Infigers

[CanadaAotS]

A very interesting problem I found...

 

http://www.umanitoba.ca/faculties/science/mathematics/new/seminars/html/net62mathsumanitobacaJan31407082002.html

"Infigers"

ABSTRACT

 

The inhabitants of a distant planet call themselves "Endians"' date=' because they live by the creed that "All good things must come to an end". Mathematically, they have evolved similarly to us, with positional notation for whole numbers in base 10, fractions expressed as ratios of integers, and knowledge of some irrational quantities, such as square root of 2. How to represent such numbers as decimals, however, is problematic, because such decimals have no end (see creed above).

 

A young mathematician proposes a different apporoach: decimals that have no beginning, instead (this is not a violation of the above creed). Since the smallest place value is a unit, such objects resemble integers, but since an infinite number of digits may be nonzero, they appear to be infinitely large; so they are called infigers.

 

For example, one can write 1/3 = ...6667. To convince yourself that this is appropriate, multiply this "number", in either order, by 3. You will observe that the result is ...0001 (=1). Similarly ...999 = -1, and ...3334 = -2/3.

 

Unlike ordinary decimals, there is no ambiguity arising from numbers that can be represented by two decimal strings --- but other problems arise.

Does arithmetic with infigers makes sense? Can all real numbers be so represented? (...and: do all infigers represent real numbers?) What about complex numbers? Since place value loses its meaning, how does one judge relative size among infigers? How do our favourite functions behave under infiger arithmetic? How is this all connected to p-adic valuations and quotients of infinite dimensional rings?

[/quote']

 

This is actually very interesting...

 

You can multiply, divide, add, subtract them. I'm sure about other operators.

 

an example is 30 * 1/3

30 = ...00030

1/3 = ...6667

 

...6667

x...0030

--------

...000

...0001

...000

--------

...00010 = 10

 

Also, if you start figuring out using the math above, what some other numbers are you see a pattern with them.

However, there are some problems with the system. decimals that do not repeat have no infiger representation.

 

Also, long division's really hard lol. you have to work backwards or something...

 

Anyways, thought this was really interesting to find out the properties of these "infigers"

[uncool]

This can be seen as what happens when you take mod 10^n, and limit as n -> infinity.

I just have to wonder, what happens for 1/5? Does it become 0.5?

=Uncool=

[CanadaAotS]

...decimals that do not repeat have no infiger representation.

 

Yup, 1/5 (.2) would be a non-repeating decimal.

This was actually part of a math problem, and the problem was how the "endians" could represent non-repeating decimals while still keeping to their way of life.

 

In other words, being able to display say 1/2, 1/5 etc. in such a way to be compatible with ...6667 = 1/3 (you show a representation of them that allows you to still divide, subtract etc.)

 

If someone can figure out how to do long division, just divide 2/3 into 1/3 to get the 1/2.

[uncool]

Well, let us assume that there is a normal representation, that is, such that it takes the form ......ABC = 1/2. Then 1 = .....(2*A)(2*B)(2*C) with all the carry overs. But then the last digit is divisible by 2 - meaning an impossibility.

This will happen in any base - so it cannot be represented *normally*. However, we could perhaps do it by putting the normal part before and the rest after.

This will always have an end as anythign that wouldn't have an end can be represented beforehand.

Example: 8/15 = 1/3 + 1/5

= 67 + 0.2 = 67.2

1/15 = -5/15 + 6/15 = -1/3 + 2/5 = 6 + 0.4

=6.4

=Uncool=

[matt grime]

This is 10-adic arithmetic. Look up p-adic and find out all about it.