Express 4/7 in decimal form and state the kind of decimal expansion.
The Correct Answer and Explanation is:
There are two main types of decimal expansions for rational numbers:
- Terminating Decimals: These decimals end after a finite number of digits (e.g., 1/4 = 0.25). This occurs when the prime factors of the denominator are only 2s and/or 5s.
- Non-Terminating and Repeating (or Recurring) Decimals: These decimals continue infinitely but have a block of digits that repeats forever. This occurs when the denominator has prime factors other than 2 or 5.
Since the denominator of 4/7 is 7 (a prime number other than 2 or 5), its decimal expansion does not terminate. As shown by our division, it produces a repeating block of six digits. We use a bar (vinculum) over the repeating sequence to represent this:
0.571428‾0.571428. Therefore, 4/7 has a non-terminating and repeating decimal expansion.thumb_upthumb_down
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