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Bass
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Infinite = 1
« on: 2006-06-20 12:02:28 » |
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Indeterminations of Infinite
infinate - infinate = Indeterminable ?
infinate / infinate = Indeterminable ?
infinate * 0 = Indeterminable ?
1 ^ infinate = Indeterminable ?
Well, mathmaticians say (and teach) that Infinite minus Infinite is indeterminable (and a bunch of other indeterminations as the ones listed above).
I think (to be true: I know I know) that's crap. For two reasons:
Simple reason: Anything minus itself is cero. (as anything divided by itself is one, and anything multiplied by cero is cero, and one to the power of no matter what, will be one). I think that those who think otherwise, haven't really understood what minus means, nor multiply, nor divide... and of course not Infinite.
Simplier reason (harder to understand though): Infinite equals one, but not completely as one as a number, but as a whole. So, Infinite is a whole. So Infinite is the times that that whole can be splitted into.
I would say that Universal Maths don't need to go beyond 1. Ever.
By the way, of course there is -1. But that's 1 too.
So, as I know they teach that at the University, I would like to know how many people think as I do, and how many are wrong (just kidding)... and how many think wrong... God I just can't stop saying they're wrong! They are still wondering what cero divided by cero would be...
if 22/7 equals pi, then 3 1/7 also equals pi which would make 1/7 the unresolved expansive component theoretically equalling infinity.
lets test it 1/7 =0.142 857 142 857 143 000 recurring
Hmm 0 recurring sounds like infinity to me, so it must be constent and equal a constant of 1.
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Hermit
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Re:Infinite = 1
« Reply #1 on: 2006-06-20 14:53:06 » |
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Welcome to the CoV Bass
I think you are suffering from some (not entirely uncommon) confusions.
Firstly, infinity is not a number. Infinity can be a mapping, an algorithm or process, a limit, a destination, even a singularity. Infinity can be a lot of things, but is never a number or equivalent to a number. We can assign infinity a symbol, and this can make it look like a number. Confusion results when you try to treat it like one. So all those natty little rules you attempted to apply to infinity don't work. Because those rules are part of the identity of the operators - which relate to normal numbers. Not to a symbol conveniently representing infinity (With acknowledgement to the Prince previously known as Symbol).
Secondly, infinity is not just a single class of thing. It can mean different classes of things in different places, and these classes of things may have very different properties. For example there are countable infinities (like the set of whole numbers), and uncountable infinities (like the set of rational numbers). These classes of infinity have rather different properties.
Thirdly, if you ever get to studying calculus, you will discover that indeterminate doesn't mean that an answer cannot be found, but rather that the answer depends on what the question is. This means that we can't say what an indeterminate expression means unless we are given a specific problem to solve. Most examples I can think of require calculus, but if you knew calculus you wouldn't be having this problem. Perhaps the easiest example is to consider the simple case of 1/0 which is an indeterminate value. If we examine the limit of the sequence of the recipricals of the recipricals of the whole numbers, i.e. 1/(1/2), 1/(1/3), 1/(1/4) etc, the denominator tends towards zero as the terms of the sequence tend towards infinity. Now consider the limit of the sequence of the recipricals of the negative recipricals of the whole numbers, i.e. 1/(-1/2), 1/(-1/3), 1/(-1/4). This sequence also tends towards zero, but as the terms of the sequence tend towards negative infinity. So we can say that the ultimate term of either sequence tends towards 1/0, but this can resolve to either positive or negative infinity depending on which sequence you are evaluating, or as I said before, the value of the answer depends the question you ask. Or, more formally, the value of 1/0 yields an indeterminate result. (BTW welcome to the joys of calculus).
Finally, 22/7 is not "equal to pi." If it were, pi would not be an irrational number. Rather 22/7 is an approximation to pi. And quite a good one for all practical purposes, as in a decimal representation this results in 3.142857142857142857142857142857 where pi is more accurately approximated by a decimal representation of 3.14159265358979323846264338327950288419716939937510... (non-recurring). Or a difference of around .001264489 (and if you remember to subtract .00126 from 22/7 you can get to within -.0000044 or so of the true approximation for pi - which is even better and still trivial to remember). But it certainly invalidates your treatment of the irrational approximation which is pi as something which can be described as a representational rounding issue.
I hope this makes some sense.
Hermit
Calculations performed with bc set to a scale of 50.
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Fox
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Re:Infinite = 1
« Reply #2 on: 2006-06-20 15:55:37 » |
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Quote from: Bass on 2006-06-20 12:02:28 mathmaticians say (and teach) that Infinite minus Infinite is indeterminable (and a bunch of other indeterminations as the ones listed above).
I think (to be true: I know I know) that's crap. |
Welcome, Bass.
I weyken you’re confused by the notation commonly used to represent infinity, ?. By convention, this symbol usually refers to “the count (cardinality) of any set for which any member can be assigned (mapped to) a natural (1, 2, 3 …) number”. ? doesn’t refer to a specific number, but one of many numbers.The statement“? - ? is indeterminable”,is similar to the statement“Give A, B are members of the set of natural numbers, A - B is indeterminable”.The statement“? - ? = ?”,is similar to the statement“A - B = C, where C is a member of the set of integers” (the cardinality of the set of integers is the same transfinite number as the cardinality of the set of natural numbers, named “Aleph-null” in the late 19th century by Cantor.
In Math, the study of infinite, also called transfinite, numbers, is interesting mostly in deciding if a particular infinite set can be mapped to another. It’s more about mapping algorithms than arithmetic.
Quote from: Bass on 2006-06-20 12:02:28 if 22/7 equals pi, then 3 1/7 also equals pi which would make 1/7 the unresolved expansive component theoretically equalling infinity.
lets test it 1/7 =0.142 857 142 857 143 000 recurring
Hmm 0 recurring sounds like infinity to me, so it must be constent and equal a constant of 1.
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Allow me to try and elucidate the circumstance you propose.
Any rational number in any base system has a repeating part (when the repeating sequence of digits consists of exactly one zero, we call it a terminating, but this is just a convention - those zeros do repeat forever). When the remainder of the numerator divided by the denominator is one, the repeating sequence always begins in the first place after the point.
It’d be no fun to leave this bit of textbook Math as is, so consider this:Here is 22/7 long divided by the first 99 bases (which I worked with some time ago).
([]s surround the repeating portion, ()s surround numerals too big to be represented in base 10):
Code: (some food for thought)
3.[001] base 2, 3 repeating digits
3.[010212] base 3, 6 repeating digits
3.[021] base 4, 3 repeating digits
3.[032412] base 5, 6 repeating digits
3.[05] base 6, 2 repeating digits
3.1 base 7, 1 repeating digits
3.[1] base 8, 1 repeating digits
3.[125] base 9, 3 repeating digits
3.[142857] base 10, 6 repeating digits
3.[163] base 11, 3 repeating digits
3.[186(10)35] base 12, 6 repeating digits
3.[1(11)] base 13, 2 repeating digits
3.2 base 14, 1 repeating digits
3.[2] base 15, 1 repeating digits
3.[249] base 16, 3 repeating digits
3.[274(14)9(12)] base 17, 6 repeating digits
3.[2(10)5] base 18, 3 repeating digits
3.[2(13)(10)(16)58] base 19, 6 repeating digits
3.[2(17)] base 20, 2 repeating digits
3.3 base 21, 1 repeating digits
3.[3] base 22, 1 repeating digits
3.[36(13)] base 23, 3 repeating digits
3.[3(10)6(20)(13)(17)] base 24, 6 repeating digits
3.[3(14)7] base 25, 3 repeating digits
3.[3(18)(14)(22)7(11)] base 26, 6 repeating digits
3.[3(23)] base 27, 2 repeating digits
3.4 base 28, 1 repeating digits
3.[4] base 29, 1 repeating digits
3.[48(17)] base 30, 3 repeating digits
3.[4(13)8(26)(17)(22)] base 31, 6 repeating digits
3.[4(18)9] base 32, 3 repeating digits
3.[4(23)(18)(28)9(14)] base 33, 6 repeating digits
3.[4(29)] base 34, 2 repeating digits
3.5 base 35, 1 repeating digits
3.[5] base 36, 1 repeating digits
3.[5(10)(21)] base 37, 3 repeating digits
3.[5(16)(10)(32)(21)(27)] base 38, 6 repeating digits
3.[5(22)(11)] base 39, 3 repeating digits
3.[5(28)(22)(34)(11)(17)] base 40, 6 repeating digits
3.[5(35)] base 41, 2 repeating digits
3.6 base 42, 1 repeating digits
3.[6] base 43, 1 repeating digits
3.[6(12)(25)] base 44, 3 repeating digits
3.[6(19)(12)(38)(25)(32)] base 45, 6 repeating digits
3.[6(26)(13)] base 46, 3 repeating digits
3.[6(33)(26)(40)(13)(20)] base 47, 6 repeating digits
3.[6(41)] base 48, 2 repeating digits
3.7 base 49, 1 repeating digits
3.[7] base 50, 1 repeating digits
3.[7(14)(29)] base 51, 3 repeating digits
3.[7(22)(14)(44)(29)(37)] base 52, 6 repeating digits
3.[7(30)(15)] base 53, 3 repeating digits
3.[7(38)(30)(46)(15)(23)] base 54, 6 repeating digits
3.[7(47)] base 55, 2 repeating digits
3.8 base 56, 1 repeating digits
3.[8] base 57, 1 repeating digits
3.[8(16)(33)] base 58, 3 repeating digits
3.[8(25)(16)(50)(33)(42)] base 59, 6 repeating digits
3.[8(34)(17)] base 60, 3 repeating digits
3.[8(43)(34)(52)(17)(26)] base 61, 6 repeating digits
3.[8(53)] base 62, 2 repeating digits
3.9 base 63, 1 repeating digits
3.[9] base 64, 1 repeating digits
3.[9(18)(37)] base 65, 3 repeating digits
3.[9(28)(18)(56)(37)(47)] base 66, 6 repeating digits
3.[9(38)(19)] base 67, 3 repeating digits
3.[9(48)(38)(58)(19)(29)] base 68, 6 repeating digits
3.[9(59)] base 69, 2 repeating digits
3.(10) base 70, 1 repeating digits
3.[(10)] base 71, 1 repeating digits
3.[(10)(20)(41)] base 72, 3 repeating digits
3.[(10)(31)(20)(62)(41)(52)] base 73, 6 repeating digits
3.[(10)(42)(21)] base 74, 3 repeating digits
3.[(10)(53)(42)(64)(21)(32)] base 75, 6 repeating digits
3.[(10)(65)] base 76, 2 repeating digits
3.(11) base 77, 1 repeating digits
3.[(11)] base 78, 1 repeating digits
3.[(11)(22)(45)] base 79, 3 repeating digits
3.[(11)(34)(22)(68)(45)(57)] base 80, 6 repeating digits
3.[(11)(46)(23)] base 81, 3 repeating digits
3.[(11)(58)(46)(70)(23)(35)] base 82, 6 repeating digits
3.[(11)(71)] base 83, 2 repeating digits
3.(12) base 84, 1 repeating digits
3.[(12)] base 85, 1 repeating digits
3.[(12)(24)(49)] base 86, 3 repeating digits
3.[(12)(37)(24)(74)(49)(62)] base 87, 6 repeating digits
3.[(12)(50)(25)] base 88, 3 repeating digits
3.[(12)(63)(50)(76)(25)(38)] base 89, 6 repeating digits
3.[(12)(77)] base 90, 2 repeating digits
3.(13) base 91, 1 repeating digits
3.[(13)] base 92, 1 repeating digits
3.[(13)(26)(53)] base 93, 3 repeating digits
3.[(13)(40)(26)(80)(53)(67)] base 94, 6 repeating digits
3.[(13)(54)(27)] base 95, 3 repeating digits
3.[(13)(68)(54)(82)(27)(41)] base 96, 6 repeating digits
3.[(13)(83)] base 97, 2 repeating digits
3.(14) base 98, 1 repeating digits
3.[(14)] base 99, 1 repeating digits
3.[(14)(28)(57)] base 100, 3 repeating digits
If you inspect this list, you’ll notice something peculiar - the number of repeating digits is either 1, 2, 3 or 6, depending on the base. This remains true even for dividing in the first million bases! Odder still, the number of bases for which 22/7 (or 1/7 - the part before the decimal isn’t critical to this problem) repeats in 1, 2, 3 or 6 digits appears to have the same ratio: 2 each for 1, 3, or 6 digits to 1 for 2 digits.
Consider that as x goes to infinity x^3 goes to infinity. So does x^2. However, x^3 grows much faster. Subtracting the two, in any case, doesn't yield 0.
Such tantalizing patterns seemingly spring up everywhere numbers are examined. Is a particular one useful (other than for winning bets in bars)? I’ve no guess.
I Think thats all right, hope it helps.
Fox
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Bass
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Re:Infinite = 1
« Reply #3 on: 2006-06-20 20:44:25 » |
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Thankyou for your kind welcomes, Hermit and White Fox. I appreciate your answers. It is exactly what I was looking for as a minimum to start discussing with.
Hermit
I think I see where you are coming from, but Im having trouble seeing the bigger picture. The problem with Infinite, I think is the name we gave it. It's not easily understandable. We hear it so many times and still can't understand its meaning. I would call it something more like "whole", and I would even make 2 concepts out of Infinite and call them differently, and that would be something like Whole, and Loop.
Infinite doesn't mean "constantly growing and unreachable end". We can understand the whole, and it simply loops. In all directions (to make a graphic analogy). Infinite is 1, where there's no 2. This is just how I see it, although I still just may not be understanding it right.
White Fox
In your example, A - B is indeterminable because A nor B are determined.
I determined Infinite to be a whole, a kind of one (but not one). Therefore Infinite - Infinite = 0 without any doubt for it's like 1 - 1 = 0. The only thing that can confuse one, once he has understood that, is, I think, if he thinks like about apples and pears. Then, he shall remember his first day at Maths, where we learn that apples can only be substracted from apples. Numbers aren't just numbers. Numbers are to represent things. Things that aren't numbers. So we calculate variations of these "things" represented by numbers. Numbers are still the only way our brains have to understand these variations, and comunicate them too.
Given a house, if you substract the same house to it, you will have nothing.
That is:
1 - 1 = 0.
House - House = 0.
Infinite - Infinite = 0.
Cheers.
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Fox
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Re:Infinite = 1
« Reply #5 on: 2006-06-21 09:20:31 » |
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Hmm, okay let me try and elucidate myself in a different light.
If you have five apples and someone swipes two, you have three left, that's why 5 - 2 is 3.The cardinality of the counting numbers is one of the infinities, commonly called aleph-null. The cardinality of the odds is also aleph-null (ask Cantor). Pull the odds away from the rest, and you're left with the evens, also of cardinality aleph-null. So,infinity - infinity = infinity. And all three are the same level of infinity, not "big" infinity minus "smaller" infinity - as I think is what you may be implying or perhaps where your confusion is coming from.
Hang in there a moment:
the cardinality of all the counting numbers greater than three {4, 5, 6, ...} is also aleph-null. Pull those away from the counting numbers, and you're left with just {1, 2, 3}, a set of cardinality, well, three.
So, Infinity - Infinity = 3 (sets)
Infinity - infinity can be anything you can rig it to be. That's why it's indeterminate.
Infinity is not a real number but it is an element of the extended real number line, in which arithmetic operations involving infinity (and minus infinity) may be performed. In an example of this system, infinity has the following arithmetic properties:
Infinity with itself: (Infinity = ?)
? + ? = ?
? . ? = ?
-? . (-?) = ?
-? + (-?) = -?
? . (-?) = -?
x / ? = 0 is not equivalent to 0 . ? = x If the second were true, it would have to be true for every x, and, by transitivity of the equals relation, all numbers would be equal. This is what is meant by 0 . ? being undefined, or indeterminate. Note that in some contexts, such as measure theory, 0 . ? = 0
Infinity is often used not only to define a limit but as if it were a value in the extended real numbers in real analysis; if f(t) >_ 0 then
 means that f(t) does not bound a finite area from 0 to 1
 means that the area under f(t) is infinite
 means that the area under f(t) equals 1
In saying anything else I think I'd just be repeating myself and Hermit.
Hope this works better for you.
Fox
p.s From a personal perspective, an irrational number raised to the power of the product of an imaginary number, and where an irrational should yield -1 sounds mind-blowing, and suggests an order to the universe that is scary.
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Bass
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Re:Infinite = 1
« Reply #6 on: 2006-06-21 22:46:35 » |
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Thanks guys the info really helped!
Cheers.
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