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Consider numbers of the form $a+b\sqrt{D}$. Particularly when $D=-1$, these numbers are called complex numbers and $i=\sqrt{D}$ is the imaginary unit. Further, $a$ is called the real part, and $b$ is called the imaginary part (of the complex number).

My question is: how is the $D$ generally called? And a supplementary question: how should the $\sqrt{D}$, $a$, $b$, and the numbers of the form $a+b\sqrt{D}$ be called in the general case?

DaBler
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    $D$ is called the radicand, and $a$ and $b$ are sometimes called the rational and irrational part - though the component names are not in very wide use. See also this earlier question about notation for the parts. – Bill Dubuque Aug 15 '17 at 15:07
  • Note also that many number theorists and algebraists don't restrict "irrational" to reals but instead use the general definition irrational $:=$ not rational (see this answer) so the above names work also for complex numbers. – Bill Dubuque Aug 15 '17 at 15:21
  • @BillDubuque: This is actually the answer I was looking for (along with the lhf's answer below). I found that it really is being used, e.g. here. – DaBler Aug 16 '17 at 20:51

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If you don't restrict $a,b$, then the numbers of the form $a+b\sqrt{D}$ are exactly the real numbers when $D \ge 0$ and exactly the complex numbers when $D < 0$.

The numbers of the form $a+b\sqrt{D}$ with $a,b \in \mathbb Q$ and $\sqrt{D} \notin \mathbb Q$ are sometimes called quadratic numbers.

The numbers of the form $a+b\sqrt{D}$ with $a,b \in \mathbb Z$ are called quadratic integers. (Actually, there is a small difference when $D \equiv 1 \bmod 4$.)

$D$ is closely related to the discriminant of the quadratic field.

The components $a,b$ have no standard name when $D>0$.

lhf
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    Don't forget that for quadratic integers, we assume $D$ is square-free. – Arthur Aug 15 '17 at 12:24
  • Thank you for the clarification. However, the point of my question is: how are the individual components (especially $D$, possibly $\sqrt{D}$, $a$, and $b$) called in such cases? If necessary, we can restrict ourselves to quadratic integers. – DaBler Aug 15 '17 at 12:36
  • @DaBler, see my edited answer. – lhf Aug 15 '17 at 12:53
  • @lhf: I see. Could you further extend your answer to include the terms mentioned in Bill Dubuque's comments? Then I would accept it as the answer. Thanks. – DaBler Aug 16 '17 at 21:01