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I've just thought about this.

All the textbooks I've been looking at for pre-calc,

the domains are always written as 'all real numbers', whereas my calculus textbooks would rather write them as '$(-\infty, \infty)$.

Is there any difference using the terms vice versa (implicitly)?

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  • $\begingroup$ The easiest way to represent that set is a single character – $\mathbb R$. $\endgroup$ – JustAskin Jul 12 '15 at 23:22
  • $\begingroup$ $(-\infty,\infty)=\mathbb R$. $\endgroup$ – Akiva Weinberger Jul 12 '15 at 23:33
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There is no difference: $\Bbb R = \left]-\infty,+\infty\right[$. Writing it like this serves to get you used with the symbol $\infty$, I guess (mostly psychological reasons?). Also, there will be a time when you'll need to use concepts dealing with the extended real line, so it will be natural to talk about: $$\left]-\infty,+\infty\right], \quad \left[-\infty,+\infty\right[, \text{ and } \left[-\infty,+\infty\right].$$

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    $\begingroup$ (In case OP doesn't know: $]a,b[$ is another notation for $(a,b)$; also, $]a,b]$ is another notation for $(a,b]$.) $\endgroup$ – Akiva Weinberger Jul 12 '15 at 23:32
  • $\begingroup$ I should have thought of that. Thanks for pointing! $\endgroup$ – Ivo Terek Jul 12 '15 at 23:33
  • $\begingroup$ So in the context of all real numbers, the definitions specifies all real numbers to be in the domain of -infinity to +infinity all the times? $\endgroup$ – hs2345 Jul 12 '15 at 23:42
  • $\begingroup$ Yes. The point is that $+\infty$ and $-\infty$ are not real numbers. $\endgroup$ – Ivo Terek Jul 12 '15 at 23:45
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There is no difference. The notation $(-\infty, \infty)$ in calculus is used because it is convenient to write intervals like this in case not all real numbers are required, which is quite often the case. eg. $(-1,1)$ only the real numbers between -1 and 1 (excluding -1 and 1 themselves).

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As far as precalculus is concerned, there is no difference.

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    $\begingroup$ Is there a difference in some other context? $\endgroup$ – Antonio Vargas Jul 12 '15 at 23:21
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    $\begingroup$ Just in case there was, I made it clear there was none in the world of precalc or calc. $\endgroup$ – Race Bannon Jul 12 '15 at 23:22
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    $\begingroup$ This answer is correct and the author was just explaining where they were sure about their knowledge. I don't think it should be downvoted, and my comment definitely wasn't intended to point out any kind of inadequacy. $\endgroup$ – Antonio Vargas Jul 13 '15 at 1:13
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    $\begingroup$ Yes. Because if we are talking about $\mathbb{N}$ for example, then the interval $ (-\infty,\infty)$ is not the same as the real numbers. $\endgroup$ – Race Bannon Jul 13 '15 at 1:16

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