1. What Are Repeating Decimals?

A repeating decimal is a decimal number that after some point continues infinitely with a repeating pattern of digits. For example, 1/3 = 0.333... is represented as 0.(3) where the digit 3 repeats infinitely.

2. How Does the Calculator Work?

The calculator uses long division to find the decimal representation of a fraction:

The algorithm tracks remainders during division to identify repeating patterns:

• Divide numerator by denominator using long division
• Track remainders at each step
• If a remainder repeats, the decimal starts repeating from that point
• The repeating portion is marked with parentheses

3. Importance of Repeating Decimals

Details: Understanding repeating decimals is fundamental in mathematics, helping with exact representations of fractions, number theory concepts, and avoiding rounding errors in calculations.

4. Using the Calculator

Tips: Enter any positive integers for numerator and denominator. The calculator will show either:

• A terminating decimal (e.g., 1/2 = 0.5)
• A repeating decimal with the repeating portion in parentheses (e.g., 1/3 = 0.(3))

5. Frequently Asked Questions (FAQ)

Q1: What makes a fraction have a repeating decimal?
A: A fraction has a repeating decimal if its denominator (in simplest form) has prime factors other than 2 or 5.

Q2: How can I tell if a decimal will terminate or repeat?
A: If the denominator's prime factors (in simplest form) are only 2 and/or 5, it terminates. Otherwise, it repeats.

Q3: What's the longest possible repeating sequence?
A: For denominator d, the maximum length is d-1 digits (when d is prime).

Q4: Can all repeating decimals be converted to fractions?
A: Yes, all repeating decimals are rational numbers and can be expressed as fractions.

Q5: How are repeating decimals used in real life?
A: They're used in precise calculations, computer science, cryptography, and whenever exact fractional values are needed.