On 0.999...

While trying to explain to someone that 0.999... aka 0.(9) is equal to 1 - not almost equal, but equal - one of the "arguments" he used was that recurring decimals are not really numbers, you can't do mathematical operations with them. While the idea is mind-boggling - 0.(1) is 1/9 which is a rational number - I thought of a proof that doesn't use recurring decimals: using base 9.

0.(1)10 = (0.1)9

(0.0)9 + (0.1)9 = (0.1)9
(0.1)9 + (0.1)9 = (0.2)9
(0.2)9 + (0.1)9 = (0.3)9
(0.3)9 + (0.1)9 = (0.4)9
(0.4)9 + (0.1)9 = (0.5)9
(0.5)9 + (0.1)9 = (0.6)9
(0.6)9 + (0.1)9 = (0.7)9
(0.7)9 + (0.1)9 = (0.8)9

Finally,

(0.8)9 + (0.1)9 = ?

Well... in any numeration base, when you finished the digits you restart at 0 and add 1 to the next order, so:

(0.8)9 + (0.1)9 = (1.0)9

Recasting the above in base 10, we have

0.(8) + 0.(1) = 1.0

However, in base 10 we still have a digit left, so it is equally valid to write

0.(8) + 0.(1) = 0.(9)

just as it is valid to write 0.(7) + 0.(1) = 0.(8).

Given that the left side is identical, it follows that 1 = 0.(9). QED.

Comments

Post a Comment

Popular posts from this blog

Posting dynamic Master / Detail forms with Knockout

Image
I have the following dynamic form in my GitHub Inventory project : I created the form with KnockoutJS , using some ideas from this article . The detail view looks like this: <table id="acquisition_items" class="jtable"> <thead> <tr> <th>Product Name</th> <th>Quantity</th> <th>Price</th> <th>Value</th> <th><button data-bind="click: addItem">[+]</button></th> </tr> </thead> <tbody data-bind="foreach: items"> <tr> <td> <input data-bind="value: ProductName" type="text" value="" /> </td> <td> <input data-bind="value: Quantity" type="text" value="" /> </td> <td>
Read more

Comparing Excel files, take two

Image
I already wrote this project once, on GitHub , but I can't say that I like the result. I wrote that code without TDD, mostly to see if I can still work in the "normal" fashion. I can, but I had a bug that took me a while to even discover and then was difficult to write tests for. I also hate the temporal dependency in that design (if you don't call the SortBy method first, the ExcludeRecords method will return incorrect results). Anyway; here's attempt number two, writing that code using TDD. Structure I start by creating my usual structure for projects of this type - a library for the logic, a console application for the main program and a test project for tests. I also use the Enable NuGet Package Restore option (right-click on the solution node) to add the three NuGet files that will allow someone to automatically download all the packages I'm referencing without wasting space in the source code repository. (Right now I've only added the Moq pa
Read more

EF Code First: seeding with foreign keys

This is just a quick tip, more for future reference for myself than anything else. When using Entity Framework Code First, one of the things you might want to do while developing the application is to seed your database with some test data. This can be something as easy as: protected override void Seed(InventoryDB context) { context.Products.AddOrUpdate( p => p.Name, new Product { Name = "Hammer", SalePrice = 11.99m, }, new Product { Name = "Nail Pack x100", SalePrice = 0.05m, }, new Product { Name = "Saw", SalePrice = 19.99m, }, new Product { Name = "Toolkit", SalePrice = 39.99m, } ); //... } (I am using the AddOrUpdate method so the code detects whether those products already exist and doesn't add them again if they do.) The Product class is quite simple and – most importantly – doesn't depend on any other: public class Product { public int Id { g
Read more