Is infinity a real number?
If not, why not?
I want some very good arguments.
Thanks.
$$\rightarrow\leftarrow\Huge\Huge\Huge\boldsymbol\infty$$
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8No. By definition. As good and short as possible, imo. – DonAntonio Apr 12 '14 at 16:08
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1$\infty-\infty=0$? – npisinp Apr 12 '14 at 16:14
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4Why are people marking this question down? It's a fair question. – Stephen Montgomery-Smith Apr 12 '14 at 16:18
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@npisinp No, $\infty-\infty$ is an indeterminate form. Google: Hilbert's Hotel or check this answer to understand why. – Hakim Apr 12 '14 at 16:18
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2@StephenMontgomery-Smith I couldn't agree more with you. It's a fair question. – Tunk-Fey Apr 12 '14 at 16:20
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@StephenMontgomery-Smith Because having a big random $\infty$ symbol and random tags is not good – dani_s Apr 12 '14 at 16:21
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Yes, the old tag "wreath-products" was completely wrong. But someone fixed it. – Stephen Montgomery-Smith Apr 12 '14 at 16:22
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This question was tagged "wreath product." If you want people to take time to answer your question, take time to ask it. – Jonathan Hebert Apr 12 '14 at 16:23
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I thought it would be creative to do so and that people would ''enjoy'' it :/ – user142652 Apr 12 '14 at 16:23
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I think the reason for the question is that in high school math we are taught that infinity is a non-concept. It is correct that the concept of infinity should be used with care. – Stephen Montgomery-Smith Apr 12 '14 at 16:24
3 Answers
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No. If you look up the definition of the real numbers, you will not find any of its elements called "infinity".
However, the extended real numbers has two numbers called $+\infty$ and $-\infty$, which become the endpoints of the number line in the extended reals.
There are other structures that have an element or places named "infinity" that are similar but often different.
There are other situations where there are "infinite" objects, although we would never use the noun "infinity" to refer to them; e.g. the infinite cardinal numbers and ordinal numbers, or the unlimited hyperreal numbers.
7
Infinity $(\infty)$ is not a real number, it is merely an abstract concept describing something that has no end. Boundless! We could sometimes use infinity like it is a number, but infinity does not behave like a real number. To help you understand, think "endless" whenever you see the infinity symbol "$\infty$". For example: $$\infty+1=\infty$$ Which says that infinity plus one is still equal to infinity, even $$\infty+\infty=\infty$$ If something is already endless, you can add $1$ or any number you want and it will still be endless.
Infinity cannot be measured, even the universe itself cannot compete with infinity. Most things we know have an end, but infinity does not.
Tunk-Fey
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Infinity isn't a real number by definition. This definition is sensible because adding $\infty$ to $\mathbb{R}$ would break its field structure, and the fact that $\mathbb{R}$ is a field brings a lot of nice properties to real numbers
Alessandro Codenotti
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Well that's the problem with wikipedia - it's math definitions tend to be a bit lengthy. Try some other google search for "field mathematics." – Stephen Montgomery-Smith Apr 12 '14 at 16:14
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2But one thing a field must satisfy is that if $a+c = b+c$ then $a = b$. This wouldn't work if we were to allow $c = \infty$, because as other posts have said, that would imply $1 = 2$. – Stephen Montgomery-Smith Apr 12 '14 at 16:16
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A commutative ring is a set (for example, the real numbers) with two operations (for example, multiplication and addition) that interact in a special way (you use these interactions all the time to solve equations). Adding infinity to the set of real numbers makes these interactions not work anymore. – Jonathan Hebert Apr 12 '14 at 16:16
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