โฒ 233 โผ Since Pi is infinite and non-repeating, would that mean any finite sequence of non-repeating numbers should appear somewhere in Pi? (lemmy.dbzer0.com) submitted 2 years ago* by Melatonin@lemmy.dbzer0.com to c/asklemmy@lemmy.ml 148 comments fedilink hide all child comments How about ANY FINITE SEQUENCE AT ALL?
[โ] flashgnash@lemm.ee 5 points 2 years ago (2 children) You can't prove that there isn't one somewhere permalink fedilink source parent hideshow 4 child comments replies: [โ] Danitos@reddthat.com 11 points 2 years ago (1 child) You can, it's literally the way the number is defined. permalink fedilink source parent hideshow 2 child comments replies: [โ] mukt@lemmy.ml 3 points 2 years ago (1 child) Defined where ? permalink fedilink source parent hideshow 2 child comments replies: [โ] cosecantphi@hexbear.net 7 points 2 years ago* (2 children) It's implicitly defined here by its decimal form: 0.101001000100001000001 . . . The definition of this number is that the number of 0s after each 1 is given by the total previous number of 1s in the sequence. That's why it can't contain 2 despite being infinite and non-repeating. permalink fedilink source parent hideshow 4 child comments replies: [โ] mukt@lemmy.ml 1 point 2 years ago (1 child) 0.101001000100001000001 . . . Might very well be : 0.101001000100001000001202002000200002000002 ... Real life, is different from gamified questions asked in student exams. permalink fedilink source parent hideshow 2 child comments replies: [โ] cosecantphi@hexbear.net 1 point 2 years ago* (1 child) Implicitly defining a number via it's decimal form typically relies on their being a pattern to follow after the ellipsis. You can define a different number with twos in it, but if you put an ellipsis at the end you're implying there's a different pattern to follow for the rest of the decimal expansion, hence your number is not the same number as the one without twos in it. permalink fedilink source parent hideshow 2 child comments replies: [โ] mukt@lemmy.ml 1 point 2 years ago (1 child) assumption โ definition permalink fedilink source parent hideshow 2 child comments replies: [โ] cosecantphi@hexbear.net 1 point 2 years ago* (1 child) Math kind of relies on assumptions, you really can't get anywhere in math without an assumption at the beginning of your thought process. permalink fedilink source parent hideshow 2 child comments replies: [โ] mukt@lemmy.ml 1 point 2 years ago (1 child) Obviously. But still maths avoids stuff like "I assume the answer is X. QED." permalink fedilink source parent hideshow 2 child comments replies: [โ] cosecantphi@hexbear.net 1 point 2 years ago* (1 child) Right and the point of defining this number as a non-repeating infinite sequence of 0s and 1s is just to show that non-repetition of digits alone is not sufficient to say a number contains all finite sequences. permalink fedilink source parent hideshow 2 child comments replies: [โ] mukt@lemmy.ml 1 point 2 years ago That trivial point is not the one we (you and me) are contending. The issue is that OP hasn't actually defined the sequence, just given some properties (which does not lead to any definition or determination of the location of the number/s on the number line, by itself). Assuming that he has defined it, doesn't change anything as any other commentator can assume something different, which consistent with OP's post. permalink fedilink source parent [โ] Hawk@lemmy.dbzer0.com 1 point 2 years ago (1 child) Pi is often defined as 3.141 592 653... Does that mean Pi does not contain any 7s or 8s? permalink fedilink source parent hideshow 2 child comments replies: [โ] cosecantphi@hexbear.net 1 point 2 years ago* That's a decimal approximation of Pi with an ellipsis at the end to indicate its an approximation, not a definition. The way the ellipsis is used above is different. It's being used to define a number via the decimal expansion by saying it's an infinite sum of negative powers of 10 defined by the pattern before the ellipsis. So we have: 0.101001000100001000001 . . . = 10^-1 + 10^-2 + 10^-3 + 10^-4 +10^-5+ . . . Pi, however, is not defined this way. Pi can be defined as twice the solution of the integral from -1 to 1 of the square root of (1-x^2), a function defining a unit semi-circle. permalink fedilink source parent [โ] xigoi@lemmy.sdf.org 4 points 2 years ago (1 child) Why couldnโt you? permalink fedilink source parent hideshow 2 child comments replies: [โ] flashgnash@lemm.ee 2 points 2 years ago Because you'd need to search through an infinite number of digits (unless you have access to the original formula) permalink fedilink source parent
[โ] Danitos@reddthat.com 11 points 2 years ago (1 child) You can, it's literally the way the number is defined. permalink fedilink source parent hideshow 2 child comments replies: [โ] mukt@lemmy.ml 3 points 2 years ago (1 child) Defined where ? permalink fedilink source parent hideshow 2 child comments replies: [โ] cosecantphi@hexbear.net 7 points 2 years ago* (2 children) It's implicitly defined here by its decimal form: 0.101001000100001000001 . . . The definition of this number is that the number of 0s after each 1 is given by the total previous number of 1s in the sequence. That's why it can't contain 2 despite being infinite and non-repeating. permalink fedilink source parent hideshow 4 child comments replies: [โ] mukt@lemmy.ml 1 point 2 years ago (1 child) 0.101001000100001000001 . . . Might very well be : 0.101001000100001000001202002000200002000002 ... Real life, is different from gamified questions asked in student exams. permalink fedilink source parent hideshow 2 child comments replies: [โ] cosecantphi@hexbear.net 1 point 2 years ago* (1 child) Implicitly defining a number via it's decimal form typically relies on their being a pattern to follow after the ellipsis. You can define a different number with twos in it, but if you put an ellipsis at the end you're implying there's a different pattern to follow for the rest of the decimal expansion, hence your number is not the same number as the one without twos in it. permalink fedilink source parent hideshow 2 child comments replies: [โ] mukt@lemmy.ml 1 point 2 years ago (1 child) assumption โ definition permalink fedilink source parent hideshow 2 child comments replies: [โ] cosecantphi@hexbear.net 1 point 2 years ago* (1 child) Math kind of relies on assumptions, you really can't get anywhere in math without an assumption at the beginning of your thought process. permalink fedilink source parent hideshow 2 child comments replies: [โ] mukt@lemmy.ml 1 point 2 years ago (1 child) Obviously. But still maths avoids stuff like "I assume the answer is X. QED." permalink fedilink source parent hideshow 2 child comments replies: [โ] cosecantphi@hexbear.net 1 point 2 years ago* (1 child) Right and the point of defining this number as a non-repeating infinite sequence of 0s and 1s is just to show that non-repetition of digits alone is not sufficient to say a number contains all finite sequences. permalink fedilink source parent hideshow 2 child comments replies: [โ] mukt@lemmy.ml 1 point 2 years ago That trivial point is not the one we (you and me) are contending. The issue is that OP hasn't actually defined the sequence, just given some properties (which does not lead to any definition or determination of the location of the number/s on the number line, by itself). Assuming that he has defined it, doesn't change anything as any other commentator can assume something different, which consistent with OP's post. permalink fedilink source parent [โ] Hawk@lemmy.dbzer0.com 1 point 2 years ago (1 child) Pi is often defined as 3.141 592 653... Does that mean Pi does not contain any 7s or 8s? permalink fedilink source parent hideshow 2 child comments replies: [โ] cosecantphi@hexbear.net 1 point 2 years ago* That's a decimal approximation of Pi with an ellipsis at the end to indicate its an approximation, not a definition. The way the ellipsis is used above is different. It's being used to define a number via the decimal expansion by saying it's an infinite sum of negative powers of 10 defined by the pattern before the ellipsis. So we have: 0.101001000100001000001 . . . = 10^-1 + 10^-2 + 10^-3 + 10^-4 +10^-5+ . . . Pi, however, is not defined this way. Pi can be defined as twice the solution of the integral from -1 to 1 of the square root of (1-x^2), a function defining a unit semi-circle. permalink fedilink source parent
[โ] mukt@lemmy.ml 3 points 2 years ago (1 child) Defined where ? permalink fedilink source parent hideshow 2 child comments replies: [โ] cosecantphi@hexbear.net 7 points 2 years ago* (2 children) It's implicitly defined here by its decimal form: 0.101001000100001000001 . . . The definition of this number is that the number of 0s after each 1 is given by the total previous number of 1s in the sequence. That's why it can't contain 2 despite being infinite and non-repeating. permalink fedilink source parent hideshow 4 child comments replies: [โ] mukt@lemmy.ml 1 point 2 years ago (1 child) 0.101001000100001000001 . . . Might very well be : 0.101001000100001000001202002000200002000002 ... Real life, is different from gamified questions asked in student exams. permalink fedilink source parent hideshow 2 child comments replies: [โ] cosecantphi@hexbear.net 1 point 2 years ago* (1 child) Implicitly defining a number via it's decimal form typically relies on their being a pattern to follow after the ellipsis. You can define a different number with twos in it, but if you put an ellipsis at the end you're implying there's a different pattern to follow for the rest of the decimal expansion, hence your number is not the same number as the one without twos in it. permalink fedilink source parent hideshow 2 child comments replies: [โ] mukt@lemmy.ml 1 point 2 years ago (1 child) assumption โ definition permalink fedilink source parent hideshow 2 child comments replies: [โ] cosecantphi@hexbear.net 1 point 2 years ago* (1 child) Math kind of relies on assumptions, you really can't get anywhere in math without an assumption at the beginning of your thought process. permalink fedilink source parent hideshow 2 child comments replies: [โ] mukt@lemmy.ml 1 point 2 years ago (1 child) Obviously. But still maths avoids stuff like "I assume the answer is X. QED." permalink fedilink source parent hideshow 2 child comments replies: [โ] cosecantphi@hexbear.net 1 point 2 years ago* (1 child) Right and the point of defining this number as a non-repeating infinite sequence of 0s and 1s is just to show that non-repetition of digits alone is not sufficient to say a number contains all finite sequences. permalink fedilink source parent hideshow 2 child comments replies: [โ] mukt@lemmy.ml 1 point 2 years ago That trivial point is not the one we (you and me) are contending. The issue is that OP hasn't actually defined the sequence, just given some properties (which does not lead to any definition or determination of the location of the number/s on the number line, by itself). Assuming that he has defined it, doesn't change anything as any other commentator can assume something different, which consistent with OP's post. permalink fedilink source parent [โ] Hawk@lemmy.dbzer0.com 1 point 2 years ago (1 child) Pi is often defined as 3.141 592 653... Does that mean Pi does not contain any 7s or 8s? permalink fedilink source parent hideshow 2 child comments replies: [โ] cosecantphi@hexbear.net 1 point 2 years ago* That's a decimal approximation of Pi with an ellipsis at the end to indicate its an approximation, not a definition. The way the ellipsis is used above is different. It's being used to define a number via the decimal expansion by saying it's an infinite sum of negative powers of 10 defined by the pattern before the ellipsis. So we have: 0.101001000100001000001 . . . = 10^-1 + 10^-2 + 10^-3 + 10^-4 +10^-5+ . . . Pi, however, is not defined this way. Pi can be defined as twice the solution of the integral from -1 to 1 of the square root of (1-x^2), a function defining a unit semi-circle. permalink fedilink source parent
[โ] cosecantphi@hexbear.net 7 points 2 years ago* (2 children) It's implicitly defined here by its decimal form: 0.101001000100001000001 . . . The definition of this number is that the number of 0s after each 1 is given by the total previous number of 1s in the sequence. That's why it can't contain 2 despite being infinite and non-repeating. permalink fedilink source parent hideshow 4 child comments replies: [โ] mukt@lemmy.ml 1 point 2 years ago (1 child) 0.101001000100001000001 . . . Might very well be : 0.101001000100001000001202002000200002000002 ... Real life, is different from gamified questions asked in student exams. permalink fedilink source parent hideshow 2 child comments replies: [โ] cosecantphi@hexbear.net 1 point 2 years ago* (1 child) Implicitly defining a number via it's decimal form typically relies on their being a pattern to follow after the ellipsis. You can define a different number with twos in it, but if you put an ellipsis at the end you're implying there's a different pattern to follow for the rest of the decimal expansion, hence your number is not the same number as the one without twos in it. permalink fedilink source parent hideshow 2 child comments replies: [โ] mukt@lemmy.ml 1 point 2 years ago (1 child) assumption โ definition permalink fedilink source parent hideshow 2 child comments replies: [โ] cosecantphi@hexbear.net 1 point 2 years ago* (1 child) Math kind of relies on assumptions, you really can't get anywhere in math without an assumption at the beginning of your thought process. permalink fedilink source parent hideshow 2 child comments replies: [โ] mukt@lemmy.ml 1 point 2 years ago (1 child) Obviously. But still maths avoids stuff like "I assume the answer is X. QED." permalink fedilink source parent hideshow 2 child comments replies: [โ] cosecantphi@hexbear.net 1 point 2 years ago* (1 child) Right and the point of defining this number as a non-repeating infinite sequence of 0s and 1s is just to show that non-repetition of digits alone is not sufficient to say a number contains all finite sequences. permalink fedilink source parent hideshow 2 child comments replies: [โ] mukt@lemmy.ml 1 point 2 years ago That trivial point is not the one we (you and me) are contending. The issue is that OP hasn't actually defined the sequence, just given some properties (which does not lead to any definition or determination of the location of the number/s on the number line, by itself). Assuming that he has defined it, doesn't change anything as any other commentator can assume something different, which consistent with OP's post. permalink fedilink source parent [โ] Hawk@lemmy.dbzer0.com 1 point 2 years ago (1 child) Pi is often defined as 3.141 592 653... Does that mean Pi does not contain any 7s or 8s? permalink fedilink source parent hideshow 2 child comments replies: [โ] cosecantphi@hexbear.net 1 point 2 years ago* That's a decimal approximation of Pi with an ellipsis at the end to indicate its an approximation, not a definition. The way the ellipsis is used above is different. It's being used to define a number via the decimal expansion by saying it's an infinite sum of negative powers of 10 defined by the pattern before the ellipsis. So we have: 0.101001000100001000001 . . . = 10^-1 + 10^-2 + 10^-3 + 10^-4 +10^-5+ . . . Pi, however, is not defined this way. Pi can be defined as twice the solution of the integral from -1 to 1 of the square root of (1-x^2), a function defining a unit semi-circle. permalink fedilink source parent
[โ] mukt@lemmy.ml 1 point 2 years ago (1 child) 0.101001000100001000001 . . . Might very well be : 0.101001000100001000001202002000200002000002 ... Real life, is different from gamified questions asked in student exams. permalink fedilink source parent hideshow 2 child comments replies: [โ] cosecantphi@hexbear.net 1 point 2 years ago* (1 child) Implicitly defining a number via it's decimal form typically relies on their being a pattern to follow after the ellipsis. You can define a different number with twos in it, but if you put an ellipsis at the end you're implying there's a different pattern to follow for the rest of the decimal expansion, hence your number is not the same number as the one without twos in it. permalink fedilink source parent hideshow 2 child comments replies: [โ] mukt@lemmy.ml 1 point 2 years ago (1 child) assumption โ definition permalink fedilink source parent hideshow 2 child comments replies: [โ] cosecantphi@hexbear.net 1 point 2 years ago* (1 child) Math kind of relies on assumptions, you really can't get anywhere in math without an assumption at the beginning of your thought process. permalink fedilink source parent hideshow 2 child comments replies: [โ] mukt@lemmy.ml 1 point 2 years ago (1 child) Obviously. But still maths avoids stuff like "I assume the answer is X. QED." permalink fedilink source parent hideshow 2 child comments replies: [โ] cosecantphi@hexbear.net 1 point 2 years ago* (1 child) Right and the point of defining this number as a non-repeating infinite sequence of 0s and 1s is just to show that non-repetition of digits alone is not sufficient to say a number contains all finite sequences. permalink fedilink source parent hideshow 2 child comments replies: [โ] mukt@lemmy.ml 1 point 2 years ago That trivial point is not the one we (you and me) are contending. The issue is that OP hasn't actually defined the sequence, just given some properties (which does not lead to any definition or determination of the location of the number/s on the number line, by itself). Assuming that he has defined it, doesn't change anything as any other commentator can assume something different, which consistent with OP's post. permalink fedilink source parent
[โ] cosecantphi@hexbear.net 1 point 2 years ago* (1 child) Implicitly defining a number via it's decimal form typically relies on their being a pattern to follow after the ellipsis. You can define a different number with twos in it, but if you put an ellipsis at the end you're implying there's a different pattern to follow for the rest of the decimal expansion, hence your number is not the same number as the one without twos in it. permalink fedilink source parent hideshow 2 child comments replies: [โ] mukt@lemmy.ml 1 point 2 years ago (1 child) assumption โ definition permalink fedilink source parent hideshow 2 child comments replies: [โ] cosecantphi@hexbear.net 1 point 2 years ago* (1 child) Math kind of relies on assumptions, you really can't get anywhere in math without an assumption at the beginning of your thought process. permalink fedilink source parent hideshow 2 child comments replies: [โ] mukt@lemmy.ml 1 point 2 years ago (1 child) Obviously. But still maths avoids stuff like "I assume the answer is X. QED." permalink fedilink source parent hideshow 2 child comments replies: [โ] cosecantphi@hexbear.net 1 point 2 years ago* (1 child) Right and the point of defining this number as a non-repeating infinite sequence of 0s and 1s is just to show that non-repetition of digits alone is not sufficient to say a number contains all finite sequences. permalink fedilink source parent hideshow 2 child comments replies: [โ] mukt@lemmy.ml 1 point 2 years ago That trivial point is not the one we (you and me) are contending. The issue is that OP hasn't actually defined the sequence, just given some properties (which does not lead to any definition or determination of the location of the number/s on the number line, by itself). Assuming that he has defined it, doesn't change anything as any other commentator can assume something different, which consistent with OP's post. permalink fedilink source parent
[โ] mukt@lemmy.ml 1 point 2 years ago (1 child) assumption โ definition permalink fedilink source parent hideshow 2 child comments replies: [โ] cosecantphi@hexbear.net 1 point 2 years ago* (1 child) Math kind of relies on assumptions, you really can't get anywhere in math without an assumption at the beginning of your thought process. permalink fedilink source parent hideshow 2 child comments replies: [โ] mukt@lemmy.ml 1 point 2 years ago (1 child) Obviously. But still maths avoids stuff like "I assume the answer is X. QED." permalink fedilink source parent hideshow 2 child comments replies: [โ] cosecantphi@hexbear.net 1 point 2 years ago* (1 child) Right and the point of defining this number as a non-repeating infinite sequence of 0s and 1s is just to show that non-repetition of digits alone is not sufficient to say a number contains all finite sequences. permalink fedilink source parent hideshow 2 child comments replies: [โ] mukt@lemmy.ml 1 point 2 years ago That trivial point is not the one we (you and me) are contending. The issue is that OP hasn't actually defined the sequence, just given some properties (which does not lead to any definition or determination of the location of the number/s on the number line, by itself). Assuming that he has defined it, doesn't change anything as any other commentator can assume something different, which consistent with OP's post. permalink fedilink source parent
[โ] cosecantphi@hexbear.net 1 point 2 years ago* (1 child) Math kind of relies on assumptions, you really can't get anywhere in math without an assumption at the beginning of your thought process. permalink fedilink source parent hideshow 2 child comments replies: [โ] mukt@lemmy.ml 1 point 2 years ago (1 child) Obviously. But still maths avoids stuff like "I assume the answer is X. QED." permalink fedilink source parent hideshow 2 child comments replies: [โ] cosecantphi@hexbear.net 1 point 2 years ago* (1 child) Right and the point of defining this number as a non-repeating infinite sequence of 0s and 1s is just to show that non-repetition of digits alone is not sufficient to say a number contains all finite sequences. permalink fedilink source parent hideshow 2 child comments replies: [โ] mukt@lemmy.ml 1 point 2 years ago That trivial point is not the one we (you and me) are contending. The issue is that OP hasn't actually defined the sequence, just given some properties (which does not lead to any definition or determination of the location of the number/s on the number line, by itself). Assuming that he has defined it, doesn't change anything as any other commentator can assume something different, which consistent with OP's post. permalink fedilink source parent
[โ] mukt@lemmy.ml 1 point 2 years ago (1 child) Obviously. But still maths avoids stuff like "I assume the answer is X. QED." permalink fedilink source parent hideshow 2 child comments replies: [โ] cosecantphi@hexbear.net 1 point 2 years ago* (1 child) Right and the point of defining this number as a non-repeating infinite sequence of 0s and 1s is just to show that non-repetition of digits alone is not sufficient to say a number contains all finite sequences. permalink fedilink source parent hideshow 2 child comments replies: [โ] mukt@lemmy.ml 1 point 2 years ago That trivial point is not the one we (you and me) are contending. The issue is that OP hasn't actually defined the sequence, just given some properties (which does not lead to any definition or determination of the location of the number/s on the number line, by itself). Assuming that he has defined it, doesn't change anything as any other commentator can assume something different, which consistent with OP's post. permalink fedilink source parent
[โ] cosecantphi@hexbear.net 1 point 2 years ago* (1 child) Right and the point of defining this number as a non-repeating infinite sequence of 0s and 1s is just to show that non-repetition of digits alone is not sufficient to say a number contains all finite sequences. permalink fedilink source parent hideshow 2 child comments replies: [โ] mukt@lemmy.ml 1 point 2 years ago That trivial point is not the one we (you and me) are contending. The issue is that OP hasn't actually defined the sequence, just given some properties (which does not lead to any definition or determination of the location of the number/s on the number line, by itself). Assuming that he has defined it, doesn't change anything as any other commentator can assume something different, which consistent with OP's post. permalink fedilink source parent
[โ] mukt@lemmy.ml 1 point 2 years ago That trivial point is not the one we (you and me) are contending. The issue is that OP hasn't actually defined the sequence, just given some properties (which does not lead to any definition or determination of the location of the number/s on the number line, by itself). Assuming that he has defined it, doesn't change anything as any other commentator can assume something different, which consistent with OP's post. permalink fedilink source parent
[โ] Hawk@lemmy.dbzer0.com 1 point 2 years ago (1 child) Pi is often defined as 3.141 592 653... Does that mean Pi does not contain any 7s or 8s? permalink fedilink source parent hideshow 2 child comments replies: [โ] cosecantphi@hexbear.net 1 point 2 years ago* That's a decimal approximation of Pi with an ellipsis at the end to indicate its an approximation, not a definition. The way the ellipsis is used above is different. It's being used to define a number via the decimal expansion by saying it's an infinite sum of negative powers of 10 defined by the pattern before the ellipsis. So we have: 0.101001000100001000001 . . . = 10^-1 + 10^-2 + 10^-3 + 10^-4 +10^-5+ . . . Pi, however, is not defined this way. Pi can be defined as twice the solution of the integral from -1 to 1 of the square root of (1-x^2), a function defining a unit semi-circle. permalink fedilink source parent
[โ] cosecantphi@hexbear.net 1 point 2 years ago* That's a decimal approximation of Pi with an ellipsis at the end to indicate its an approximation, not a definition. The way the ellipsis is used above is different. It's being used to define a number via the decimal expansion by saying it's an infinite sum of negative powers of 10 defined by the pattern before the ellipsis. So we have: 0.101001000100001000001 . . . = 10^-1 + 10^-2 + 10^-3 + 10^-4 +10^-5+ . . . Pi, however, is not defined this way. Pi can be defined as twice the solution of the integral from -1 to 1 of the square root of (1-x^2), a function defining a unit semi-circle. permalink fedilink source parent
[โ] xigoi@lemmy.sdf.org 4 points 2 years ago (1 child) Why couldnโt you? permalink fedilink source parent hideshow 2 child comments replies: [โ] flashgnash@lemm.ee 2 points 2 years ago Because you'd need to search through an infinite number of digits (unless you have access to the original formula) permalink fedilink source parent
[โ] flashgnash@lemm.ee 2 points 2 years ago Because you'd need to search through an infinite number of digits (unless you have access to the original formula) permalink fedilink source parent