Calculating Digits of Pi in Base-16 Using the Wolfram Formula

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In summary, the formula provided by Wolfram for calculating the base-16 representation of π involves a series of fractions that need to be dealt with in order to obtain the nth digit. This process can be done with the help of a computer program as it is not easily done by hand. Further information on how to do this can be found at the provided link.
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According to this link at Wolfram, the following formula can be used to calculate any digit of the base-16 representation of π:

[tex]
\pi = \sum_{n=0}^{\infty} \
\left(
\frac{4}{8n+1} - \frac{2}{8n+4} - \frac{1}{8n+5} - \frac{1}{8n+6}
\right)
\cdot \frac{1}{16^n}
[/tex]

But apparently it is not as straightforward as simply taking the nth term in the series to get the nth digit. For example, the 0th term is 3.1333..., and not simply 3 as it must be.

So my question is, just how does one use this formula to calculate a digit? Or am I missing something in my above reasoning?
 
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The problem is that the expression between brackets isn't an integer between 0 and 15, so it's not going to give you the nth digit. You're going to have to deal with fractional and integral parts and do some cleaning up before you actually get the nth digit to pop out. You can read up on how to do this here. This is something that's not so easily done by hand, but very easily done by a computer program.
 
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morphism said:
You can read up on how to do this here.
Ah, thanks.

For the most part I'll have to be satisfied with "it's not so simple", but now I have a better appreciation and some sense of how it can be done.

Thanks again!
 

1. What is the significance of using base 16 in calculating digits of pi?

Base 16, also known as hexadecimal, is commonly used in computer programming and digital systems. Using base 16 allows for a more efficient and accurate representation of pi, as it is a rational number in this base. This means that the digits in the sequence will eventually repeat, making it easier to calculate and store.

2. How many digits of pi can be accurately calculated in base 16?

Currently, over 13 trillion digits of pi have been calculated in base 16 using various supercomputers and algorithms. However, it is estimated that only about 39 digits are needed for most scientific calculations, and any additional digits are considered unnecessary for practical use.

3. Is there a pattern in the digits of pi in base 16?

There is no known pattern in the digits of pi in any base, including base 16. This is because pi is an irrational number, meaning it cannot be expressed as a finite or repeating decimal. Therefore, the digits of pi in base 16 (or any other base) are considered random and unpredictable.

4. How is calculating pi in base 16 different from calculating it in other bases?

Calculating pi in different bases is essentially the same process, but the resulting digits will be different. In base 16, the digits range from 0 to 9 and A to F, while in other bases, such as base 10, the digits range from 0 to 9. The main difference is the efficiency and accuracy of representing pi in different bases.

5. Are there any practical applications for calculating digits of pi in base 16?

While calculating digits of pi in base 16 may not have any immediate practical applications, the process of finding and testing new algorithms for calculating pi can lead to advancements in computer science and mathematics. Additionally, having a more accurate representation of pi in base 16 can aid in certain calculations used in fields such as engineering and physics.

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