# Project proposal: Intro to numbers

https://arbital.com/p/numbers_proposal

by Eric Rogstad Aug 16 2016 updated Sep 2 2016

Should Arbital's first "project" be a guide to numbers?

The purpose of this page is to sketch out what an Intro to Numbers project might look like, so we can evaluate whether it would make a good project.

## Goal of the project

• Provide a guide to numbers that would be enlightening to Math 1 and Math 2 readers.

## Outline

1. Basic taxonomy -- provide quick intuitive definitions for each of the following types of number.

• [45h $\mathbb N$]:
• History of zero
• Definition from successor function
• [48l $\mathbb Z$]:
• History of negative numbers
• [4zq $\mathbb Q$]:
• Did you know that there are the same number of rational numbers as there are integers?
• [54z $\mathbb I$]:
• ??
• Transcendental numbers:
• ?? %%note:Eric Rogstad notes that he, as a Math 2 reader (engineering major, professional programmer), recently realized that he didn't know the difference between an irrational number and a transcendental number. (And a quick survey indicates that the rest of the development team also doesn't know the difference.) So it should be easy to find something to say that's new for Math 2 readers.%%
• [4bc $\mathbb R$]:
• Did you know that although there are the same number of rational numbers as natural numbers, there are provably more real numbers than rational numbers?
• [4zw $\mathbb C$]:
• http://simplifience.com/sample/identity.html

1. Exercises to test of understanding of each type of number.
• Set of questions where a number is presented, and the reader has to say which type it belongs to.

## Plan

Much of the work has already been done -- we have pages on each of these types of number. Two main pieces of work remaining:

1. Polish the existing pages up to A-class or B-class (including testing the explanations on Math 1/2 readers and incorporating feedback.)

Patrick Stevens

The non-existence of a total order on $\mathbb{C}$ is fun and interesting, I think, and also not very difficult. An excellent exercise in proof by contradiction.

Eric Bruylant

eric_b [2:39 AM]
I'd add a "what is a 'number' anyway"-type page with an explanation of the general constructive, formal definitions of different types of number, for people who've been confused by the education system's tendency to be informal and be taught by people who don't have a clear idea what a number is.

Edit: this is maybe just a lens on "Number" (edited)

eric_b [2:46 AM]
I'd consider replacing irrational and transcendental with just real to reduce the scope, it's still ~15 pages, if we want one for each math level

[2:47]
(15 new pages, and a bunch of existing ones to work on)

[2:48]
oh, math 1 and 2? (edited)

[2:48]
I think 0 is the most important audience here, and 3 is the cheapest set of pages to create

[2:49]
I'd go for 0, 3, and have 1 as a stretch goal

[2:49]
especially since many of the pages already have math 0 lenses

[2:51]
that'd mean we need 5 new math 3 pages (shortish), 2 new math 0 pages, and reviews of a bunch of existing pages. (edited)

[2:52]
and, optionally, 6 math 1 pages

[2:53]
(we'd try and do those after the math 0 pages were made great, since copying a good math 0 page and adding algebraic shorthand is cheap and gets much of the way from math 0 to 1) (edited)

[2:55]
another good stretch goal could be external resource pages.

eric_b [3:01 AM]
11 pages to polish / finish + 7-10 to create. Even without irrational and transcendental, and even accounting for 5 of the new pages being short/mathy, that's fairly ambitious for a week. (edited)

[3:01]
project generally seems good, though.

[3:02]
it'd be a collection of pages linked to from loads of places, and a neat set of things to have tied together.

[3:07]
it's also small enough that it seems doable, has the engagement of many of our most active contribs, and has enough potential for interesting facts to keep it engaging

Plus some discussion with ER being keen to do math 2 and me hesitant, me being keen to do 0 and him hesitant. I now like 0 everywhere plus a mix of non-0 levels, and think 2 everywhere is not optimal, but am fine so long as we have 0s.