# Is there any difference between 'all real numbers' and '$(-\infty, \infty)$'

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)?

• The easiest way to represent that set is a single character – $\mathbb R$. – JustAskin Jul 12 '15 at 23:22
• $(-\infty,\infty)=\mathbb R$. – Akiva Weinberger Jul 12 '15 at 23:33

## 3 Answers

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: $$\left]-\infty,+\infty\right], \quad \left[-\infty,+\infty\right[, \text{ and } \left[-\infty,+\infty\right].$$

• (In case OP doesn't know: $]a,b[$ is another notation for $(a,b)$; also, $]a,b]$ is another notation for $(a,b]$.) – Akiva Weinberger Jul 12 '15 at 23:32
• I should have thought of that. Thanks for pointing! – Ivo Terek Jul 12 '15 at 23:33
• 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? – hs2345 Jul 12 '15 at 23:42
• Yes. The point is that $+\infty$ and $-\infty$ are not real numbers. – Ivo Terek Jul 12 '15 at 23:45

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).

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

• Is there a difference in some other context? – Antonio Vargas Jul 12 '15 at 23:21
• Just in case there was, I made it clear there was none in the world of precalc or calc. – Race Bannon Jul 12 '15 at 23:22
• 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. – Antonio Vargas Jul 13 '15 at 1:13
• Yes. Because if we are talking about $\mathbb{N}$ for example, then the interval $(-\infty,\infty)$ is not the same as the real numbers. – Race Bannon Jul 13 '15 at 1:16