Ch 04 - Another Peek Beyond the Point7th Ganita Prakash Part 2

• Concept: Decimals are the natural mathematical extension of the Indian place value system.
• Mechanism: They represent fractional parts where denominators are strict powers of 10 (e.g., tenths $\\frac{1}{10}$, hundredths $\\frac{1}{100}$, thousandths $\\frac{1}{1000}$).
• Anatomy: In 27.53, 5 represents strictly 5 tenths, and 3 represents strictly 3 hundredths.

• The Rule: Any number expressed as a simple fraction $p/q$ (where $p, q \\in \\mathbb{Z}$, and $q \\neq 0$) mathematically guarantees one of exactly TWO outcomes when subjected to long division.

1. Terminating Decimal: It ends cleanly with a remainder of 0 (e.g., 1/4 = 0.25).
2. Non-Terminating Repeating Decimal: The remainder cycles, causing an infinite repeating loop of digits (e.g., 10/3 = 3.333... or 1/7 = 0.142857...).

• The Absolute Trap: A standard integer fraction will never produce a non-terminating, non-repeating decimal sequence.

• Concept: Fractions where the denominator is strictly a power of 2 or a power of 5 will mathematically ALWAYS resolve into a clean, terminating decimal.
• The Pattern for $1/2^n$:
  $1/2 = 0.5$
  $1/4 = 0.25$
  $1/8 = 0.125$
• The Pattern for $1/5^n$:
  $1/5 = 0.2$
  $1/25 = 0.04$
  $1/125 = 0.008$
• Reason: The numbers 2 and 5 are the exclusive prime factors of base-10. Thus, any fraction composed entirely of 2s and 5s in the denominator perfectly expands into $10^k$.

Boundary Conditions / Domain:

• Condition holds for $n \\ge 0$, where $n \\in \\mathbb{Z}$.