Question on Infinite

[Jet2]

Can any maths experts explain to me about this?

Are infinite large and infinite small the same?

 

Jet2

[sanctus]

sorry never heard of infinite large or infinite small, but since it is infinite it must be the same. Can you elaborate a bit more?

[Pyrotex]

Can any maths experts explain to me about this?

Are infinite large and infinite small the same?

Jet2 :hihi:

I can explain this.

 

Mathematicians and/or Physicists find themselves dealing with a new phenomenon, or an old one from a new direction, and they coin a word for it. The word (like "infinite") represents (to them) a very specific, well-defined concept.

 

Then the word leaks out into common usage. And ordinary, non-technical folks sling the word around in any fashion that sounds neat or useful to them. The word finds a "niche" in common usage far removed from its original, pristine concept. Sometimes it gets merged with other words in phrases that are so far removed from its origins that it becomes unrelated.

 

"Infinite large" or more properly, "infinitely large" is redundant. It just means "Hey, Dude, it's bigger than... than... than... you know, just bigger than I can even think of!"

 

"Infinite small" or more properly, "infinitely small" is self contradictory and its ONLY meaning is whatever context it is found in. It might mean as small as an amoeba/atom/electron [pick one], or just smaller than one can imagine. Neither phrase has any technical, mathematical meaning whatsoever. You will not find them in math books.

[Turtle]

I can explain this.

 

Mathematicians and/or Physicists find themselves dealing with a new phenomenon, or an old one from a new direction, and they coin a word for it. The word (like "infinite") represents (to them) a very specific, well-defined concept.

 

Being Low Shoe, I'm the second to drop. :hihi::hihi: The specific concept Pyrotex refers to is 'growth without bound'. :hihi:

 

Can any maths experts explain to me about this?

Are infinite large and infinite small the same?

 

In this context all that is the same mathematically speaking is 'growth without bound', not the value of what is growing. Similar to referring to the similarity of all the people in a foot race as 'runners', but without regard to individual differences. :hihi: :hihi:

[CraigD]

Can any maths experts explain to me about this?

Are infinite large and infinite small the same?

I’d say “no”, but note that one can be written as an arithmetic expression using terms of the other.

 

Since the most popular numbers have a sign – positive or negative – you can speak of [math]+ \infty[/math] or [math]- \infty[/math]. Since most of the most interesting math about involving infinity tends to consider it a cardinal number – the count of elements of a set – most mathematicians consider [math]+ \infty[/math] and [math]- \infty[/math] to be just slightly different “flavors” of the same concept.

 

“Infinitely small” might be taken as a synonym for “infinitesimal”, and written something like [math]\frac1{\infty}[/math]. In ordinary language, an infinitesimal is a number closer to zero than any other, yet not equal to zero. They’re a very important concept in calculus – you might reasonably call them “the essence of calculus”, where they’re written (sort of) with “d”, as in [math]dX[/math] or [math]\frac{dY}{dX}[/math]. Infinitesimals, and whether it’s a good idea to use the word or think of them as somehow conceptually real, have provoked centuries of discussion among mathematicians.

 

Infinity is one of the central, deep concepts of math and philosophy, so doesn’t lend itself to the sort of short answers I’ve given above. In my experience, you have to read and think about it a good bit to begin appreciating its history and significance – and will find doing so a pleasant and rewarding experience - though YMMV.

[Jay-qu]

sorry never heard of infinite large or infinite small, but since it is infinite it must be the same. Can you elaborate a bit more?

I could explain infinite to you, but it would just take forever..

:hihi:

[Fatstep]

Do you mean like infinitely close to zero(infinitely small), and infinitely far from zero(infinitely large)?

[Jet2]

I could explain infinite to you, but it would just take forever..

:hihi:

 

This is great!

Thanks Jay-qu.

I can wait forever.

 

:hyper:Jet2

[Jet2]

......

Infinity is one of the central, deep concepts of math and philosophy, so doesn’t lend itself to the sort of short answers I’ve given above. In my experience, you have to read and think about it a good bit to begin appreciating its history and significance – and will find doing so a pleasant and rewarding experience - though YMMV.

 

Thanks CraigD!

Just found a book called "The Infinite Book - A short guide to the boundless, timeless and endless". You are right, it's very interesting indeed.

 

Jet2:)

[Jay-qu]

This is great!

Thanks Jay-qu.

I can wait forever.

 

:hyper:Jet2

well that backfired :doh: I dont have time to ;)

[von Faulkenstein]

Could the infinite small contain the infinite large?;)

[Tormod]

Would the infinitely stupid contain the infinitely smart? ;)

 

Here's a good article on Cantor, who basically revolutionized the concept of infinities:

Georg Cantor - Wikipedia, the free encyclopedia

 

...and an article on actual infinities:

Actual infinity - Wikipedia, the free encyclopedia

[Qfwfq]

Are infinite large and infinite small the same?

Infinitely small is just a loose way of saying infinitesimal.

 

In calculus, a term is said to be infinitesimal in a given limit if this limit is zero. For the same limit, the reciprocal of the term is infinity. Infinite cardinality is yet another definition; a set has infinite cardinality if and only if a bijection is possible between it and some proper subset of it.

[Jet2]

If the limit of infinitely small is infinity, then does zero ever exist?

 

Jet2

[von Faulkenstein]

Could one consider the infinite number of transcendental numbers between two fractions--even if these two fractions are extremely close together?

[ughaibu]

This page is pretty good: Peter Suber, "Infinite Sets"

[Jay-qu]

yes - there are just as many numbers between 0-1 as there are between 0-1,000,000,000,000,000