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

Question

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)?

Answer 1

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

Answer 2

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).

Answer 3

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