Converting common fractions to infinite, periodic and non-periodic decimals
2Converting Fractions to Decimal Expansions
🔹 1. What is a decimal expansion?
The decimal expansion of a number is its representation as a number with a decimal point. It can be obtained by dividing the numerator by the denominator. The result can be:
finite expansion
recurring infinite expansion
non-recurring infinite expansion
Here we will focus on infinite decimal expansions.
🔸 2. Periodic infinite decimal expansion
✔️ Definition:
This is a decimal expansion in which, after a certain point, a cycle of digits begins to repeat, continuing indefinitely. This cycle is called a period.
✔️ Features:
The digits after the decimal point form a repeating sequence.
It can be written as a fraction.
Every periodic decimal expansion corresponds to a rational number.
🔸 3. Non-Periodic Infinite Decimal Expansion
✔️ Definition:
This is a decimal expansion that does not terminate and does not contain any repeating pattern.
✔️ Features:
The digits after the decimal point do not form a regular pattern.
Such an expansion cannot be written as a fraction.
It corresponds to an irrational number.
🔸 4. When does a fraction give a periodic expansion, and when does it give a non-periodic expansion?
Every common fraction (i.e., every rational number) has a decimal expansion:
finite – if the denominator (after reduction) contains only factors 2 and/or 5,
infinite periodic – if the denominator (after reduction) contains factors other than 2 and 5.
Irrational numbers (e.g., irrational roots, the number π) have infinite, non-periodic decimal expansions.