Number Systems - Real Numbers and their Decimal Expansions

Real Numbers and the Number Line: Real numbers are the set containing all rational and irrational numbers. Visually, every real number corresponds to a unique point on a continuous horizontal number line. Rational numbers occupy specific positions, while irrational numbers like $\sqrt{2}$ fill the 'gaps' between them, ensuring the line is entirely covered without any empty spaces.

Terminating Decimal Expansion: A rational number $\frac{p}{q}$ has a terminating decimal expansion if the prime factorization of the denominator $q$ (when the fraction is in its simplest form) consists only of the factors $2$ and $5$ (i.e., $q = 2^n 5^m$). Visually, this means the long division process finishes with a remainder of zero after a finite number of decimal places.

Non-terminating Repeating (Recurring) Decimals: Rational numbers that do not terminate will have a block of digits that repeat infinitely in a specific pattern. For example, $\frac{1}{3} = 0.333...$, which is represented by a bar over the repeating digit as $0.\overline{3}$. On the number line, these are represented by points that can be identified by their repeating fractional nature.

Irrational Numbers and Decimal Expansion: Numbers whose decimal expansion is non-terminating and non-recurring are called irrational numbers. Examples include $\pi \approx 3.14159...$ and $\sqrt{2} \approx 1.4142...$. Unlike rational numbers, their digits never settle into a repeating cycle. Visually, they represent points on the number line that cannot be measured as a simple ratio of two whole lengths.

Identifying Irrationality from Decimal Patterns: A decimal that follows a non-repeating but structured pattern, such as $0.101101110...$, is irrational. Even if the digits follow a logical rule (increasing the number of 1s), the lack of a constant repeating block (periodicity) means the number cannot be expressed as a fraction $\frac{p}{q}$.

Successive Magnification: This is a visual technique used to locate a real number on the number line by repeatedly 'zooming in' on smaller sub-intervals. Imagine the segment between $2$ and $3$ divided into $10$ equal parts; to find $2.665$, you first focus on the segment between $2.6$ and $2.7$, then divide that into $10$ parts to find the segment between $2.66$ and $2.67$, repeating until the specific point is reached.