
Title: Infinity

Series: Advent of Mathematical Symbols

YouTubeTitle: Advent of Mathematical Symbols  Part 24  Infinity

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Timestamps

Subtitle in English
1 00:00:00,586 –> 00:00:05,271 Hello and welcome to the last video of this series in this year.
2 00:00:06,043 –> 00:00:11,386 Therefore it’s the perfect moment thanking all the nice supporters on Steady and Paypal.
3 00:00:11,502 –> 00:00:13,914 You made all these videos possible.
4 00:00:14,814 –> 00:00:19,980 and now i can tell you the mathematical symbol of today is the infinity symbol.
5 00:00:20,986 –> 00:00:24,081 It’s a symbol that is used throughout mathematics,
6 00:00:24,281 –> 00:00:27,914 because it represents the concept of infinity.
7 00:00:28,714 –> 00:00:33,866 Of course then the specific meaning, when it is used, depends on the context.
8 00:00:34,800 –> 00:00:41,077 Indeed most of the time it just should remind us that some infinity is at work.
9 00:00:41,543 –> 00:00:50,393 For example if you want to write down the limit of a sequence, you would write limit n goes to infinity.
10 00:00:51,571 –> 00:00:57,730 and here in our example if the sequence is 1 over n, you know the limit is 0.
11 00:00:58,600 –> 00:01:03,226 Now here you should see there is no need to use the symbol infinity here
12 00:01:03,426 –> 00:01:05,729 because we know the definition of a limit.
13 00:01:07,014 –> 00:01:13,629 However this symbol is there to tell us that a number n should increase and increase without stopping.
14 00:01:14,300 –> 00:01:18,729 So in this case infinity is not a number we just put into the sequence.
15 00:01:19,786 –> 00:01:23,186 Rather infinity is used in a symbolic way here.
16 00:01:24,343 –> 00:01:30,885 In fact this is often the case, but sometimes we need to calculate with the symbol infinity.
17 00:01:31,786 –> 00:01:35,055 and then of course it acts more like a number.
18 00:01:36,143 –> 00:01:40,101 One important application of this you find in measure theory,
19 00:01:41,000 –> 00:01:46,004 because there we need to calculate with numbers from 0 to infinity.
20 00:01:46,204 –> 00:01:49,714 and 0 and infinity are included in the interval.
21 00:01:50,671 –> 00:01:55,749 One short explanation here is that in measure theory we deal with volumes.
22 00:01:56,543 –> 00:02:02,514 and of course it could happen that the volume is 0, but we also deal with infinite volumes.
23 00:02:03,514 –> 00:02:09,385 and then in calculations that occur we need to add volumes and multiply them.
24 00:02:10,229 –> 00:02:16,088 Therefore we need a meaning for “a + infinity” and “a times infinity”.
25 00:02:16,971 –> 00:02:23,941 So lets start with the first one. So “a + infinity” should of course be the same as “infinity + a”.
26 00:02:24,457 –> 00:02:31,357 and now because we see infinity as the largest number in our interval, this should be infinity again.
27 00:02:32,529 –> 00:02:37,186 Indeed this should also hold when “a” was infinity at the beginning.
28 00:02:37,929 –> 00:02:43,343 Or to put it in other words. Infinity + infinity should be infinity again.
29 00:02:44,514 –> 00:02:50,730 and there you already see the only meaningful thing for two times infinity is infinity again.
30 00:02:51,729 –> 00:02:55,771 In fact this should be our definition for all numbers “a”.
31 00:02:55,857 –> 00:02:57,700 Including infinity.
32 00:02:58,486 –> 00:03:03,170 So infinity times infinity is again infinity.
33 00:03:04,143 –> 00:03:09,514 However here one question remains: what to do with 0 times infinity?
34 00:03:10,471 –> 00:03:15,707 Usually for example when you deal with limits this expression is not well defined,
35 00:03:16,629 –> 00:03:20,657 but in measure theory it makes sense to define it.
36 00:03:21,443 –> 00:03:25,402 Indeed there 0 times infinity should be 0.
37 00:03:26,171 –> 00:03:30,179 This definition makes some formulas in measure theory simpler.
38 00:03:30,786 –> 00:03:32,805 Therefore it’s very helpful there.
39 00:03:33,800 –> 00:03:41,713 Now having these definitions now you can check that we still have commutative laws associative laws and distributive laws.
40 00:03:42,514 –> 00:03:49,023 Or in other words some calculation rules still hold when we deal with infinity in this way.
41 00:03:49,586 –> 00:03:55,230 Of course if you want to know more about measure theory, you can look at my whole series about it.
42 00:03:55,957 –> 00:04:00,214 Ok and with this i really hope you learned something in this series here.
43 00:04:01,229 –> 00:04:03,576 I wish you all the best and see you next time.
44 00:04:03,776 –> 00:04:04,586 Bye! :)