...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:

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INFIGERS

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R. Craigen

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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!)

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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.

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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.

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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!

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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!

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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”.

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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

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...6667 x 3 = 1, and ...9999 + 1 = 0 .

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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,

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-2 = ...9998, and -157 = ...99843 .

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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:

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...6667
x 23
...0001
...33340
...333341,

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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:

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...6667
x ...6667
...6669
...00020
...000200
...0002000
: : :
...8889.

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This should be the infiger for 1/9, which you can check easily by
multiplying by 9.

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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!

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http://www.umanitoba.ca/science/mathematics/new/Issue2.pdf

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Edited by arachnophilia, : typo, in what little i wrote.

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