Above is a diagram I made to illustrate the common number types. There apparently is no agreement on the definition of “number” and the meaning has changed over the centuries. I will be using this diagram to assist in other posts on Math & the Religious Mind. Let me know if your favorite type of number is not on the chart. Meanwhile, go ahead, I dare you, find a better chart than this one out there on the web! 🙂

**Related Posts**: Math & the Religious Mind: an index post

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Nice. I deal with Quaternions a lot (I have a chapter on them in my book), they are slightly out beyond the complex. And recently have been doing stuff with Geometric algebra, which is out again. But when you get out there a) the idea of a number isn’t much use. and b) there are different ways to generalize, so you’d have to draw a tree.

So I think your diagram is fine 🙂

@ Ian,Ooops, I missed this comment. Thanks mate. I wish I understood what a quaternion is! (What book, btw? Link?)

I also wish I understood how “number” is not of much use. I wonder if any modern science/math writers have written anything easy enough for even me to understand on the issue. Glad you liked this even if it is rudimentary! 🙂

Nice diagram. There are also hyperreal numbers (which capture, in an iconoclastic manner) the notion of infinitesimal and haven’t found much use in mainstream mathematics. Also, there are surreal numbers which capture infinities beyond the “integer infinities”–whatever this means. They have found applications in 2-person games (games are surreal numbers, so they say).

As for “imaginary” numbers, yet another unfortunate terminology. For a constructivist, there is nothing imaginary about \sqrt(-1). It is, simply, rotation in the plane around the origin by +90 degrees. (And other, equivalent, representations–depends on one’s taste.). We just choose to call rotations, and other such transformations, numbers, because they behave as numbers. For example, squaring a rotation by +90 degrees means applying it twice, i.e. rotating by 180 degrees. So, a vector with coordinates (x,y), rotated twice by 90 degrees becomes the vector with coordinates (-x, -y), that is, the original vector multiplied by -1. It is for this reason that (rotation by 90)^2 = -1, and so (rotation by 90) = \sqrt(-1).

Also, we should remember that there is nothing real about real numbers. Long time ago, legend has it, people used to lose their lives if they contended that the length of the hypotenuse of an isosceles right-angle triangle divided by the length of one of the equal sides is not a fraction. It took long time for humanity to accept real numbers so much so that, nowadays, it is not too hard, even at an intuitive level, to “explain” them to a child. Also, we should keep in mind that real numbers can be constructed (just like I attempted to construct imaginary numbers above); and that there are many constructions.

In a sense, “number” is anything which can be treated by operations like addition and multiplication, and for which we can formulate equations with unknowns, e.g., x, y,… and solve them–or ask for their solution. For example, the equation x^2 + x = 6 makes sense if we think of x as real (and we can find two solutions, x=2, and x=-3), but the equation x^2=2x-2 still makes sense if we think of x as real–but the answer is: there are no solutions–but also makes sense when x is thought as some kind of transformation of the plane with the property that applying the transforation twice (this is the meaning of x^2) is the same as subtracting the identity to the transformation (that is, x-1), followed by a change of scale by a factor of 2 (this is 2(x-1)). However, instead of thinking of all this “complicated geometry”, just because these tranformations behave as numbers, we make the problem “abstract”, forget their interpretation of x as a geometric transformation, call it, instead, a complex number, and treat the equation within the realm of complex numbers. This enables us to find the solution(s). In the end, we can re-interpret the solutions as geometric objects–if we so wish.

Another use of complex numbers is in order to design electric power supply networks. Before the era of computers, even practical electricians had learned (by rote) ways to manipulate complex numbers.

Nothing complex. Nothing imaginary. As always, mathematics introduces new languages in order to treat real objects. But the language (a) has to be learned and understood and (b) the interpretation should not be remembered at all times: abstraction aids our compactification of accumulated knowledge.

Apologies for the rant. Just had a few minutes available. In fact, I’m procrastinating from real work… 🙂

Excellent additions and clarifications, Takis — nothing better than hearing from the Mathematicians. May I ask how you suddenly happened on this post? Today I just the pic to my side column. Were you just looking over the blog?

Indeed, indeed. I was looking at your blog, the diagram caught my attention, clicked on it, and found myself on your older posting. And (as usual) I got diverted and started reading it and felt like adding some comments.

I said, towards the end: “Nothing complex. Nothing imaginary.” I meant, of course, there is nothing complex about complex numbers and nothing imaginary about imaginary numbers. I also wanted to add: there is nothing real about real numbers. (And hence make my posting a bit tao-istic.) But it’s true: one (i.e., a mathematician) shouldn’t cling too much to one point of view, one particular idea, one interpretation. Ability to switch sides and ways of viewing things is so important! Further, language does constrain the meaning, and this is why it’s important to define terms 🙂

A few years ago, a French minister of education said that children in schools learn irrelevant mathematics, for example, they are being taught useless things such as how to take the square root of minus one. (He was, I think, a geologist himself); rather than learning imaginary things, they should learn real things. He was put in place by a mathematician, a member of the academy of sciences, Marc Yor, whom I happened to know personally (he died recently). Story has it that the minister took back his sayings and probably resigned.

You said:

…. nothing better than hearing from the Mathematicians.

Thank you. But to that, I would like to add: 1) there are (many) mathematicians who don’t want to explain, or do not want to explain, or have never wondered about the process that made them learn; and so, I think you should say “nothing better than hearing from those mathematicians who have thought about what it means to know mathematics and what mathematics is”. Actually, we should further adjust this. There are many non-mathematicians who also know how to think about mathematics. Sometimes, better than self-proclaimed mathematicians (because the latter ones may have pressure to say they’re mathematicians because they work in an environment which judges them from a mathematical point of view). So, a further adjustment would transform your claim into: “nothing better than hearing from those people who have thought about what it means to know mathematics and what mathematics is”.

OK, I’m off to a birthday party of a one-year old girl.

Great additions and caveats, Takis. Enjoy the party

Another fun subset of the reals: the computable numbers, which include any number whose digits finite, terminating algorithm can compute. Essentially any real number one would run across is computable–all algebraic numbers, pi, e, etc–however, the set of computable numbers is countable, since the set of all finite algorithms is countable, while the reals are countable.

I’m also rather interested on how you chose the sizes of the various sets of numbers. Most make sense intuitively (although the implication that the whole numbers contain something the natural numbers don’t, other than zero, is off even intuitively), but none do mathematically. Of course, it is rather hard to illustrate even the relative sizes of infinite sets with finite shapes.

Also, the algebraic numbers blob should be spilling into the complex and imaginary blobs.

Hey CRL:

Fun! I’d not heard of computable numbers, countable sets — this stuff goes over my head quickly, but thanx for enlightening.

Good point about size of the ovals — I paid no attention to them at all.

My challenge to you: make a diagram you are proud of, use IMGUR or some such thing and share it! (if you have time, of course)

The diagram may happen at some point, but right now, I can’t say I’ve thought of a way to make a more accurate diagram which would be worth 1000 words; visually representing infinite sets is hard. Meanwhile, a friend of mine sent me a related link a few days ago. While it’s nominally about the “no self defeating object” argument, it touches a lot on what I was talking about above.

@CRL

“Visually representing infinite sets is hard.”

The difficulty in representing relations between sets has nothing to do with their size.

@ CRL & Takis,Both of you guys, and several other commentors on this thread, are leagues beyond me in your understanding of mathematics. I drew this graph because I was not satisfied with the ones I found on the web. So, I invite all of you, if you please do make another or a supplementary one and I will add it here.

I’d be curious to hear what Takis thought of CRL’s linked article. I read a little but it was over my head quickly. I’d love to hear Takis and CRL talk about its significance, errors, insights …..

Thanx folks. Love this stuff. Wish I knew more.

I’m not sure I agree with you, Takkis. It’s pretty easy to have pictures representing the relative size of finite sets–a circle representing the size of {1,*, potato} should be three times the size of {potato}, and the circle representing {potato} should inside the circle for {1,*,potato}. Here’s a visual representation of those sets.

Then it seems like {integers} and {integers divisible by three} could be represented by the same diagram, since 1/3 of the integers are divisible by three. But, also, the cardinality (“size”) of the sets are the same, so it seems off that one set’s picture is 1/3 the size of the other. So if we’re representing countably infinite sets as pictures, we need to choose between making relative sizes of circles represent cardinality, and making relative sizes of circles represent what fraction of elements of a set are in a subset of that set.

@CRL:

I was talking about logical relations between sets, not diagrams which reflect other aspects of them such as their size.

Fair enough.

transcendental numbers

Thanx, jsmunroe

They were there, but just not labeled — I fixed that!