Connections Puzzle #328 Sat 4 May
Quizzes & Puzzles0 min ago
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...
16:27 Sun 17th Feb 2013
We don't 'allow' it to exist; we haven't got any choice, it's just there!
Pi is defined as the ratio of the circumference of a circle to its diameter; we don't choose that ratio, we just have to work with it.
An infinite number of decimal places doesn't make a number infinite (as you seem to be assuming). As an analogy, think about what happens if you add ½ to 1 and, in particular, how close you are to 2. Clearly the 'gap' is ½. Then add ¼ to the 1½ you've already got and consider your answer in terms of how far away it is from 2. The gap is now ¼. Then add 1/8. The gap is now 1/8, and so on. So, as you continue to add terms which are all half of what you previously added you keep getting closer to 2 but you never quite get there (because the gap is always equal to the last term you added). You could keep going for a billion, billion, billion terms (and more) and you still wouldn't get to 2. So adding an infinite number of terms doesn't create an infinite number.
Chris
Pi is defined as the ratio of the circumference of a circle to its diameter; we don't choose that ratio, we just have to work with it.
An infinite number of decimal places doesn't make a number infinite (as you seem to be assuming). As an analogy, think about what happens if you add ½ to 1 and, in particular, how close you are to 2. Clearly the 'gap' is ½. Then add ¼ to the 1½ you've already got and consider your answer in terms of how far away it is from 2. The gap is now ¼. Then add 1/8. The gap is now 1/8, and so on. So, as you continue to add terms which are all half of what you previously added you keep getting closer to 2 but you never quite get there (because the gap is always equal to the last term you added). You could keep going for a billion, billion, billion terms (and more) and you still wouldn't get to 2. So adding an infinite number of terms doesn't create an infinite number.
Chris
Cannot answer your question directly, except to point out, as someone else has, that it is an infinite series after the decimal point, so has no influence over other whole numbers.
Thought you might find this link interesting though. The use of Pi in TV shows and movies, with clips. I quite like the one from "Person of Interest", with one of the protaganists posing as a substitute teacher and explaining the magic of Pi :)
http://
Thought you might find this link interesting though. The use of Pi in TV shows and movies, with clips. I quite like the one from "Person of Interest", with one of the protaganists posing as a substitute teacher and explaining the magic of Pi :)
http://
I'm unsure what you are getting at when you say "allow". Human beings didn't design the universe, we merely observe and try to understand it.
As for saying 4,5,6,7 wouldn't exist, how do you work that out ?
It's not infinitely big, it is an infinitely long string of digits as you try for precision.
You just have to accept the concept since the human mind is not infinitely capable.
As for saying 4,5,6,7 wouldn't exist, how do you work that out ?
It's not infinitely big, it is an infinitely long string of digits as you try for precision.
You just have to accept the concept since the human mind is not infinitely capable.
OK the term you want isn't infinite but *irrational*
an irrational number is one that can't be expressed by a simple fraction like 1/2 or even 1/3
That means that the numbers in the decimal point continue forever without pattern
It's not that easy to prove why - it was 1761 before someone proved that was the case wikipedia has an overview but it's somewhat complex - If it was easy it wouldn't have taken 3,000 years to get to!
http://
As a matter of interest there is the Fynman point in pi which is a sequence of 6 9s 762 decimals in ...4999999837...
It got it's name because Richard Fynman joked that he'd like to be able to memorise PI to this point so he could quote it and then say " 4999999 and so on" implying that it continued like that! - a bit of scientific humour!
an irrational number is one that can't be expressed by a simple fraction like 1/2 or even 1/3
That means that the numbers in the decimal point continue forever without pattern
It's not that easy to prove why - it was 1761 before someone proved that was the case wikipedia has an overview but it's somewhat complex - If it was easy it wouldn't have taken 3,000 years to get to!
http://
As a matter of interest there is the Fynman point in pi which is a sequence of 6 9s 762 decimals in ...4999999837...
It got it's name because Richard Fynman joked that he'd like to be able to memorise PI to this point so he could quote it and then say " 4999999 and so on" implying that it continued like that! - a bit of scientific humour!
quite right Ludwig
There was an attempt in Indiana in1897 to legislate on this very subject.
http://
There was an attempt in Indiana in1897 to legislate on this very subject.
http://
Thank you all for your input, I've marked the answer i found most useful. Many of you refer to my term "allow." I understand mathematics as the language of the universe, physics, chemistry and all sciences, and I would agree to the idea that we 'discovered it', opposed to 'inventing it', so mathematicians (seemingly) shrug off the consequences of the term 'infinity' as a mere oddity of the fact that is reality governed by the physical laws... I am extremely uncomfortable with the idea that we cannot quantify this number, ESPECIALLY as the neat, ordered world of maths and theorems and formula and equations ALL demand a quantifiable set of perameters to bring meaning, context and to make physically applicable the otherwise lifeless chalk dust scribbled on a board...
I fear I've rambled here but does anyone understand my problem. I have given great thought to the notion of Infinity, and I am humble enough to admit that it may be impossible for my human brain to fathom, however it is my brain that refuses to accept this.
I hope this thread interests you enough for a reply.
Infinity
I fear I've rambled here but does anyone understand my problem. I have given great thought to the notion of Infinity, and I am humble enough to admit that it may be impossible for my human brain to fathom, however it is my brain that refuses to accept this.
I hope this thread interests you enough for a reply.
Infinity
My apologies for a third post. I must learn to be more concise. When I said "how can we allow this number to exist...", I should really have said, how do we allow infinity in this number to exist. When physicists resort to using infinity, they're work is never taken seriously, as it is seen by many as a cheat to make equations fit....
Infinity
Infinity
I don't always get what it is your are getting at. But would just like to say that although renormalization doesn't sit comfortably with everyone, I don't think it means work using it isn't taken seriously; just that there's a hope one can eventually find a better, mathematically rigorous path, to the same result.
The infinity bit, I suggest, is a property of the decimal numbering system not of pi. The ratio of the circumference to the diameter is an exact and finite amount. As is 1/9th which in decimal terms is 1.1 recurring, that is to say another infinite series describing a finite amount, which was the basis for Zeno's "proof" that Achilles could never overtake the tortoise..
Late to the game, but, using the generally accepted definition of "infinite", no number, including our onomastically challenged Pi can be infinite, simply because it has a beginning. Nothing infinite can have a beginning or end.
Fact is, Pi exists only (as already explained by both Chris and JTP) as a convenient expression of a distance. I can take a piece of dowel, lay it with each end touching the circumference of a given circle and thereby obtain an accurate mesurement of its diameter. Inconvenient, granted, but it's only when a ratio is expressed that anyone actually requires Pi... or if it becomes neccessary to define the length of dowel as it relates to the circle...
Fact is, Pi exists only (as already explained by both Chris and JTP) as a convenient expression of a distance. I can take a piece of dowel, lay it with each end touching the circumference of a given circle and thereby obtain an accurate mesurement of its diameter. Inconvenient, granted, but it's only when a ratio is expressed that anyone actually requires Pi... or if it becomes neccessary to define the length of dowel as it relates to the circle...
Thanks all for your contributions.
I think Clanad has hit the nail on the head though:
"Late to the game, but, using the generally accepted definition of "infinite", no number, including our onomastically challenged Pi can be infinite, simply because it has a beginning. Nothing infinite can have a beginning or end."
My quandary exactly. For something to be infinite, it must have always and will always be total. If something has a beginning, it sacrifices its ability to be or become infinite because it was necessarily finite to start with.
As you can guess from my chosen username, I am driven mad with the comfort our greatest minds (many of whom have spoken on this thread) have with the notion of infinity. I refuse to accept it is a failure of my mind to grasp the concept of an infinite space or number, alternatively I seek to eradicate it from modern mathematics and sciences. I can accept that an equation (e.g. 22/7) returns a result that appears so irrational that 'the only easy' explanation is to resort to infinity but maths is all about logic, and I dare someone to logically justify the existence of a never-ending number derived from a perfectly logical equation...
Clanad has come the closest. Thank you for your input sir.
Infinity
I think Clanad has hit the nail on the head though:
"Late to the game, but, using the generally accepted definition of "infinite", no number, including our onomastically challenged Pi can be infinite, simply because it has a beginning. Nothing infinite can have a beginning or end."
My quandary exactly. For something to be infinite, it must have always and will always be total. If something has a beginning, it sacrifices its ability to be or become infinite because it was necessarily finite to start with.
As you can guess from my chosen username, I am driven mad with the comfort our greatest minds (many of whom have spoken on this thread) have with the notion of infinity. I refuse to accept it is a failure of my mind to grasp the concept of an infinite space or number, alternatively I seek to eradicate it from modern mathematics and sciences. I can accept that an equation (e.g. 22/7) returns a result that appears so irrational that 'the only easy' explanation is to resort to infinity but maths is all about logic, and I dare someone to logically justify the existence of a never-ending number derived from a perfectly logical equation...
Clanad has come the closest. Thank you for your input sir.
Infinity
Despite Clanad's valiant attempt, I cannot make the slightest sense of your ramblings about pi. As others have explained pi is what it is. It is not infinite; it is irrational.
If you're that obsessed with whole numbers then choose a counting system based on pi instead of on ten. Then pi will = 1. It'll make a dog's dinner of decimal numbers, mind you.
If you're that obsessed with whole numbers then choose a counting system based on pi instead of on ten. Then pi will = 1. It'll make a dog's dinner of decimal numbers, mind you.
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