Many people searching online for “what is the 300th digit of 0.0588235294117647” are often surprised to discover that this simple-looking decimal actually hides a fascinating repeating pattern. At first glance, the number 0.0588235294117647 appears to be just a list of digits, but when examined more closely, it reveals a precise, recurring sequence that can be extended infinitely. Understanding how to find the 300th digit of this decimal requires a combination of pattern recognition, knowledge of repeating decimals, and a bit of modular arithmetic.
In this detailed guide, we will explain:
- Why the decimal repeats
- The exact repeating cycle
- How to compute the 300th digit
- Why this number is mathematically interesting
- How long-range digit questions like this are solved
By the end, you will clearly understand not only the answer but also the mechanism behind it—giving you confidence to solve similar problems.
Understanding the Decimal 0.0588235294117647
The number 0.0588235294117647 is not random—it comes from a specific mathematical fraction. This decimal is actually part of the full repeating representation of:
1 ÷ 17 = 0.0588235294117647 0588235294117647 …
In other words:
- The decimal repeats every 16 digits
- The repeating block is:
0588235294117647
This repeating block is exactly 16 digits long, and it continues forever.
When a decimal repeats, we can determine any digit—whether it is the 20th, 100th, or even the 300th—by understanding where it falls inside the repeating cycle.
Step 1: Identify the repeating block
The repeating part is:
0 5 8 8 2 3 5 2 9 4 1 1 7 6 4 7
If we count them:
- 0
- 5
- 8
- 8
- 2
- 3
- 5
- 2
- 9
- 4
- 1
- 1
- 7
- 6
- 4
- 7
This 16-digit sequence repeats infinitely.
Step 2: Understand how digit positions work in repeating decimals
To find any digit in a repeating decimal:
- Determine the position number you want—here, 300.
- Divide that number by the length of the repeating block (which is 16).
- Find the remainder.
- The remainder tells you which digit in the repeating block matches the position.
If the remainder is 0, that means the digit is the last digit in the sequence.
Step 3: Calculate where the 300th digit falls
We compute:
300 ÷ 16 = 18 remainder ?
16 × 18 = 288
300 − 288 = 12
So:
The 300th digit is the 12th digit in the 16-digit repeating pattern.
Now we simply locate the 12th digit.
Step 4: Identify the 12th digit in the repeating sequence
Here is the numbered list again:
- 0
- 5
- 8
- 8
- 2
- 3
- 5
- 2
- 9
- 4
- 1
- 1
- 7
- 6
- 4
- 7
The 12th digit is:
1
Final Answer: The 300th digit of 0.0588235294117647 is 1
This results from the repeating nature of the decimal expansion of 1/17.
Why the Number 0.0588235294117647 Is So Interesting
Many decimals repeat, but 1/17 is especially notable because its repeating block is:
- Lengthy (16 digits long)
- Perfectly cyclical
- Mathematically elegant
When multiplied by numbers from 1 to 16, this repeating block shifts in orderly and predictable patterns. This makes the decimal part of 1/17 a classic example used in number theory and recreational mathematics.
How Repeating Decimals Work
When you divide numbers, there are three possibilities:
- The decimal terminates (like 1/8 = 0.125)
- The decimal repeats (like 1/3 = 0.333…)
- The decimal is irrational (like π)
A repeating decimal happens when the denominator contains prime factors other than 2 or 5.
Since 17 is a prime number that is not 2 or 5, the result must repeat.
Finding Large Digit Positions Without Writing the Decimal Out
If you wanted the 10,000th digit or the 1,000,000th digit, the same method applies:
- Compute the remainder when the position is divided by 16
- Map the remainder to the repeating sequence
This is why such problems are very popular in competitive exams, aptitude tests, and number theory challenges.
Why the Repeating Cycle of 1/17 Is 16 Digits Long
Decimals repeat because the division process loops. The length of the repeating block is related to:
- The denominator
- The multiplicative order of 10 modulo the denominator
For 1/17:
- The longest possible repeating cycle is 16 digits
- And indeed, 1/17 achieves that maximum
This makes 17 one of the full-reptend primes, meaning the decimal expansion of 1/p reaches the maximum length p−1.
A Deeper Look at the Repeating Pattern
Let’s list the repeating block again:
0588235294117647
This block has some interesting properties:
1. It is symmetrical in structure
- It begins and ends with 0 and 7
- It contains an equal distribution of larger and smaller digits
2. Doubling, tripling, and multiplying it cycles the pattern
For example:
- 1/17 = 0.0588235294117647
- 2/17 = 0.1176470588235294
- 3/17 = 0.1764705882352941
Each fraction is the same digits, just shifted.
3. The pattern wraps like a loop
This makes finding distant digits incredibly easy.
Step-by-Step Example: Finding Another Digit
Let’s find the 1000th digit to reinforce the concept.
- 1000 ÷ 16 = 62 remainder 8
- The 8th digit in the cycle is 2
So the 1000th digit would be 2.
This shows how fast the process is once you understand the pattern.
Common Questions Related to This Topic
1. Why does 1/17 have such a long repeating decimal?
Because 17 is a full-reptend prime for the base 10 number system.
2. Does every fraction repeat eventually?
All rational numbers (fractions of integers) either:
- Terminate, or
- Repeat
They never form non-repeating infinite sequences; those only occur in irrational numbers.
3. Could the 300th digit ever be zero?
Yes. The repeating sequence contains 0 as its first digit.
For digit positions where the remainder equals 1, the digit is 0.
SEO-Focused Summary
For anyone searching “what is the 300th digit of 0.0588235294117647”, the key points are:
- The decimal comes from 1/17
- It repeats every 16 digits
- To find any digit, divide the position by 16
- The remainder gives the location in the repeating block
- For the 300th digit, the remainder is 12
- The 12th digit in the repeating block is 1
Thus, the 300th digit is:
1
This structured approach ensures clear understanding and allows you to compute any future digit efficiently.
Conclusion
The question “what is the 300th digit of 0.0588235294117647” at first appears complicated, but once the structure of the repeating decimal is understood, the solution is straightforward. Because 1/17 produces a 16-digit repeating cycle, every position in the decimal expansion corresponds neatly to one of those 16 digits.
