What is the 300th Digit of 0.0588235294117647?

If you’re here searching for the answer to the question, “What is the 300th digit of 0.0588235294117647?”, you’re not alone. This curious-sounding mathematical query delves into the intriguing world of repeating decimals and rational numbers. In this article, we’ll break down the origin of 0.0588235294117647, explain recurring decimals, and unveil how to find the 300th digit with ease and precision.

Understanding Decimal Numbers

Before addressing the 300th digit directly, it’s essential to grasp what decimal numbers are and why some of them seem to stretch infinitely—either repeating or terminating.

Types of Decimal Numbers

• Terminating Decimals: These decimals stop after a certain number of digits. Example: 0.5, 0.125
• Non-Terminating, Repeating Decimals: These go on forever but with a predictable repeating pattern. Example: 0.333… (repeats 3).
• Non-Terminating, Non-Repeating Decimals: These go on infinitely without a predictable pattern, such as irrational numbers like π (pi) or √2.

The number 0.0588235294117647 is particularly interesting because at first glance it appears to stop, but actually represents a rational number with a repeating decimal pattern. Let’s explore this further.

Origin of 0.0588235294117647

You might be wondering: where does this number come from? The answer is quite fascinating—it is the decimal representation of 1 divided by 17. That’s right:

1 ÷ 17 = 0.0588235294117647...

This number repeats in a cycle of 16 digits. If you take your calculator and compute 1/17, you’ll observe this repeating block:

0.0588235294117647 0588235294117647 0588235294117647…

This predictable pattern lets us determine the value at any digit position, whether the 10th, 100th, or even the 300th digit.

Why Do Some Decimals Repeat?

A decimal will repeat if it is formed from a rational number—a fraction made of integers. When you divide two integers (numerator and denominator) and the denominator is not a multiple of only 2 or 5, the resulting decimal repeats. In our case, 17 is neither a multiple of 2 nor 5, making 1/17 a repeating decimal.

The Repeating Cycle in 1/17

The complete repeating sequence of 1/17 is as follows:

0588235294117647 (16 digits)

You’ll notice that after every 16 digits, the sequence starts over. This means we can predict any digit using modular math.

Finding the 300th Digit of 0.0588235294117647

Now we get to the heart of the question: What is the 300th digit in the decimal expansion of 1/17, i.e., the number 0.0588235294117647 repeated?

Step-by-Step Guide

1. Identify the repeating block: 16 digits long — “0588235294117647”
2. Exclude the “0.” part: Since we’re talking only about the digits after the decimal point, we begin counting from the first digit after the dot (i.e., 0).
3. Determine position within the repeating cycle: Use modular arithmetic to compute the position of the 300th digit in the cycle.
   • Cycle length = 16
   • 300 ÷ 16 = 18 remainder 12
4. Find the 12th digit in the repeating sequence: Count in the block “0588235294117647”

The repeating sequence is:

1. 0
2. 5
3. 8
4. 8
5. 2
6. 3
7. 5
8. 2
9. 9
10. 4
11. 1
12. 1
13. 7
14. 6
15. 4
16. 7

Therefore, the 300th digit is: 1

The Final Answer

To put it simply:

The 300th digit of 0.0588235294117647 is 1

Why Is This Mathematically Interesting?

This question touches on numerical patterns, division, and cyclic repetition within rational numbers. It showcases how math allows for predictable infinite behaviors in certain decimal forms. In fact, dividing 1 by a prime number often results in interesting and sometimes lengthy repeating decimal cycles, and 17 is no exception.

Examples of Other Repeating Decimal Patterns

• 1/3 = 0.333… — repeats every 1 digit
• 1/7 = 0.142857 142857… — repeats every 6 digits
• 1/13 = 0.076923 076923… — repeats every 6 digits
• 1/17 = 0.0588235294117647… — repeats every 16 digits

How To Calculate the Nth Digit of a Repeating Decimal

If you ever want to calculate the Nth digit of any repeating decimal, use this general approach:

1. Identify the repeating block.
2. Count the number of digits in the block (length).
3. Subtract any non-repeating digits — in this case, “0.” is not counted.
4. Do N mod (length of block) to get the index position.
5. Use the result to get the digit at that position in the repeating block.

Applications of Repeating Decimal Patterns

Understanding repeating patterns in decimals isn’t just a brain teaser—it has practical applications in:

• Computer Programming: Where precise infinite decimal approximations are needed.
• Cryptography: Using primes and cyclic patterns in algorithm design.
• Data Compression: By predicting repeating sequences.
• Math Education: Teaching decimals, fractions, and rational numbers effectively.

Fun Fact: Decimal Repetends

The number of digits in the repeating portion of a decimal is called its repetend length. For 1/17, the repetend is 16 digits long. Interestingly, since 17 is a prime number, its repetend length is determined by the smallest number x such that 10^x ≡ 1 (mod 17). That x here is 16, making it a full reptend prime!

Conclusion

In conclusion, the seemingly complicated question: “What is the 300th digit of 0.0588235294117647?” becomes easy once you understand that this is simply the decimal expansion of 1/17, which repeats every 16 digits. Once you’ve established the repeating cycle and its length, you can find any digit at any position using simple modular arithmetic.

To recap:

• 1 divided by 17 = 0.0588235294117647…
• This decimal repeats every 16 digits.
• The 300th digit lies at position 12 of the repeating cycle.
• Hence, the answer is: 1

We hope this explanation has demystified the fascinating world of repeating decimals and helped you understand not only the answer but the logic behind the calculation. Keep exploring the beauty of mathematics!