# How do we draw the number hierarchy from natural to complex in a Venn diagram?

I want to make a Venn diagram that shows the complete number hierarchy from the smallest (natural number) to the largest (complex number). It must include natural, integer, rational, irrational, real and complex numbers.

How do we draw the number hierarchy from natural to complex in a Venn diagram?

Edit 1:

I found a diagram as follows, but it does not include the complex number.

My doubt is that shoul I add one more rectangle, that is a litte bit larger, to enclose the real rectangle? But I think the gap is too large enough only for i, right?

Edit 2:

Is it correct if I draw as follows?

• The complex numbers would simply surround the whole thing, while the reals should be split between the irrationals and rationals. So here the green part itself is irrational, while everything the green contains is real. Commented Oct 20, 2012 at 2:16
• There are a lot of number systems between the natural and the complex numbers. Most of those will hardly be mentioned explicitly outside courses for math students, though. Commented Oct 20, 2012 at 9:37
• Related question: math.stackexchange.com/questions/216177/… Commented Oct 21, 2012 at 6:52
• ガベージ, Your second diagram is what I originally considered as a further edit to mine, but if we want to be totally accurate, there's a problem. See how the circles that represent $\mathbb{Q}$ and $\mathbb{R}$ have small slivers outside of {0} that intersect $\mathbb{I}$? Technically, we shouldn't have that. Commented Oct 21, 2012 at 22:32
• maybe instead of trying to cram everything into a cute-but-limited 2D venn blob-graph thingy, we stop limiting our self and start by listing out the known sets (including computable numbers and other things) and then draw lines connecting stuff on a separate sheet of paper Commented Apr 12, 2020 at 17:26

Emmad's second link is just perfect, IMHO. For something right in front of you, here's this:

• Do not forget the outer rectangle for the quaternions!! Commented Oct 20, 2012 at 2:18
• Can you draw it? I cannot understand it without a real diagram. Commented Oct 20, 2012 at 2:25
• Sigur: working on it, and adding some characterizations. Commented Oct 20, 2012 at 2:38
• The union of the imaginary and real numbers is not the complex numbers!!! Commented Oct 20, 2012 at 2:47
• There's an infinite tower of larger non-associative real algebras in dimension powers of $2$ going on above $\mathbb{O},$ beginning with the sedenions $\mathbb{S}$, but they're of no use to anybody, as far as I know. Commented Oct 24, 2012 at 3:16

There is a good picture at: number-set-venn-diagram. For detailing Complex Numbers, you can see this one: Complex Numbers Venn Diagram.

You may decide to combine the two to get a very complex picture!

• The first link depicts real numbers and imaginary numbers as disjoint, while the second one shows (correctly, in my opinion) that their intersection is $\{0\}$.
– user856
Commented Oct 20, 2012 at 10:14
• @RahulNarain, its nice that the two worlds have something in common ;) Commented Oct 20, 2012 at 11:44
• Sorry, is each set open? I mean that whether or not points on the boundary of the set (in this case the rectangle) are included. Commented Oct 20, 2012 at 16:15
• In Venn diagrams, objects live within the boundaries not on the boundaries themselves (I think). The pictures are meant to show the general idea, something like a country's high level map. Commented Oct 20, 2012 at 18:39