Number Systems - Introduction to Number Systems

Natural Numbers and Whole Numbers: Natural numbers ($N$) are counting numbers starting from $1$ ($1, 2, 3, \dots$). Whole numbers ($W$) include zero along with all natural numbers ($0, 1, 2, 3, \dots$). On a number line, these are visualized as discrete points starting from $0$ or $1$ and extending infinitely to the right.

Integers: Integers ($Z$) comprise all whole numbers and their negative counterparts ($\dots, -3, -2, -1, 0, 1, 2, 3, \dots$). Visually, the number line extends in both directions from the origin ($0$), with positive integers to the right and negative integers to the left.

Rational Numbers: A number is rational if it can be written in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. This set includes integers, terminating decimals (like $0.5 = \frac{1}{2}$), and non-terminating repeating decimals (like $0.333\dots = \frac{1}{3}$). On a number line, rational numbers are densely packed between any two integers.

Irrational Numbers: These are numbers that cannot be expressed in the form $\frac{p}{q}$. Their decimal expansions are non-terminating and non-repeating. Examples include $\sqrt{2}$, $\sqrt{3}$, and $\pi$. Visually, they occupy specific positions on the number line that 'fill the gaps' left by rational numbers.

Real Numbers: The collection of all rational and irrational numbers together forms the set of Real numbers ($R$). Every real number is represented by a unique point on the number line, and every point on the number line represents a unique real number. This continuous line is often called the Real Number Line.

Decimal Expansion Characteristics: Rational numbers have decimal expansions that either terminate (e.g., $\frac{7}{8} = 0.875$) or are non-terminating repeating (e.g., $\frac{10}{3} = 3.333\dots$). Irrational numbers always have non-terminating, non-repeating decimal expansions (e.g., $\sqrt{2} = 1.4142135\dots$).

Density Property: Between any two given rational numbers, there are infinitely many rational numbers. This can be visualized by repeatedly zooming into any segment of the number line; no matter how small the interval, more numbers can always be found using the average method or the common denominator method.