize this as a large box (Real Numbers) containing two separate compartments that do not overlap.

Rational Numbers ($\mathbb{Q}$): A rational number is any number that can be expressed in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. In decimal form, these numbers either terminate (e.g., $0.75 = \frac{3}{4}$) or repeat in a specific pattern (e.g., $0.333... = \frac{1}{3}$ or $0.142857142857... = \frac{1}{7}$).

Irrational Numbers ($\mathbb{I}$): These numbers cannot be written as a simple fraction of two integers. Their decimal expansions are non-terminating and non-repeating. Common examples include $\pi$, $e$, and square roots of non-perfect squares like $\sqrt{2}$ or $\sqrt{3}$. Visually, if you try to plot these on a number line, they occupy the 'gaps' between rational numbers.

The Set Hierarchy (Concentric Circles): The number system is nested. Natural numbers ($\mathbb{N} = \{1, 2, 3, ...\}$) are inside Whole numbers (including $0$), which are inside Integers ($\mathbb{Z} = \{..., -2, -1, 0, 1, 2, ...\}$), which are inside Rational numbers ($\mathbb{Q}$). All of these, along with Irrational numbers, sit inside the set of Real numbers ($\mathbb{R}$). Think of this as a Venn diagram with $\mathbb{N}$ at the center and $\mathbb{R}$ as the outermost circle.

Density Property: Between any two distinct rational numbers, there is always another rational number, and also an irrational number. This means the number line is 'dense'. You can visualize this by zooming into the segment between $0$ and $1$; no matter how much you zoom, you can always find a midpoint like $\frac{1}{2}$, then $\frac{1}{4}$, and so on.

Surds and Radicals: A surd is an irrational number expressed using a root sign. While $\sqrt{9} = 3$ is a rational integer, $\sqrt{5}$ is a surd because its value is approximately $2.23606...$ and never repeats or ends. On a right-angled triangle with sides of $1$ unit, the hypotenuse is $\sqrt{2}$, which is a classic visual proof of an irrational length existing in geometry.

Converting Recurring Decimals: Recurring decimals (marked with a bar, like $0.\overline{5}$) are always rational. To visualize the conversion, we use algebra to 'shift' the repeating part across the decimal point and subtract it away, leaving a clean fraction.