Infinity – often denoted by the symbol , ∞, or (in unicode) ∞ – represents something that is boundless or endless, or else something that is larger than any real or natural number.[1] Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions among philosophers. In the 17th century, with the introduction of the infinity symbol and the infinitesimal calculus, mathematicians began to work with infinite series and what some mathematicians (including l'Hôpital and Bernoulli) regarded as infinitely small quantities, but infinity continued to be associated with endless processes As mathematicians struggled with the foundation of the calculus, it remained unclear whether infinity could be considered as a number or magnitude and, if so, how this could be done.[2]
At the end of the 19th century, Georg Cantor enlarged the mathematical study of infinity by studying infinite sets and infinite numbers, showing that they can be of various sizes.For example, if a line is viewed as the set of all of its points, their infinite number (i.e. the cardinality of the line) is larger than the number of integers.[6] In this usage, infinity is a mathematical concept, and infinite mathematical objects can be studied, manipulated, and used just like any other mathematical object.
The mathematical concept of infinity refines and extends the old philosophical concept, in particular by introducing infinitely many different sizes of infinite sets. Among the axioms of Zermelo–Fraenkel set theory, on which most of modern mathematics can be developed, is the axiom of infinity, which guarantees the existence of infinite sets.The mathematical concept of infinity and the manipulation of infinite sets are used everywhere in mathematics, even in areas such as combinatorics that may seem to have nothing to do with them. For example, Wiles's proof of Fermat's Last Theorem implicitly relies on the existence of very large infinite sets for solving a long-standing problem that is stated in terms of elementary arithmetic.
In physics and cosmology, whether the Universe is infinite is an open question.
| Infinity ... | |
| ... it's not big ... | |
| ... it's not huge ... | |
| ... it's not tremendously large ... | |
| ... it's not extremely humongously enormous ... | |
| ... it's ... |
How strange is that? Can you imagine? Mm. I was thinking about the passage of time today and how far the universe can possible extend in terms of time. Now of course, if you believe in the explanation that science provides, about how time and space came about by the big bang or quantum fluctuation or any other possible means that is yet to be hypothesized or thought up of, the question then lies was there really ever a beginning before our own universe's idea of time came about? Was there time before our version of time came about, or at least some sense of measurement of a start and end, or something?
Maybe it is hard for us mortals to grasp because we see everything in a limited scope. We see it limited within our own time span, within our own observation of the world, etc, we see time as something that we can generally see as consistent. And we believe, or at least most of us mortals do, that if we had a beginning, if the stars had a beginning, if everything had a beginning, then the universe too must have a beginning.
But what if that isn't true? What if there is something beyond that beginning? Beyond the concept of time and nothing. If there is something that transcends everything we think about.
If the universe is infinite, with no ends but also no beginning, then what do we call this?
Can you imagine the possibilities of that universe though? That world? The rules of laws that dominate that place?
It wouldn't be a world of constants but rather a world of uncertainty, one that breaks the fathom of what we can imagine.
Look. We apply time as a constant. A constant where everything can then be defined into a state. (Generally, or at least through how I think).
We believe in life and death, the beginning and end, etc.
But if infinity had no beginning, then what is there?
So the only idea I can grasp with my mind was that it was stuck in a place in between not existing and a beginning, a world of uncertainty.
A world where everything is stuck within this hypothetical space of nothing and something at the same time.
Urghh. I can only imagine such a world. A world that exists in between that space is certainly beyond fathomable for me right now, and thinking about it is causing my head to expand trying to figure it out.
"Infinity is the idea of something that has no end.
In our world we don't have anything like it. So we imagine traveling on and on, trying hard to get there, but that is not actually infinity.
So don't think like that (it just hurts your brain!). Just think "endless", or "boundless".
If there is no reason something should stop, then it is infinite."
2)Infinity doesn't grow. ... However, this independent of the mathematical Platonic concept of infinity in the sense that there needs to be no relationship between mathematical ideas and the physical world. On a side note, time might not be infinite anyway; it may be that nothing is infinitely large or small.
Infinity is not "getting larger", it is already fully formed.
Sometimes people (including me) say it "goes on and on" which sounds like it is growing somehow. But infinity does not do anything, it just is.
3)Yes! It is actually simpler than things which do have an end. Because when something has an end, we have to define where that end is.
Example: in Geometry a Line has infinite length.
A Line goes in both directions without end.
When there is one end it is called a Ray, and when there are two ends it is called a Line Segment, but they need extra information to define where the ends are.
So a Line is actually simpler then a Ray or Line Segment.
More Examples: | |
{1, 2, 3, ...} | The sequence of natural numbers never ends, and is infinite. |
 | OK, 1/3 is a finite number (it is not infinite). But written as a decimal number the digit 3 repeats forever (we say "0.3 repeating"): 0.3333333... (etc) There's no reason why the 3s should ever stop: they repeat infinitely. |
| 0.999... | So, when we see a number like "0.999..." (i.e. a decimal number with an infinite series of 9s), there is no end to the number of 9s. You cannot say "but what happens if it ends in an 8?", because it simply does not end. (This is why 0.999... equals 1). |
| AAAA... | An infinite series of "A"s followed by a "B" will NEVER have a "B". |
 | There are infinite points in a line. Even a short line segment has infinite points. |
Big Numbers
There are some really impressively big numbers.
A Googol is 1 followed by one hundred zeros (10100) :
10,000,000,000,000,000,000,000,000,000,000,000,
000,000,000,000,000,000,000,000,000,000,000,
000,000,000,000,000,000,000,000,000,000,000
A Googol is already bigger than the number of elementary particles in the known Universe, but then there is the Googolplex. It is 1 followed by Googol zeros. I can't even write down the number, because there is not enough matter in the known universe to form all the zeros:
10,000,000,000,000,000,000,000,000,000,000,000,000, ... (Googol number of Zeros)
And there are even larger numbers that need to use "Power Towers" to write them down.
For example, a Googolplex can be written as this power tower: 
That is ten to the power of (10 to the power of 100),
But imagine an even bigger number like 
(which is a Googolplexian).
And we can easily create much larger numbers than those!
Finite
All of these numbers are "finite", we could eventually "get there".
But none of these numbers are even close to infinity. Because they are finite, and infinity is ... not finite!
Using Infinity
We can sometimes use infinity like it is a number, but infinity does not behave like a real number.
To help you understand, think "endless" whenever you see the infinity symbol "∞":
Undefined Operations
All of these are "undefined":
| "Undefined" Operations |
|---|
| 0 × ∞ |
| 0 × -∞ |
| ∞ + -∞ |
| ∞ - ∞ |
| ∞ / ∞ |
| ∞0 |
| 1∞ |
Infinite Sets
If you continue to study this subject you will find discussions about infinite sets, and the idea of different sizes of infinity.
That subject has special names like Aleph-null (how many Natural Numbers), Aleph-one and so on, which are used to measure the sizes of sets.
For example, there are infinitely many Whole numbers{0,1,2,3,4,...},
But that is an advanced topic, and goes beyond the simple concept of infinity we discuss here.watch a video:CLICK HERE
Conclusion:
0 Comments:
Post a Comment