6

I came across this question here: Difference between imaginary and complex numbers

The top answer contains this diagram:

enter image description here

Here we see numbers like $e - \pi i$ and $\pi + 3i$ existing outside of transcendental and algebraic numbers but within the realm of complex numbers. Is this accurate or should they technically be in the transcendental area? Are there any complex non-transcendental non-algebraic numbers?

We also see $0$ as a whole number, an integer, a rational number, a real number, an algebraic number, and a complex number, but is it not also a pure imaginary number?

user525966
  • 5,631

1 Answers1

2

They're transcendental, but the diagram distinguishes only transcendental real numbers for whatever reason. Non algebraic imaginary numbers are transcendental, and those are represented but not named in the diagram.

$0$ is a real number. Whether or not it's pure imaginary depends on your definition. If you define it as having a real part of $0$, then yes, it is.

Matt Samuel
  • 58,164
  • So the diagram would be more accurate if we pulled in $e -\pi i$ and $\pi + 3i$ inside the transcendental section? What about the pure imaginary numbers $e \pi$ and $\pi i$? Are these non-algebraic, non-transcendental complex numbers? – user525966 Oct 28 '18 at 20:46
  • 1
    @user The diagram is $100%$ accurate. In math, if something isn't $100%$ accurate, it's garbage. The diagram simply doesn't name transcendental complex numbers. They should not be included in the transcendental section since that is contained inside the reals. – Matt Samuel Oct 28 '18 at 20:49
  • It appears to name $e$ and $\pi$ as transcendental complex numbers? (contained within both). Anything involving $i$ simply not being within the "real" bubble? – user525966 Oct 28 '18 at 20:52
  • @user They are transcendental complex numbers that happen to be real, but notice they're in the green circle, which is the real numbers. – Matt Samuel Oct 28 '18 at 20:54
  • Transcendental numbers can be either real or non-real? So we could wrap a bubble aroud the lefthand numbers there ($pi, e, e - \pi i, \pi + 3i$) and call those "transcendental" instead of only localizing transcendental as if it were just part of the "real" bubble? – user525966 Oct 28 '18 at 20:56
  • (I'm phrasing it this way because the diagram seems to label everything as a "complex number" and so I am trying to be precise when you mention that "the diagram isn't naming transcendental complex numbers"), unless you are using "complex" to refer to "non-real"? – user525966 Oct 28 '18 at 20:57
  • @user Maybe they prefer to call transcendental imaginary numbers "nonalgebraic." They're entitled. But you could extend it past the real bubble if you want. – Matt Samuel Oct 28 '18 at 20:58
  • @user I meant imaginary when I said complex. – Matt Samuel Oct 28 '18 at 20:59