An Infinite Number That's Smaller Than Four

[I_Hate_Infinity]

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

[I_Hate_Infinity]

Pi is not true issue. The irrationality lies with it being defined as having an infinite amount of non-repeating decimal places. (Which is a contradiction in that if IT DOES have infinite decimals then it must repeat itself.... Infinitely) Infinity is a deceptive tool that creates false closure to mathematical disaster.

Physicists talk about the conditions for the singularity, the creation of the Universe. Modern physics can track the sequence in reverse from today, way back to when the Universe was a fraction of a second old. But any further back in time, the laws of physics simply arrive at infinity. Gravity becomes infinite, density of matter becomes infinite, and that's when we know something has gone wrong.

I hope this clarifies why I was looking for answers to why Pi is classed as infinite... Hap hazardly

Infinity

[factor-fiction]

Sorry, I_hate_infinity but I still don't see your point.
What is ⅓ as a decimal? It's 0.333333... with an infinite number of 3s after the decimal point. so what is three thirds- is it 0.999999.... with an infinite number of 9s after the decimal point? Or is it 1

What about 1/7. As a fraction it's 0.14285714285.... and goes on forever. Yet 7/7 is 1.

An infinite number of decimal place doesn't make the number bigger or infinite

[Old_Geezer]

I do not understand the claim that nothing infinite can have a beginning or an end. Surely the defining thing is it can't have both. One of the two is fine. The set of positive integers has a beginning in as much as there is a lowest value one, but it still an infinite number of members as far as I am aware.

[Clanad]

I'm not sure, nor can anyone be, that "... any further back in time, the laws of physics simply arrive at infinity..." The farthest "back" we can measure is Planck Time[i (alternately, [i]Planck Distance] ).

JTP holds the position (at least as far as I interpret his expressed position; I would not put words in his pixels)) that there is nothing "before" that point. However, it seems to me to be conceptually relevant that being able to measure time to Planck Time implies momentum. For any quantum particle to have arrived at Planck and therefore measurability, simply means that it was moving prior to that point; we simply cannot conceive of the theorem and attendant equation to explain that acceleration. But, since the immeasurable was moving, indicates a beginning and therefore, again, lacks the quality of infinite.

[ludwig]

// 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 //

I always go straight in with the tape measure.

[chakka35]

You still seem unable to grasp that the fact that pi needs an infinite number of digits to express it does not mean that pi itself is infinite. It is still less than 3.2.

[Prudie]

Pi is not infinite, it's an irrational as has been said. It is a naturally occurring ratio (like the golden one). bear in mind it's only 3.1417... in Base 10. In other bases (although I haven't investigated) it would look different - binary for example. Consider a right angle triangle. The hypotenuse is always the square root of the sum of the squares of the other 2 sides. This will often not be a number with an ending of decimal places yet clearly the hypotenuse itself is a finite length.

[mibn2cweus]

Someone once said, "You can't have your pi and eat it too" . . . or was that cake?

Oh well, what's the point in racking your brains over the precise meaning of something that is inherently . . . indefinite. Pi is limited by its practicality.

[vascop]

Infinity://The irrationality lies with it being defined as having an infinite amount of non-repeating decimal places. (Which is a contradiction in that if IT DOES have infinite decimals then it must repeat itself.... Infinitely) Infinity is a deceptive tool that creates false closure to mathematical disaster. //
You've got your facts wrong here. The actual statement is that pi never settles into a permanent repeating pattern of digits. A completely different thing.
By the way we need infinity at least as much as we need zero.
As you have probably already realised I completely disagree with you about infinity. It's a very necessary and useful concept in mathematics.

[vascop]

Also meant to say what is meant by that phrase about repeating:
So pi never gets to a point like this
3.1415...........4567456745674567.... repeating 4567 say, for ever.

[jim360]

Another thing -- it's not "believed to be infinitely long and never repeating. That was proved over a century ago. This entirely lies within the accepted framework of Mathematics. Nor must it repeat infinitely if it has an infinity of decimal digits. The nature of infinity, at least outside the realms of what it means for physics, is a well-understood concept.

[mibn2cweus]

Considering the significance of the 'final' digit of pi, perhaps your making a mountain out of a mole hill . . . to put it mildly.

[bert_h]

There's a vast amount of misunderstanding caused by people who aren't (yet) able to disentangle 'a number' from 'the representation of that number in any particular system of notation'. You've been - incorrectly - supposing from the infinite length of the decimal representation of Pi that there's something infinite about the number itself. There isn't, any more than there is about the square root of 2, which we represent with 1.41421356... which goes on and on and on but, like the representation of Pi, never repeats and never ends.

[I_Hate_Infinity]

I'm glad I've attracted such people with great knowledge of mathematics and the concept of Pi and it's useful applications in measurement and theory. I've taken on board your comments and have reworked my question to better achieve an answer to my dilemma.

The dictionary I have at my disposal, 'Chambers Twentieth Century Dictionary' (with suppliment, W&R Chambers Ltd 1977), defines infinite as:
"adj. withouth end or limit: greater than any quantity that can be assigned (math):extending to infinity: vast: in vast numbers: inexhaustible: infinitated"

If this is the definition of the term infinite by the chronicles of our language as our standard explanation of its meaning, then why is it NEVER repeating? An inexhaustible chain of numbers must, by the dictionary's definition, have all possible combinations of repeating patterns of numbers which are themselves repeated infinitely... Right?

I accept Pi is not an infinite number in itself, as chakka35 said "pi is less that 3.2." But is there a different definition of 'infinite' for a mathematician than for an etymologist? Is it just a concept inexpressible by digits or words, or is it actually a reality of nature which I'm just struggling to understand?

Please continue to express your thoughts, you've been most useful so far...

Inifnity

[jim360]

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 yourself. However as I understand it every infinite recurring decimal expansion can be written in a closed form as a fraction, e.g 1/3 = 0.3333... and 22/7 = 3.142857142857... . Then if you can prove that a particular number cannot be written a/b for two whole numbers a and b, then it never repeats. And for pi, such a proof exists (Lambert, 1761).

Infinity, anyway, has several related meanings in mathematics that are often qualified. For example, things can be "countably" or "uncountably" infinite -- where the difference is (VERY) loosely related to the fact that if you counted 1, 2, 3, 4... and so you would never be able to stop counting, but in a sense you are "counting" your way to infinity. On the other hand, counting all the real numbers is impossible -- you would die of old age before you had even got past 0.0000000000...

So infinity anyway is weird, and seems to me to split into several different types. The infinity in pi is one of having no closed fraction form, so that its decimal expansion never ends and never repeats.

[mibn2cweus]

Apparent contradictions invariably arise as a consequence of an ambiguous definition. Pi, is not a number but rather a term denoting the concept of the relationship that exists between a circle and it radius. The number given for pi (or the sqrt of 2) is to represent its quantitative value to a specified number of decimal places. By their very nature, there is no such animal as an infinitely precise value that can be assigned to irrational numbers. It is not rational to dictate that reality can not be 'messy'.

[I_Hate_Infinity]

Right, so there are different meanings to the term?
"The infinity in pi is one of having no closed fraction form..."

[jim360]

@ Hate Infinity : Yes, there are different meanings, at least to me. That said I'd be careful to check things I say for yourself, I'm not sure I've quite got it all right!

That said, @mibn, I have to disagree with you. Pi IS a number, included in the Real numbers. It's just that any decimal expansion of it is number exactly equal to the "true" value of pi, because its exact value cannot be written down in any finite form. But it's still a number, and you just have to extend the definition of numbers to include the weird ones like pi, sqrt(2), and e = 2.718...

[I_Hate_Infinity]

Why is it 'never repeating' then? Is it that the decimals never fall into a pattern which recurrs for ever e.g. 3.14 ... 357 357 357 -- for ever? Or does it mean that sequence never falls into any pattern of repetition? Surely an infinite amount of possible combination predicts that it must infinity repeat all possible lengths of combinations of numbers that have been expressed before (counting away from 3.14). Isn't maths definition of infinite=totality? Every possible expression, repeatedly expressed an infinite number for times?

Infinity

[jim360]

I'm not going to be able to produce a full proof here but take the case of a number going, say, 3.141 ...357 357 357 ... where after some finite length of digits the pattern finally emerges and you get a repeating loop.

Now multiply this be 1,000, and you would get 3141. ... 357 357 357 ...

Subtracting the two gives say 3138. ... Note that there are as many 357's at the end as there were before, because there is an infinite number of them. And more importantly all of these cancel each other so that now our decimal expansion comes to an end somewhere.

So we have the formula: (1000 -1)*pi = 999 pi = 3138. ... Coming to a stop. You could then write pi in the form (3138. ... ) / 999. This can become a fraction if you multiply top and bottom by enough ten's to move all of the decimal places to the left of the decimal point.

So, eventually, you can find a finite, closed form of pi. This argument could extend to any repeating pattern at the end of pi, of any length. And the point is that this leads to an equation that says that pi = a/b, for possibly stupendously big but nevertheless finite numbers. And 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.

Yes, at some points patterns emerge, e.g there is a 01234567890 somewhere in the string. But no, these patterns are never more than one-offs, and the cycle of digits goes on and on, never repeating itself overall. Proven fact.