As I prepare my classroom for the upcoming start of school, I have decided to create fifteen or so mathematics posters for display around the room. These will complement ten motivational posters I purchased and posted already. As I am not a classically trained mathematician, I want to make sure I do not erroneously depict the mathematics on the posters I create.

One such poster, below, consists of a Venn diagram of number set types spanning the natural numbers to the complex numbers and including the integers, the rational numbers, the real algebraic numbers, the real numbers, and the imaginary numbers. The counting, measuring, irrational, and transcendental numbers are also noted.

My question for my PLN follows. Is this a proper depiction of the different number sets in mathematics? I took liberty omitting the algebraic set containing only imaginary numbers for two reasons: 1) difficult to depict using the model I used and 2) I do not plan to discuss the algebraic numbers at all – I only included the real algebraic numbers since my earlier use of irrational numbers was too limited.

My classroom spans 9th grade algebra 1 students to 12th grade AP calculus AB students. My intent is to help all students better understand the concept of number in all of its forms, both historically, and mathematically. If anyone believes my depiction has a gross error, please let me know. I plan to print this as large as possible in the next two days.

Thanks for your help!

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[Tue 8/2/11 8:30 PM CST] Added revised diagrams further below based on feedback so far. Let me know which one seems best. I prefer version 1.1 below.

Revised diagrams follow.

**Version 1.1:**

**Version 2.0:**

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[Wed 8/3/11 2:30 PM CST] Two more, revised diagrams added below. My plan is to print version 1.2 as a 20″ x 30″ poster to mount at the front of my classroom. I elected to use this version since I understand that the complement of the real algebraic numbers is the set of the transcendental numbers. The wrinkle with this version is depicting the set of irrational numbers in version 1.2, which is why I created version 1.3. However, I feel that version 1.3 does not imply visually that the transcendentals far exceed the real algebraic in number, if I recall that fact correctly.

I must say, pure (aka esoteric) math is quite challenging, at least to this former electrical engineer.

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Latest revised diagrams follow.

**Version 1.2:**

**Version 1.3:**

I believe the two terms “counting numbers” and “natural numbers” to be synonymous. “0” may be occasionally included among “counting numbers”, but negative numbers never are.

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Hi Alexander. Thanks for your comment. Do you know if their is a definitive, primary source for mathematical terms? The reason I ask is I’ve seen negative integers referred to as the “counting backwards” numbers, which are part of the “all counting numbers” set. See the following for that author’s depictions and definitions. http://www.mathventures.com/mathed/TheNumbers.htm

Dave

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As a Venn diagram this would seem to imply that the sets of Natural/Whole/Integers/Rational numbers are within the sets of Irrational Numbers (Real algebraic & transcendental). However, rational and irrational numbers are themselves separate and distinct sets with unique properties within the set of Real numbers. It would seem more appropriate to physically separate rational and irrational number sets in your diagram.

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Hi Blair. I see what you mean. I was trying to show that only the numbers outside of the prior, nested sets applied to the arrowed annotations (e.g. irrational and transcendental designations). I agree that it could be misinterpreted as drawn. My understanding is that the rational and the irrational numbers fall within the real algebraic set. With that assumption, I will create a diagram that separates out the transcendentals from the real algebraic set.

BTW, I have seen diagrams of the irrational numbers them similar to how I have depicted them. See the following links for examples. I am curious if there is a definitive source for mathematical set definitions. The one root source for all mathematical truth…

http://www.mathventures.com/mathed/TheNumbers.htm

http://thinkzone.wlonk.com/Numbers/NumberSets.htm

I appreciate your comments. They help me attain greater insight into the subject I teach.

Dave

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I think if you’re talking about “what makes sense in the graph” I’d say you probably want to remove duplicate numbers. Since integers already includes (whole numbers) you wouldn’t need to include 1,2,3. BUT, I do think it’s a lot more clear from a student point of view to double them. In that case though, you might want to add a few examples from inner circles to maintain consistency. So real numbers would also include, 1, 2, 3.

So…actually I’m not much help. Because I probably would prefer you didn’t repeat the numbers. But either way, I’m thinking consistency is the important thing.

I probably like the third one the best, if you could line up that last arrow that’d be great. The first one made it unclear that irrational wasn’t point at all real numbers.

PS – My brain is starting to hurt because now I just started thinking about how all of these sets are infinite and so they’d all be the same size. Except that they’re not. But they are. Because INFINITY IS WEIRD.

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I feel your pain, Jason. I went through a similar experience creating it…thanks to Cantor.

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May I ask your method of poster-making? What program do you use, where do you have them printed, and how much does it cost? I’ll also suggest a Greek letter poster. The kids liked it when I posted it much like in an elementary classroom with the handwriting alphabet.

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Hi Jill. I have 15 some odd poster ideas, one of which includes Greek letters and their common usage in math, science, & engineering, so I agree its a cool posters for students. I plan to post some of my other posters when I get a moment.

I create all of my “posters” using MS PowerPoint initially. Then I save them as TIFF files if I plan to create a large poster. I then upload them to Costco’s photo site to create either 16″x20″ or 20″x30″ prints. The prices are quite reasonable for the prints, somewhere between $5 and $10 for the smaller and larger prints. It is a little tedious, even a lot tedious, but the end result can be pretty powerful; I’m having a little difficulty not getting the edges of my images cropped so I need to tweak my process a bit. For a slightly higher charge, there is an option to have them mounted as well.

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Thanks for the info! I have a “service learning” student, so if I can manage to give up some control, she could do some of the tedious work for me. Would love to see more of your ideas!

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Love your diagram and thanks for the tip on creating posters! I have been searching for some disambiguation regarding Whole & Natural numbers (and where zero actually resides) and here is what the Wiki has to say about Whole vs. Natural numbers makes it about as clear as before 😉

“There is no universal agreement about whether to include zero in the set of natural numbers: some define the natural numbers to be the positive integers {1, 2, 3, …}, while for others the term designates the non-negative integers {0, 1, 2, 3, …}. The former definition is the traditional one, with the latter definition first appearing in the 19th century. Some authors use the term “natural number” to exclude zero and “whole number” to include it; others use “whole number” in a way that excludes zero, or in a way that includes both zero and the negative integers.”

Wouldn’t you think the most ‘well defined’ area of academic studies would be able to decide on a definition for for it’s most rudimentary and foundational concepts? Humans, rational? Or irrational? Trick question, BOTH 😉

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I guess I missed this post.

I think I like V1.2 the best, but V1.3 works except for the size issues I discussed. I.e. rationals are countably infinite, but irrationals are uncountably infinite. However, real numbers are a subset of the imaginaries.

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Don’t you mean real numbers are a subset of the complex, which are comprised of the real and the imaginary?

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Could I use your poster to publish on my classroom webpage?

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Absolutely, Tere. Thanks for asking. Please do so with proper attribution.

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I’m a bit late for a question from 2011, but feel I should contribute after reading a few of the listed posts. I did not study each version carefully, but my first thought was that you needed to distinguish the difference between the rationals and irrationals. Only in version 1.3 are they separate, which they need to be. The other versions make it appear that a number can be both irrational and rational at the same time as they are nested. This is where I throw in an English lesson about the prefix in- (I include how it changes due to the word beginning with an r).

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Also several years late, but I must agree with you. Representing the numbers in any version but 1.3, it utterly incorrect.

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Reblogged this on Reflections of a Second-career Math Teacher.

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Very nice. But sin(pi/3) is not transcendental. It is algebraic, in fact, it’s sqrt(3/4). The sines and cosines of pi/N where N is an integer are all algebraic. Lovely graphic, by the way…

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