I have seen in some book about extend real number system the following: is |infinity| = |-infinity|= infinity ? When infinity is not a number, how can we think about modulus? Or infinity is number or not. Please discuss. I think infinity is not a number. This is my feeling only. How far I am correct, I really do not know. Please discuss, especially about the modulus of infinity how defined, when infinity is a not a number. Please...
about modulus of not a number or infinity case
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0See also: [Is Inifity a number?](http://math.stackexchange.com/questions/36289/is-infinity-a-number) I think it has more elaborate answers that related to this one. – 2016-09-09
2 Answers
Usually, when we are working with the extended real numbers, we take continuous functions that are defined on the reals (or some subset) and extended them by continuity to have values at $+\infty$ or $-\infty$ when possible.
In the case of absolute values, we define $|+\infty| = +\infty$ and $|-\infty| = +\infty$ for this reason: i.e. because
$ \lim_{x \to +\infty} |x| = +\infty \qquad \text{and} \qquad \lim_{x \to -\infty} |x| = +\infty $
Whether you want to call $+\infty$ and $-\infty$ numbers or not is not really relevant to this situation. I believe that it is very useful to consider them numbers... but maybe that should only be done once you are comfortable doing arithmetic with them.
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0!great explanation is given – 2012-12-05
Infinity can be a number if you want it to be. Mathematicians can define any sort of number system. What is important is if it's useful and interesting. We see that infinity is not considered a number in the set of real numbers, which is denoted by $\mathbb{R}$. However, in Calculus and other subjects, it helps to informally (sometimes formally) consider infinity a number with a special properties in order to evaluate limits. This number system is called the Extended Real Numbers.
I believe you feel this is simply a yes or no question, and it is not. The answer depends entirely on what you're working with. In some number systems, infinity is defined. In other number systems, infinity is not defined.
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0Great attempt! +1 for it. :) – 2012-12-07