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An Infinite Number That's Smaller Than Four
54 Answers
Can someone please explain why we can use mathematics to calculate blueprints for architecture, shape and age of the universe, the fastest possible design for an F1 car and the change you get from a morning paper and pint of milk from £5........
...And yet we allow a number, such as PI to exist...
Here's what i do know. PI is 3.141.. (etc) and it is believed to have an infinite number of decimal places (and has been calculated [reportedly] to 10 trillion.
But It can't be infinately long because then 4,5,6,7 etc wouldn't exist... surely one is restricted and unable to calculate any number below or above it? And it's not infinitely big because is's less then 4 (3.14>4)....
My mind melts down at this point... What is it that I don't know that allows this number to exist?
R.S.V.P
Infinity
...And yet we allow a number, such as PI to exist...
Here's what i do know. PI is 3.141.. (etc) and it is believed to have an infinite number of decimal places (and has been calculated [reportedly] to 10 trillion.
But It can't be infinately long because then 4,5,6,7 etc wouldn't exist... surely one is restricted and unable to calculate any number below or above it? And it's not infinitely big because is's less then 4 (3.14>4)....
My mind melts down at this point... What is it that I don't know that allows this number to exist?
R.S.V.P
Infinity
Answers
As a rule you shouldn't really turn to a dictionary to define mathematical or even some physical concepts (The Chambers 2011 definition for "Higgs boson" is particularly bad...). It is entirely possible to have an infinite string of numbers that never has a recurring pattern. Of course, because such a string is infinite, then you can't really check this for...
15:27 Sun 17th Feb 2013
Jim
Just to be clear: still talking about pi, you can get a repeat of say 3456 somewhere else along the line eg
3.1415.........346.........346............346...
but what NEVER happens with pi is that at some point it ends up repeating a pattern of digits for ever
3.14159......3444232........34222.....34222........976976976976976...
with the group 976 being repeated for ever.
Just to be clear: still talking about pi, you can get a repeat of say 3456 somewhere else along the line eg
3.1415.........346.........346............346...
but what NEVER happens with pi is that at some point it ends up repeating a pattern of digits for ever
3.14159......3444232........34222.....34222........976976976976976...
with the group 976 being repeated for ever.
Great. Though I'm still perplexed by the notion that:
"So, eventually, you can find a finite, closed form of pi...", which you explained well, yet
"...it has been proved that such a form does not exist for pi, so that there is no repeating patter of any length, anyway, for inifinitely long." How on Earth can this be logical? To have an ability to quantify an number with infinite decimal places in finite form?
I'm getting closer to understanding this, even with my inquisitive yet uneducated mind, and I'm excited. Can you refer me to any texts available online that show the proof it never repeats? Who first discovered this?
Is the problem purely practical? Pi=ratio of a circumference of a circle to it's diameter, and a circle's circumference has no beginning or end so can be expressed as having infinite length. When a finite length (diameter) has it's ratio calculated to an infinite length, the outcome is a number that is finite in size (pi
"So, eventually, you can find a finite, closed form of pi...", which you explained well, yet
"...it has been proved that such a form does not exist for pi, so that there is no repeating patter of any length, anyway, for inifinitely long." How on Earth can this be logical? To have an ability to quantify an number with infinite decimal places in finite form?
I'm getting closer to understanding this, even with my inquisitive yet uneducated mind, and I'm excited. Can you refer me to any texts available online that show the proof it never repeats? Who first discovered this?
Is the problem purely practical? Pi=ratio of a circumference of a circle to it's diameter, and a circle's circumference has no beginning or end so can be expressed as having infinite length. When a finite length (diameter) has it's ratio calculated to an infinite length, the outcome is a number that is finite in size (pi
"circle's circumference has no beginning or end so can be expressed as having infinite length".
That just doesn't make any sense to me.
A circle of diameter 1m has a circumference equal to pi metres. But it has a finite length.
Just as a circle of circumference 1m has a finite diameter of 1/pi metres
That just doesn't make any sense to me.
A circle of diameter 1m has a circumference equal to pi metres. But it has a finite length.
Just as a circle of circumference 1m has a finite diameter of 1/pi metres
I know what the problem is... i can't use the less than sign...
pi (lessthan) 3.2 therefore finite but has an infinity of decimal places because the data we input to the equation has a finite diameter and an infinite circumference? Is it our fault the maths generates this irrational number?
Infinity
pi (lessthan) 3.2 therefore finite but has an infinity of decimal places because the data we input to the equation has a finite diameter and an infinite circumference? Is it our fault the maths generates this irrational number?
Infinity
The concept of infinity is interesting, I agree. There is an infinite number of infinities.Infinity can be large or small- something can be infinitely small.
Pi has an infinite number of decimal places with no repeating pattern, but it is not the same thing as infinity- it is a number between 3.1 and 3.2
Pi has an infinite number of decimal places with no repeating pattern, but it is not the same thing as infinity- it is a number between 3.1 and 3.2
@ vascop You're right, of course, thanks for the clarification.
@ OP Well partly it's because in some senses it is better to think of the finite closed form as the "true number" and the decimal expansion as just that, i.e. an approximation that becomes arbitrarily close to the true value of the number.
So, in the classic case of 1/3 = 0.3333333333333333333... Think of the definition of the number as 1/3, which is one divided by three, rather than some infinite string of 3's after a decimal point. In fact the decimal expansion then comes "second". Certainly conceptually it should be easier to visualise what 1/3 looks like, or even what pi looks like, as it is the area of a circle of radius 1.
If you want a discussion of this I'm not aware myself of any decent sources because this is outside my area of study (I'm a physicist) -- but Wikipedia is usually a good start. And I mean that -- it's often unfairly criticised but most of the maths/ science articles are excellent.
@ OP Well partly it's because in some senses it is better to think of the finite closed form as the "true number" and the decimal expansion as just that, i.e. an approximation that becomes arbitrarily close to the true value of the number.
So, in the classic case of 1/3 = 0.3333333333333333333... Think of the definition of the number as 1/3, which is one divided by three, rather than some infinite string of 3's after a decimal point. In fact the decimal expansion then comes "second". Certainly conceptually it should be easier to visualise what 1/3 looks like, or even what pi looks like, as it is the area of a circle of radius 1.
If you want a discussion of this I'm not aware myself of any decent sources because this is outside my area of study (I'm a physicist) -- but Wikipedia is usually a good start. And I mean that -- it's often unfairly criticised but most of the maths/ science articles are excellent.
But why does this occur?
Through this thread I've come to learn that pi isn't infinite itself, as you said it's between 3.1 and 3.2.... What creates it's 'proven condition' that is must have an infinite length of non-repeating decimals? I don't think the maths is wrong. I think the linguistic expression is at fault, using the term infinity.
I'm aware I may just be cosmically stupid and looking for answers to problems we may not have the language, time or patience to satisfy me, but the thought of infinity repulses me. And if it is to be taken as a pure concept, I still struggle to grasp it's 'condition' to be 'non-repeating'...
Infinity
Through this thread I've come to learn that pi isn't infinite itself, as you said it's between 3.1 and 3.2.... What creates it's 'proven condition' that is must have an infinite length of non-repeating decimals? I don't think the maths is wrong. I think the linguistic expression is at fault, using the term infinity.
I'm aware I may just be cosmically stupid and looking for answers to problems we may not have the language, time or patience to satisfy me, but the thought of infinity repulses me. And if it is to be taken as a pure concept, I still struggle to grasp it's 'condition' to be 'non-repeating'...
Infinity
To say pi can have no repeating pattern is not correct as I have tried to point out in previous posts.
Think about it: there are ten possible digits 0,1,2,3,4,5,6,7,8,9. This means there are 10^10 possible ways of writing these digits, after that you would get repetition.
So what pi does not do is get to a point where the same set of digits in the same order keeps repeating forever. See my previous posts.
Think about it: there are ten possible digits 0,1,2,3,4,5,6,7,8,9. This means there are 10^10 possible ways of writing these digits, after that you would get repetition.
So what pi does not do is get to a point where the same set of digits in the same order keeps repeating forever. See my previous posts.
Vascop thank you for this clarity. I can accept now that it doesn't into endless repetition though it will repeat within it's structure. Is it true then that infinity doesn't truly exist here because the whole number (to the left of the decimal place) is finite? And the expression of its decimal places is unquantifiable, although we know the condition that its doesn't repeat itself, so we refer to the sequence as unending/infinitely long?
Infinity
Infinity
I'd agree with your last sentence. If 'true pi' = the ratio of a circle's circumference to its diameter, and "nearly pi" = 3.1415927...., then you can say that give any (arbitrarily tiny) number epsilon, you can write down a decimal representation of "nearly pi" such that the absolute value of (true pi - nearly pi) is less than epsilon. It may be more helpful to think of limits as the number of digits in "nearly pi" gets arbitrarily large, than to try to imagine infinity.
Infinity drove Georg Cantor nuts, after all.
Infinity drove Georg Cantor nuts, after all.
Infinity:
Here is just one possible INFINITE series for pi
pi=4/1-4/3+4/5-4/7+4/9-4/11+4/13 -.....
If you do this calculation you will find that it gives:
3.3396... approx.
If you keep going by adding and subtracting the next few terms you will find that you get a more and more accurate value for pi.
As this series is infinite in the sense that there are an infinite number of terms (eg -4/3) then this is an example of an infinite number of terms (values) which when summed result in a finite number - pi.
This was pointed out by Buenchico in the third answer to your original post.
Here is just one possible INFINITE series for pi
pi=4/1-4/3+4/5-4/7+4/9-4/11+4/13 -.....
If you do this calculation you will find that it gives:
3.3396... approx.
If you keep going by adding and subtracting the next few terms you will find that you get a more and more accurate value for pi.
As this series is infinite in the sense that there are an infinite number of terms (eg -4/3) then this is an example of an infinite number of terms (values) which when summed result in a finite number - pi.
This was pointed out by Buenchico in the third answer to your original post.
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