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Food for thought

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I was thinking about pi last night and wondering if it really does have infinite decimal places that do not repeat. Then I came across this and this today that made me curious more and want to study pi some more. Basically this blog post is about me asking questions. Can I say that the topic is pi-quant. Interestingly enough the word piquant has two meanings 1. having a pleasantly sharp taste or appetizing flavor and 2. pleasantly stimulating or exciting to the mind.

Is it a fact that pi has infinite decimals after the point that do not rep-eat? (I see what I did there). I must be thinking of f∞d. According to my friend Bard, the answer is yes and has been proven to be irrational. Is this true for all irrational numbers? Also yes. While chatting with Bard, I noticed a pattern that 1/17 has a 16 digit repeating pattern and 1/23 has a 22 digit repeating pattern and 1/97 has a 96 digit repeating pattern. An approximation to pi is 22/7 which involves 1/7 and this has a 6 digit repeating pattern.

This leads me to some questions. Is there a general rule for the relationship between numerator and denominator and the length of the repeating pattern? Can I say that 1 divided by infinity will give a repeating pattern that is infinity-1 in length? And is that useful to think about somehow? Is pi related to division by infinity or infinity in some useful way that we have not thought of already? To me the infinity symbol is really a circle that has been twisted. According to my friend Bard, the earliest known symbol resembling infinity dates back to ancient Egypt and Greece, where it appeared as a serpent or dragon eating its own tail, called Ouroboros. According to wikipedia π is the sixteenth letter of the Greek alphabet and it has a value of 80 in the Greek numeral system and it came from the Phoenician letter pē, which meant mouth.

Now some more thoughts and questions. If I have a number I can always add 1 to it to make it bigger. Is it true to say that if I have a non-repeating number pattern I can always make it longer by repeating the pattern twice and combining those and changing the last digit up or down by 1? This works if the number is base-2 or higher. But in base-1 there are no non-repeating numbers and in base-infinity every number is a non-repeating number as you would need a different symbol to represent each number.

Which leads to my final two questions. In base-2 there are certain fractions that cannot be represented accurately as they are in base-10. Does this mean that there might be a higher base that we can use to represent pi finitely and accurately? I think the answer is no because irrationality is a property of the number and not because of its representation but I still have to ask. Is it possible to create a number system where every number is represented as related to pi? Writing this blog post has given me some food for thought and enough to want a pi-ece* of pecan pi-e and learn about pi and infinity some more. It does help that pecans are shaped like brains and can be considered brain food. Share your interesting finds about pi in the comments below.

*Here ece from pi-ece of pi-e could be short for euler's constant (e) - which is a related and fun concept

*Also I discovered that my blog post has 3141 characters if I include title and delete one word at the end (below) and I ignore punctuation (73) and ignore numeric characters (34). An interesting (to me) coin-c-id-ence - id being pi if I flipped it - like flipping a coin - head and tail - which takes me back to Ouroboros. Maybe the c in coin-c-idence could be circle.

*Correction e=eulers number which is different from eulers constant (γ)
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