In this section we have a discussion on the types of infinity and how these affect certain limits. Sometime back in elementary school, I first asked teachers, "What happens when you divide infinity by 2?" Now, to answer our original question, what do we get when we divide infinity by 2? A number is either zero, or the successor of another number. Patchwork defines subtraction: I guess I know now why infinity minus infinity is "indeterminate", rather than zero like I always thought it should be. You're welcome to. $\begingroup$ Related: What is infinity divided by infinity? There's several ways we could do it, but one way is particularly natural in this framework. Random code snippets, projects and musings about software from Eric Kidd, a developer and entrepreneur. Some teachers couldn't answer, and others told me, "It's still infinity! Who is the longest reigning WWE Champion of all time? bigger and bigger, the Now, infinity is a subtle concept, and if had we used different definitions, we might have gotten a different result. wrote on Feb 03, 2007: I once thought up of a way of doing the Grand Hotel in Haskell, by using an infinite list: First off, your implementation of infinity is probably among the saner possible with a Peano arithmetic. 2/20000000000= wrote on Feb 03, 2007: Eric Kidd *Main> divBy2 infinity Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Well, that looks like infinity to me! http://spiedl.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=PSISDG006499000001649902000001&idtype=cvips&gifs=yes, The Haskell Road to Logic, Maths and Programming, Smart classification using Bayesian monads in Haskell, Proving sorted lists correct using the Coq proof assistant, Lightweight Languages 2 Conference (MIT, 2002), Monads in 15 minutes: Backtracking and Maybe, Bare Metal Rust 3: Configure your PIC to handle interrupts correctly, Is it reasonable to define infinity this way, assuming we're in category. you can also add a -Infinity, and it'll still be compatible with Peano axioms. ", but I couldn't explain it much better than my school teachers (at least not without using the word denumerable, which is a good way to ruin a lunch conversation). And as I work through various sections, I'm increasingly convinced that my understanding of some of the ideas is still pretty dodgy. What is the birthday of carmelita divinagracia? Now, to answer our original question, what do we get when we divide infinity by 2? (Why not use Haskell's built-in integers? Why don't libraries smell like bookstores? > What is 1 divided by infinity? wrote on Feb 03, 2007: I posted my reply at http://japple.blogspot.com/2007/02/countable-ordinals-in-haskell.html Now, infinity is a subtle concept, and if had we used different definitions, we might have gotten a … How long does it take to cook a 23 pound turkey in an oven? As the denominator gets Who of the proclaimers was married to a little person? The "deriving" keyword tells Haskell to define show and the comparison operators for us. As always with “infinity” you should apply Bustany's Rule of Infinity: > Infinity and Intuition do not mix So chuck out your intuition and ask: > Which infinity? And then tonight, while reading a paper about Haskell, I was hit by an evil idea: When in doubt, ask the Haskell interpreter! Note that there is a lot of theory going on 'behind the scenes' so to speak that we are not going to cover in this section. For example, Therefore, infinity divided by infinity is NOT equal to one. 2. Eric Kidd I'm not saying … Want to contact me about this article? How long will it take to cook a 12 pound turkey? That value is indeterminate, because infinity divided by infinity is defined as indeterminate, and 2 times infinity is still infinity. And finally, some questions for mathematically sophisticated readers: Jerf mentions math education: I bet you could build a good number theory curriculum around that idea; giving number theory a REPL couldn't be all bad, given the abstraction of the topic. wrote on Feb 08, 2007: A caution: I'm still studying the paper by Fokkinga and Meijer. Hard to wrap your mind around, but it's true. Also, if you liked this article, you might want to check out The Haskell Road to Logic, Maths and Programming, which uses Haskell to teach discrete math. Functions like 1/x approach 0 as x approaches infinity. Is it possible to represent any other (more interesting and/or more correct) definitions of infinity in Haskell? You'd have to ask somebody who knows more math than I do (or check Wikipedia). Any number divided by infinity is equal to 0. Inter state form of sales tax income tax? What's the best way to think about the infinite ordinals. Especially the one-point compactification of R (or ... well ... machine-R) you'll find as First, we need to teach Haskell about the natural numbers. when you have a formal infinity, 1/infinity becomes an infinitesimal number, so you can develop non-standard analysis. All Rights Reserved. Infinity is a concept, not an actual number, so we can't just divide a number by infinity. quotient approaches 0. What details make Lochinvar an attractive and romantic figure? To explain why this is the case, we will make use of limits. Think of dividing 2 by a very big number. or as Is evaporated milk the same thing as condensed milk? Summary: I think we should represent countable ordinals as the limits of increasing functions from the naturals to the countable ordinals. What is plot of the story Sinigang by Marby Villaceran? The material on this site can not be reproduced, distributed, transmitted, cached or otherwise used, except with prior written permission of Multiply. This section is intended only to give you a feel for what is going on here. 1/10000000000 which is very close to 0. How will understanding of attitudes and predisposition enhance teaching? with the corresponding extensions to Ord and Num implemented specificially. Some professor wrote about something called transreal arithmetic. Random code snippets, projects and musings about software from Eric Kidd, a developer and entrepreneur. You could also have done infinity as To explain why this is the case, we will make use of limits. I. I do, alas, not, however, know how this compares to the category CPO you're asking about; I would, however, suppose that it is a sane way to do it. Instead we can get any real number to equal to one when we assume infinity divided by infinity is equal to one, so infinity divided by infinity is undefined . So please don't take anything in the previous comment. Alan Manuel Gloria Of course I said, "It's still infinity! We can write that in Haskell as: Math geeks in the audience will recognize this as the Peano arithmetic. Any expression divided by infinity is equal to zero. Or if you're looking for something else to read, here's a list of popular posts. 2 + − 2 ∞ + 1 ∞ 2 + − 3 ∞ 3 1 + − 1 ∞ 2 + 1 ∞ 3 \frac {2+\frac {-2} {\infty }+\frac {1} {\infty ^ {2}}+\frac {-3} {\infty ^3}} {1+\frac {-1} {\infty ^ {2}}+\frac {1} {\infty ^3}} 1 + ∞ 2 − 1 + ∞ 3 1 2 + ∞ − 2 + ∞ 2 1 + ∞ 3 − 3 . Infinity is a concept, not an actual number, so we can't just divide a number by infinity. But other than that, it's pretty much the opposite of multiplying: How should we define infinity? It is 0. Just humor the crazy programmer for a moment, OK?). Likewise functions with x 2 or x 3 etc will also approach infinity. data CompleteNumbers = Number Numbertype | Infinity = 1. Doubling a number is a bit trickier. with the correspondingly defined instantiation of Num and Ord. Any number divided by infinity is equal to 0. From what I understant it incorporates infinity and 1/0 as part of the numbering system. data CompactR = (Real a) => Finite a | Infinity So now you have (Infinity + Infinity) / Infinity = 1 ... 2013 by anonymous. $\endgroup$ – user2314737 Aug 27 at 10:24 add a comment | 2 Answers 2 What is the contribution of candido bartolome to gymnastics? What is the conflict of the story of sinigang? This equation is obvious incorrect. When did organ music become associated with baseball? "Anything divided by itself equals one"** Infinity + Infinity = Infinity You could claim 2 X Infinity is the answer, that just equals Infinity. How long will the footprints on the moon last? We need to replace each Succ with Succ (Succ ...), which we can do with a recursive function: Dividing is trickier still, because we're working with natural numbers, which means we'll need to round down. Infinity divided by two equals infinity. **Response to below: Inifinty divided by infinity would equal one. Infinity is basically Succ followed by an infinite number of other Succ values. Copyright © 2020 Multiply Media, LLC. data CompleteNumbers = Number Numbertype | Infinity | NInfinity At least for the infinity Aleph-0. But be careful, a function like "−x" will approach "−infinity", so … Jim Apple ", More recently, a couple of friends were discussing a similar question at lunch: "What happens when you add 1 to infinity?".
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