# Can a number be non-imaginary and non-real?

I passed across this chart on the web and got confused. The diagram implies that there are numbers that are neither real nor imaginary. Is that possible, or is it just an incorrect diagram?

I understand the box titled "Imaginary numbers", might actually mean to say "Pure imaginary numbers"? But even then, it implies that all real numbers are imaginary (as there are examples like 4 + 2i under complex numbers, which are imaginary), which is nonsense! Kindly help me. Thank you!

• The chart is fine. – Wuestenfux Dec 27 '18 at 15:05
• It depends on definition. I have seen $3+4i$, for instance, called "imaginary"; when the real part is zero like $(0+)4i$, then it's "pure imaginary". – Oscar Lanzi Dec 27 '18 at 15:08
• There are a couple things that are useless and distracting about this chart. The chain of inclusions is perhaps the most useful part. By the way, you should probably rid yourself of the ideal of a “number” as a particular thing. There is no real definition for it. – rschwieb Dec 27 '18 at 19:29
• The chart should have $1,2,3,\ldots$ only in $N$ and not also in $W$; $0$ only in $W$ and not also in $Z$; and $\sin(\pi/3)=\sqrt3/2$ in $A$. But that seems to be unrelated to the main question. – Teepeemm Dec 27 '18 at 21:46
• If you want to know about numbers that are not complex numbers (but still make sense in some way), you can look up p-adic numbers, those are pretty interesting – Yuriy S Dec 27 '18 at 22:53

A complex number is the sum of a pure imaginary number and a real number. However, there are numbers which are neither real or imaginary: Octonions.

EDIT:

As the comments pointed out, I skipped quaternions, but octonions are so much cooler. If you are looking for even more numbers to bend your brain, there are the surreal numbers and the transfinite numbers.

• Good point, but why did you skip Quaternions? – Tobias Kienzler Dec 27 '18 at 20:06
• Because a) octonions are cooler, and b) there aren't sedecimtonions. It ends at octonions. – William Grannis Dec 28 '18 at 15:14
• There are Sedenions ;) – Tobias Kienzler Dec 28 '18 at 15:29
• Related, but AFAIK they don't behave the same. – William Grannis Dec 28 '18 at 15:51

The number $$3+4i$$ is an example of a number that is neither real nor imaginary.

EDIT
What makes you say "it implies that all real numbers are imaginary"? There are real numbers (like $$\pi$$) and there are imaginary numbers (like $$i\pi$$). These two sets are almost disjoint (they only have number $$0$$ in common), but they are both subsets of complex numbers (which also contains numbers like $$3+4i$$).

• So what's the difference between an imaginary and a pure imaginary number? – Shaan Dec 27 '18 at 15:14
• @Shaan. I consider them as being the same, perhaps with the exception of the number $0$ that might be called imaginary but not pure imaginary. – md2perpe Dec 27 '18 at 15:16
• They're not disjoint since both include $0$. – J.G. Dec 27 '18 at 15:21
• @J.G. That's correct; I missed that special case. In that way the diagram is incorrect unless "Imaginary Numbers" refer to pure imaginary numbers which exclude zero. – md2perpe Dec 27 '18 at 15:24
• So, please clarify. Real number = 1/3, Imaginary = 2i, and Complex = 1/3 + 2i – Shaan Dec 27 '18 at 15:32

The chart is correct. Its author calls imaginary numbers to the numbers of the form $$\lambda i$$, with $$\lambda\in\mathbb R$$. So, $$1+i$$ is neither real nor imaginary.

• $\lambda \ne 0$? – Daniel R. Collins Dec 28 '18 at 2:45
• Quite likely, but you would have to ask that question is the author of the chart to be sure. – José Carlos Santos Dec 28 '18 at 3:15

Those are complex numbers. You might not have seen it because the letters are as dark as the background

I understand the box titled "Imaginary numbers", might actually mean to say "Pure imaginary numbers"?

At least I learned that the term "imaginary number" means what you call "pure imaginary number":

An "imaginary number" is a number that can be written as $$yi$$ (or $$0+yi$$) with $$y\in\mathbb R$$. A number that can be written as $$x+yi$$ with $$x,y\in\mathbb R\backslash\{0\}$$ (such as $$4+2i$$) is not called "imaginary number" according to this terminology.

But even then, it implies that all real numbers are imaginary ...

No. Why?

In the diagram the real numbers are drawn as subset of "complex numbers" ($$\mathbb C$$), not as subset of "imaginary numbers".

Every real number is a "complex number".

as there are examples like 4 + 2i under complex numbers, which are imaginary

With exception of the integer numbers, the examples in a box show numbers that do not fit in a "sub-box":

The examples in the "rational numbers" box do not contain an integer number; the examples in the "real numbers" box do not contain any rational number.

This does not mean that integers numbers are not rational...

One oddity of maths that does not fit with the typical public perception is that there are not absolute fixed definitions. Many will vary from author to author. Some definitions are agreed by pretty much everyone. For others, e.g. the exponential function, not all agree on the definition but all of the popular ones are equivalent. Sometimes, there is a significant variation e.g. rings in algebra, is a multiplicative identity required? Today, I think that most people would not say that 1 is prime but some old books might. The Bourbaki school even proposed that 0 be considered both positive and negative rather than neither.

Back to your case, it seems that you regard imaginary as meaning not real but others restrict imaginary to real mutiples of $$i$$. My only issue with that chart would be the need for a tiny overlap of the imaginaires and the reals containing $$0$$,