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Author Topic:   A question of numbers (one for the maths fans)
CK
Member (Idle past 4127 days)
Posts: 3221
Joined: 07-04-2004


Message 1 of 215 (324776)
06-22-2006 8:05 AM


In this first post - you will get nothing from me but a question - why I have asked the question we can discuss later once we have had some discussion about it.
Here's the question I put to you:
is 0.9 recurring the same as 1?

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Dr Jack
Member
Posts: 3514
From: Immigrant in the land of Deutsch
Joined: 07-14-2003
Member Rating: 8.7


Message 2 of 215 (324780)
06-22-2006 8:16 AM
Reply to: Message 1 by CK
06-22-2006 8:05 AM


Not only are they equal, they are different ways of writing the exact same number. It is not unique in this, every terminating decimal has two representations.

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PaulK
Member
Posts: 17822
Joined: 01-10-2003
Member Rating: 2.2


Message 3 of 215 (324781)
06-22-2006 8:16 AM
Reply to: Message 1 by CK
06-22-2006 8:05 AM


For the record, I learned the "proper" answer to that in a 1st year undergraduate course. But anyone with an understanding of recurring decimals ought to be able to work it out.

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cavediver
Member (Idle past 3643 days)
Posts: 4129
From: UK
Joined: 06-16-2005


Message 4 of 215 (324783)
06-22-2006 8:21 AM
Reply to: Message 2 by Dr Jack
06-22-2006 8:16 AM


Not only are they equal
You try teaching that to a bunch of argumentative public school boys or an argumentative constructivist for that matter
Edited by cavediver, : No reason given.

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Modulous
Member
Posts: 7801
From: Manchester, UK
Joined: 05-01-2005


Message 5 of 215 (324791)
06-22-2006 8:50 AM
Reply to: Message 1 by CK
06-22-2006 8:05 AM


x=0.999999...
10x=9.999999...
10x-x = 9.999999... - 0.999999...
9x = 9
Therefore x=1
Edited by Modulous, : No reason given.

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subbie
Member (Idle past 1254 days)
Posts: 3509
Joined: 02-26-2006


Message 6 of 215 (324814)
06-22-2006 9:43 AM
Reply to: Message 1 by CK
06-22-2006 8:05 AM


1/3 = 0.3333333.....
2/3 = 0.6666666.....
3/3 = 0.9999999.....

Those who would sacrifice an essential liberty for a temporary security will lose both, and deserve neither. -- Benjamin Franklin

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New Cat's Eye
Inactive Member


Message 7 of 215 (324820)
06-22-2006 10:06 AM
Reply to: Message 2 by Dr Jack
06-22-2006 8:16 AM


every terminating decimal has two representations.
Whadaya mean?
nevermind.
Edited by Catholic Scientist, : stupidity

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RickJB
Member (Idle past 4990 days)
Posts: 917
From: London, UK
Joined: 04-14-2006


Message 8 of 215 (324840)
06-22-2006 11:20 AM
Reply to: Message 5 by Modulous
06-22-2006 8:50 AM


Heh. That little proof is sweet, Modulus!
Gonna crib that one for when I'm down at the pub!
Edited by RickJB, : No reason given.

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Chiroptera
Inactive Member


Message 9 of 215 (324869)
06-22-2006 1:16 PM
Reply to: Message 8 by RickJB
06-22-2006 11:20 AM


It also works for any decimal expansion that has a repeating sequence. Only instead of 10x, its (whatever)*x, where the whatever is large enough to bring the repeating parts in line. For example,
x = 0.123412341234...
10000x = 1234.123412341234...
10000x-x = 9999x = 1234
x=1234/9999
This is why every decimal that consists of a repeating segment is a rational number, that is, can be written as an ordinary fraction of integers.

"These monkeys are at once the ugliest and the most beautiful creatures on the planet./ And the monkeys don't want to be monkeys; they want to be something else./ But they're not."
-- Ernie Cline

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arachnophilia
Member (Idle past 1344 days)
Posts: 9069
From: god's waiting room
Joined: 05-21-2004


Message 10 of 215 (324962)
06-22-2006 4:50 PM
Reply to: Message 1 by CK
06-22-2006 8:05 AM


infigers
...now try it backwards. this is by far my favourite math-sturbation. i heard dr. craigen deliver this lecture at the 32nd Southeastern Conference on Combinators, Graph Theory, and Computing, in Baton Rouge, Louisianna, in 2001. (i think i got number right, i've been to so many...)
quote:

INFIGERS

R. Craigen
Not too far away, in an otherwise dull galaxy much like ours, lives the intelligent race of Endians, so called because they order their lives by the principle that all good things must come to an end. (They knock down their heritage buildings to protect them from the slow ravages of time and burn down forests before they get old and densely grown. Music groups that have more than 5 hits are disbanded by law. So, you see, they take this principle quite to the extreme!)
These clever folks are just entering their industrial revolution. In all their previous history they have never had need for any numbers other than the usual nonnegative integers, 0,1,2,3, etc., and the occasional fraction; since they have 10 fingers like us (albeit distributed somewhat unevenly among their 3 hands), they represent numbers, as we do, with place notation in base 10.
They are just now beginning to notice an increasing need for a more comprehensive system of numbers well-suited to mechanical computation, to support their technological advances, and have considered how to represent fractions (and any other numbers that might come along) in a place-type system. The first thing that was proposed was to place a decimal point after integers and continue with digits representing the fractions 1/10, 1/100, and so on, as we are accustomed to doing. But when it became clear that, in this system, some numbers, like 1/3 = 0.333..., would never come to an end, the Endians decided that the system was an abomination, for it violated the aforementioned much cherished principle, so they discarded it.
However, a forward-thinking young visionary named Antemedes discovered how to rescue some of the benefits of the idea of unending decimals without committing the ultimate heresy: the digits of his numbers had the usual integer place-values, 1, 10, 100 and so on, but his idea was to assign a value to EVERY place. So the number 7 would be thought of as ...0007; its square would be ...0049, and so on, each beginning with an infinite number of zeros. But he didn’t stop there. His system also allowed the possibility that some numbers could have an infinite number of nonzero digits!
Now, you may wonder, how is this system any better than the other, seeing as it, too, involves the use of unending strings of digits to represent numbers? Well, that is where you are wrong! For you must admit that, although these numbers do not have a beginning, they most certainly all come to an end ” in the same place, after the 1’s digit!
Now, owing to the resemblance between these numbers and the positive integers, it was easy to think of them as “integers”. However, those with an infinite number of digits seem horrendously large ” in fact, “infinite”, and are most certainly something else altogether. So the numbers of this system are called “infigers”, a compromise between “infinite” and “integers”.
Antemedes discovered that it is possible to add, subtract, and multiply any two infigers according to the usual rules of arithmetic. For example, he found that
...6667 x 3 = 1, and ...9999 + 1 = 0 .
Try these calculations yourself, to see how they work. Just multiply or add, in the same way as you are accustomed, and don’t stop until you are sure that the pattern you see will continue. This means, of course, that 1/3 = ...6667, and that ...9999 is an entirely new number (to them!), the additive inverse, or negative, of 1, which of course is what we call -1. Similarly,
-2 = ...9998, and -157 = ...99843 .
In fact, the negative of any positive integer begins with an infinite string of 9’s, and any number of this type is the negative of a positive integer. To be sure that you understand this, see if you can decide to which negative number ...999342 corresponds. Multiplication of ...6667 by 3 is not hard because 3 has only one digit; how does one multiply by, say, 23? Why, in the same way as we usually do: multiply by 3 and 2, adding a zero to the end of the second number, and add the results:
...6667
x 23
...0001
...33340
...333341,
which is, apparently, the infiger representation of 23/3. Infigers can be multiplied by continuing the above process for each of the digits of the second number in the product. Here, for example, is Antemedes’ calculation for the square of 1/3:
...6667
x ...6667
...6669
...00020
...000200
...0002000
: : :
...8889.
This should be the infiger for 1/9, which you can check easily by
multiplying by 9.
Antecedes made many other exciting discoveries. For example, no number has two DIFFERENT infiger representations, unlike in our decimal system, where 1.6999... = 1.7000... ; infiger representations of rational numbers always eventually repeat in a periodic cycle, and all infigers of this form represent rational numbers. There are many exciting discoveries to be made about infigers, and I’ll leave some of them for you. Here are some things to do:
  1. Antecedes was crushed to learn that some rational numbers, such as 1/2, do not have an infiger representation. Can you see why not? Exactly which rational numbers can, and can’t, be represented as infigers? Propose a method of incorporating these rationals into the system without committing a heresy.
  2. Try finding some of the following infigers: 2/3, 4/3, 5/3, 1/7, 2/7, 3/7, -2/3, -3/7. Do you notice any patterns? Comparing the repeating decimal part of these fractions to the repeating part in the usual representation, do you see any connection? Acalculator might be helpful.
  3. You can save some work in your calculations above if you discover how to do long division with infigers. (HINT: try working from right to left, instead of from left to right!)
  4. It would not be good if it turned out that multiplication of infigers is not commutative. Try multiplying 3 by 1/3 (that is, in the reverse order from the calculation given earlier), and see if it still comes out to 1. Can you show that multiplication is commutative, in all cases?
  5. Given two integer infigers, positive or negative, it is not hard to decide which is largest. But ” compare some of the fractional infigers you worked out in #2 above. Can you find any way, other than turning them back into fractions, to decide which is larger, or even which is positive?
  6. There are MANY infigers that represent neither positive nor negative integers or fractions. Nevertheless, they can be added, subtracted and multiplied by each other and the more familiar infigers. For example, the infiger ...010001011 whose first, second, fourth, and all digits in positions corresponding to powers of two, are 1, and all the rest are 0, is not of the familiar types. What is it? In general, what are these other numbers? Do they have any connection to interesting numbers?
  7. Things get really interesting when you try working in base systems other than 10. For example, in base 2, 1/3 (that is, 1/11 in binary notation) has infiger representation ...010101011. Some fractions with no decimal infiger representation have a binary infiger representation! What is the simplest such number? What is its binary infiger representation? Which fractions have binary infiger representations?
  8. Which fractions have base 5 infiger representations? What is 1/2 as a base 5 infiger? There is a non-repeating base 5 infiger ending in ...43212, whose square is -1. Check that this works for the digits given. Can you find the previous 3 digits? Find the last several digits of the negative of this number, and check whether it, too, is a square root of -1.
  9. Show that the square root of 2 has a base 7 infiger representation. Find its last several digits. Do you think it is possible to find a pattern?
What else can you learn about infigers? Send in your discoveries, attention the Newsletter Coordinator, and they may appear in a future newsletter!
http://www.umanitoba.ca/science/mathematics/new/Issue2.pdf
Edited by arachnophilia, : typo, in what little i wrote.


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RickJB
Member (Idle past 4990 days)
Posts: 917
From: London, UK
Joined: 04-14-2006


Message 11 of 215 (324976)
06-22-2006 5:24 PM
Reply to: Message 10 by arachnophilia
06-22-2006 4:50 PM


Re: infigers
I'm gonna have to re-read this...
I would have loved to have been born with a gift for Maths...
I've been reading Roger Penrose's "The Emporer's New Mind". I can follow the logic of his ideas pretty well as I have a reasonable amount of programming experience but my skills fail as soon as the number problems become more technical.
I just can't "visualise" numbers very well. Give me a diagram, on the other hand, and I'm much happier!
Ah well..
Edited by RickJB, : No reason given.
Edited by RickJB, : No reason given.

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riVeRraT
Member (Idle past 416 days)
Posts: 5788
From: NY USA
Joined: 05-09-2004


Message 12 of 215 (325052)
06-22-2006 10:01 PM
Reply to: Message 5 by Modulous
06-22-2006 8:50 AM


You can't subtract .9999999.... logically. That assumes infinity has an end.
9.999999 -.999999 = 8.999991

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riVeRraT
Member (Idle past 416 days)
Posts: 5788
From: NY USA
Joined: 05-09-2004


Message 13 of 215 (325054)
06-22-2006 10:02 PM
Reply to: Message 6 by subbie
06-22-2006 9:43 AM


1/3 = 0.3333333.....
2/3 = 0.6666666.....
3/3 = 0.9999999.....
It's a lie, 1/3 does not = 0.33333....
1/3 equals 60 degrees.

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ringo
Member (Idle past 412 days)
Posts: 20940
From: frozen wasteland
Joined: 03-23-2005


Message 14 of 215 (325064)
06-22-2006 10:31 PM
Reply to: Message 13 by riVeRraT
06-22-2006 10:02 PM


riVeRraT writes:
1/3 equals 60 degrees.
Fahrenheit or Celsius?

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riVeRraT
Member (Idle past 416 days)
Posts: 5788
From: NY USA
Joined: 05-09-2004


Message 15 of 215 (325067)
06-22-2006 10:34 PM
Reply to: Message 14 by ringo
06-22-2006 10:31 PM


I don't care what you call it, you can't have any of my pie

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