What is the 358th digit after the decimal point in the repeating decimal 0.3178?
Understand the Problem
The question is asking for the 358th digit after the decimal point in the repeating decimal 0.3178. To solve this, we need to identify the repeating pattern of the decimal and determine the position of the 358th digit within that pattern.
Answer
The 358th digit after the decimal point is $1$.
Answer for screen readers
The 358th digit after the decimal point in the repeating decimal 0.3178 is $1$.
Steps to Solve
- Identify the repeating pattern
The decimal 0.3178 has a repeating sequence of "3178". This sequence contains 4 digits.
- Find the position within the sequence
To determine the 358th digit, find the remainder when 358 is divided by the length of the repeating sequence (which is 4).
$$ 358 \mod 4 = 2 $$
- Determine the corresponding digit
The remainder tells us the position of the digit within the repeating sequence. If the remainder is 0, it refers to the last digit of the sequence. The sequence is indexed as follows:
- 1st digit: 3
- 2nd digit: 1
- 3rd digit: 7
- 4th digit: 8
Since the remainder is 2, the 358th digit corresponds to the 2nd digit of the sequence.
The 358th digit after the decimal point in the repeating decimal 0.3178 is $1$.
More Information
The repeating decimal is a common way to represent numbers that have a recurring pattern. Understanding how to find specific digits within repeating decimals is useful in various mathematical applications, such as number theory and fractions.
Tips
One common mistake is forgetting to use modulus correctly or miscalculating the remainder when finding which digit corresponds to the requested position. Always ensure you understand the modulus operation and how to apply it to the length of the repeating sequence.
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