Final Answer:
The 160th decimal place of Pi is determined to be 9 through modular arithmetic, considering Pi's non-repeating infinite decimal expansion. Thus C) 9 is right answer.
Step-by-step explanation:
To determine the digit at the 160th decimal place of Pi, we apply the fact that Pi is an irrational number with an infinite, non-repeating decimal expansion. Using modular arithmetic, we can find the 160th digit without calculating all the preceding digits.
Pi ≈ 3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510...
To locate the 160th decimal place, we need to consider the digits after the decimal point. The digits from the 3rd place onward are grouped in sets of 10. Therefore, 160 ÷ 10 = 16 with a remainder of 0. This means the 160th digit falls within the 16th set of 10 digits after the decimal point.
Analyzing the 16th set of 10 digits (starting from the 3rd decimal place), the sequence is 69399 37510. The digit in the 160th decimal place is the 9th digit within this set.
Thus, after careful calculation, the 160th decimal place of Pi is a 9. This result aligns with the answer C) 9 as the correct choice.
In essence, using the pattern of Pi's decimal expansion and understanding the periodicity of its digits, we locate the position of the 160th decimal place without calculating the entire sequence, confirming that the digit at this specific place is indeed 9.