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a. proove it b. pi as a decimal number is an approximation that is only as long as you calculate it to be. Moreover, decimals do not contain non-numeric infor

a. proove it b. pi as a decimal number is an approximation that is only as long as you calculate it to be. Moreover, decimals do not contain non-numeric information. That's just your own interpretation. Imagine a number system where pi is 1. Clearly no names in 1. Now imagine a number system where pi is expressed as π, like... the number system that we use. Clearly no names in π.

lutusp 14y ago
> pi as a decimal number is an approximation that is only as long as you calculate it to be. But Pi is not equal to our approximations of Pi. That's backwards. Pi owes nothing to our lame efforts to approximate it. When someone asserts that Pi may be an infinite sequence, the truth of the proposition doesn't depend on someone computing an infinite sequence of Pi's digits. > Moreover, decimals do not contain non-numeric information. Are you aware that your typing is promptly translated into numbers for transmission through the Internet? That A = 65 (in the old ASCII encoding), B = 66, etc.? So yes, decimals do contain non-numeric information if we choose to interpret them that way. And we do.
- sp332 14y ago
  You're looking for strings in the decimal expansion of pi. Pi is not a decimal number! Every decimal representation of pi is an approximation. Pi is not an infinite sequence, it is a single value a bit higher than 3.14, it is only the decimal (or other bases, besides 10) expansion that has an infinite number of digits.
  - lutusp 14y ago
    > Pi is not a decimal number! Of course it is -- although choice of base is quire arbitrary. Pi is as much a decimal number as it is a binary number. > Every decimal representation of pi is an approximation. Our inability to represent Pi doesn't constrain Pi, it only constrains us. The fact that we cannot fully represent Pi has no effect on Pi or its identity. And Pi has the same identity in any base -- the ratio of a circle's circumference to its diameter. > Pi is not an infinite sequence Pi is an infinite sequence. http://en.wikipedia.org/wiki/Pi#Infinite_series http://en.wikipedia.org/wiki/Pi#Infinite_series Quote: "The calculation of π was revolutionized by the development of infinite series techniques in the 16th and 17th centuries. An infinite series is the sum of the terms of an infinite sequence.[49] Infinite series allowed mathematicians to compute π with much greater precision than Archimedes and others who used geometrical techniques."
    - sp332 14y ago
      You're constraining yourself to a wrong definition of what a number is. The number 0xA is not a different number from 0b1010 even though they have a different number of digits. > Pi is not an infinite sequence Pi is an infinite sequence. I was talking about an infinite number of digits, but even this new definition is wrong. Pi is one single value. There are many ways to compute this value. Some of them include infinite series. Note the wording: "The calculation of π was revolutionized by the development of infinite series techniques". Other techniques were used before, and they are still valid. The only advantage of infinite series is that they are easier to calculate, and the approximations converge faster when you compute them on real (finite) hardware.
      - lutusp 14y ago
        > You're constraining yourself to a wrong definition of what a number is. That's your problem, not mine. You keep trying to say that Pi is defined by our approximations of its value. > Pi is one single value. So, tell me, which part of "Pi is an infinite sequence" are you actually disagreeing with? > There are many ways to compute this value. You persist in confusing Pi with our efforts to approximate its value. > Other techniques were used before, and they are still valid. False. 22/7 is not valid. 355/113 is not valid. This is valid: https://www.dropbox.com/s/423ieio63bsuaat/pi_w.png https://www.dropbox.com/s/423ieio63bsuaat/pi_w.png The reason? Pi is a mathematical idea that happens to have a numerical value, but the idea transcends the value. The value is a coincidence, which is why choosing to express it as 1 to the base Pi changes nothing, and why arguing about the size of its approximations changes nothing. > The only advantage of infinite series is that they are easier to calculate ... No, they are much more difficult to calculate, but they convey more meaning. Infinite series are why Pi isn't just a number, any more than e is.
        sp332 14y ago
        > False. 22/7 is not valid. 355/113 is not valid. Right, those are approximations. Those are not what we are talking about. > This is valid: https://www.dropbox.com/s/423ieio63bsuaat/pi_w.png https://www.dropbox.com/s/423ieio63bsuaat/pi_w.png I agree that it is valid. It is one of many ways of calculating the value of pi. Here is another one: http://upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Pi-unrolled-720.gif/200px-Pi-unrolled-720.gif http://upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Pi-... Also there are iterative algorithms. http://en.wikipedia.org/wiki/Pi#Computer_era_and_iterative_algorithms http://en.wikipedia.org/wiki/Pi#Computer_era_and_iterative_a... > No, they are much more difficult to calculate, but they convey more meaning. Again, I didn't mean approximations like 22/7. I was comparing them to e.g. the geometric method which took hundreds of years to extend to a few hundred digits. The geometric method conveys the same "meaning" because it is an exact description of pi, just like the infinite series and the iterative algorithms. > Pi isn't just a number, any more than e is. They are both just numbers. e is not infinite, in fact it is less than 3.

lutusp 14y ago

> pi as a decimal number is an approximation that is only as long as you calculate it to be. But Pi is not equal to our approximations of Pi. That's backwards. Pi owes nothing to our lame efforts to approximate it. When someone asserts that Pi may be an infinite sequence, the truth of the proposition doesn't depend on someone computing an infinite sequence of Pi's digits. > Moreover, decimals do not contain non-numeric information. Are you aware that your typing is promptly translated into numbers for transmission through the Internet? That A = 65 (in the old ASCII encoding), B = 66, etc.? So yes, decimals do contain non-numeric information if we choose to interpret them that way. And we do.

sp332 14y ago
You're looking for strings in the decimal expansion of pi. Pi is not a decimal number! Every decimal representation of pi is an approximation. Pi is not an infinite sequence, it is a single value a bit higher than 3.14, it is only the decimal (or other bases, besides 10) expansion that has an infinite number of digits.
- lutusp 14y ago
  > Pi is not a decimal number! Of course it is -- although choice of base is quire arbitrary. Pi is as much a decimal number as it is a binary number. > Every decimal representation of pi is an approximation. Our inability to represent Pi doesn't constrain Pi, it only constrains us. The fact that we cannot fully represent Pi has no effect on Pi or its identity. And Pi has the same identity in any base -- the ratio of a circle's circumference to its diameter. > Pi is not an infinite sequence Pi is an infinite sequence. http://en.wikipedia.org/wiki/Pi#Infinite_series http://en.wikipedia.org/wiki/Pi#Infinite_series Quote: "The calculation of π was revolutionized by the development of infinite series techniques in the 16th and 17th centuries. An infinite series is the sum of the terms of an infinite sequence.[49] Infinite series allowed mathematicians to compute π with much greater precision than Archimedes and others who used geometrical techniques."
  - sp332 14y ago
    You're constraining yourself to a wrong definition of what a number is. The number 0xA is not a different number from 0b1010 even though they have a different number of digits. > Pi is not an infinite sequence Pi is an infinite sequence. I was talking about an infinite number of digits, but even this new definition is wrong. Pi is one single value. There are many ways to compute this value. Some of them include infinite series. Note the wording: "The calculation of π was revolutionized by the development of infinite series techniques". Other techniques were used before, and they are still valid. The only advantage of infinite series is that they are easier to calculate, and the approximations converge faster when you compute them on real (finite) hardware.
    - lutusp 14y ago
      > You're constraining yourself to a wrong definition of what a number is. That's your problem, not mine. You keep trying to say that Pi is defined by our approximations of its value. > Pi is one single value. So, tell me, which part of "Pi is an infinite sequence" are you actually disagreeing with? > There are many ways to compute this value. You persist in confusing Pi with our efforts to approximate its value. > Other techniques were used before, and they are still valid. False. 22/7 is not valid. 355/113 is not valid. This is valid: https://www.dropbox.com/s/423ieio63bsuaat/pi_w.png https://www.dropbox.com/s/423ieio63bsuaat/pi_w.png The reason? Pi is a mathematical idea that happens to have a numerical value, but the idea transcends the value. The value is a coincidence, which is why choosing to express it as 1 to the base Pi changes nothing, and why arguing about the size of its approximations changes nothing. > The only advantage of infinite series is that they are easier to calculate ... No, they are much more difficult to calculate, but they convey more meaning. Infinite series are why Pi isn't just a number, any more than e is.
      - sp332 14y ago
        > False. 22/7 is not valid. 355/113 is not valid. Right, those are approximations. Those are not what we are talking about. > This is valid: https://www.dropbox.com/s/423ieio63bsuaat/pi_w.png https://www.dropbox.com/s/423ieio63bsuaat/pi_w.png I agree that it is valid. It is one of many ways of calculating the value of pi. Here is another one: http://upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Pi-unrolled-720.gif/200px-Pi-unrolled-720.gif http://upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Pi-... Also there are iterative algorithms. http://en.wikipedia.org/wiki/Pi#Computer_era_and_iterative_algorithms http://en.wikipedia.org/wiki/Pi#Computer_era_and_iterative_a... > No, they are much more difficult to calculate, but they convey more meaning. Again, I didn't mean approximations like 22/7. I was comparing them to e.g. the geometric method which took hundreds of years to extend to a few hundred digits. The geometric method conveys the same "meaning" because it is an exact description of pi, just like the infinite series and the iterative algorithms. > Pi isn't just a number, any more than e is. They are both just numbers. e is not infinite, in fact it is less than 3.