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Title: Construction
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Series: Start Learning Reals
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Parent Series: Start Learning Mathematics
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YouTube-Title: Start Learning Reals 4 | Construction
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Subtitle in English
1 00:00:00,430 –> 00:00:02,089 Hello and welcome back to
2 00:00:02,099 –> 00:00:03,480 start learning
3 00:00:03,490 –> 00:00:04,250 reals.
4 00:00:04,699 –> 00:00:05,889 And as always, I want to
5 00:00:05,900 –> 00:00:07,250 thank all the nice people
6 00:00:07,260 –> 00:00:08,369 that support this channel
7 00:00:08,380 –> 00:00:09,850 on study or paypal.
8 00:00:10,489 –> 00:00:12,079 In today’s part four, I will
9 00:00:12,090 –> 00:00:13,810 show you how we can actually
10 00:00:13,819 –> 00:00:15,449 construct the real numbers.
11 00:00:16,110 –> 00:00:17,530 This means that the starting
12 00:00:17,540 –> 00:00:18,959 point is the field of the
13 00:00:18,969 –> 00:00:19,979 rational numbers.
14 00:00:20,399 –> 00:00:22,260 And then we form a new field
15 00:00:22,270 –> 00:00:24,049 where every Cauchy sequence
16 00:00:24,059 –> 00:00:25,590 is actually convergent.
17 00:00:26,379 –> 00:00:27,690 Now because you have seen
18 00:00:27,700 –> 00:00:28,889 all the other number
19 00:00:28,899 –> 00:00:30,270 constructions, you might
20 00:00:30,280 –> 00:00:31,709 already guess that also
21 00:00:31,719 –> 00:00:33,700 here equivalence relations
22 00:00:33,709 –> 00:00:34,590 will be helpful.
23 00:00:35,340 –> 00:00:37,150 Our idea in this case should
24 00:00:37,159 –> 00:00:38,770 be that each point on the
25 00:00:38,779 –> 00:00:40,189 number line can be
26 00:00:40,200 –> 00:00:42,090 represented by a Cauchy
27 00:00:42,099 –> 00:00:42,849 sequence.
28 00:00:43,340 –> 00:00:45,150 For example, for the 0.1
29 00:00:45,159 –> 00:00:46,630 3rd, we can choose the
30 00:00:46,639 –> 00:00:48,619 sequence that has the
31 00:00:48,630 –> 00:00:50,450 members one third, one third
32 00:00:50,459 –> 00:00:51,270 and so on.
33 00:00:51,849 –> 00:00:53,349 So it’s simply a constant
34 00:00:53,360 –> 00:00:54,169 sequence.
35 00:00:54,180 –> 00:00:55,639 Therefore, in particular
36 00:00:55,650 –> 00:00:57,569 a Cauchy sequence and also
37 00:00:57,580 –> 00:00:58,889 a convergence sequence.
38 00:00:59,419 –> 00:01:00,939 However, the important part
39 00:01:00,950 –> 00:01:02,529 is that the limit is our
40 00:01:02,540 –> 00:01:04,010 0.1 3rd.
41 00:01:04,949 –> 00:01:06,319 Therefore, this infinite
42 00:01:06,330 –> 00:01:07,650 sequence of numbers
43 00:01:07,660 –> 00:01:09,559 represents one point on the
44 00:01:09,569 –> 00:01:10,309 number line.
45 00:01:10,970 –> 00:01:12,669 But now you should see that
46 00:01:12,680 –> 00:01:14,669 this representation can be
47 00:01:14,680 –> 00:01:16,589 in no way unique because
48 00:01:16,599 –> 00:01:18,230 we can just write down another
49 00:01:18,239 –> 00:01:19,830 sequence with the same limit.
50 00:01:20,610 –> 00:01:22,220 For example, we can just
51 00:01:22,230 –> 00:01:24,199 use decimals and start with
52 00:01:24,209 –> 00:01:25,800 0.3.
53 00:01:26,430 –> 00:01:27,830 Of course, you already know
54 00:01:27,849 –> 00:01:29,550 this is just another notation
55 00:01:29,559 –> 00:01:31,500 for the fraction 3/10.
56 00:01:32,199 –> 00:01:33,360 And then the next member
57 00:01:33,370 –> 00:01:34,589 in the sequence should be
58 00:01:34,599 –> 00:01:36,699 0.33.
59 00:01:37,139 –> 00:01:38,900 With this, we just continue
60 00:01:38,910 –> 00:01:40,169 a pending threes.
61 00:01:41,029 –> 00:01:41,370 OK.
62 00:01:41,379 –> 00:01:42,569 With this, we have a well
63 00:01:42,580 –> 00:01:44,129 defined sequence where you
64 00:01:44,139 –> 00:01:45,989 can also easily check that
65 00:01:46,000 –> 00:01:47,970 it is also a Cauchy sequence.
66 00:01:48,489 –> 00:01:49,620 And because we get arbitrarily
67 00:01:50,150 –> 00:01:51,849 close to 1/3,
68 00:01:51,860 –> 00:01:53,650 eventually, we also know
69 00:01:53,660 –> 00:01:55,230 we have a conversion sequence
70 00:01:55,239 –> 00:01:56,809 with limit one third.
71 00:01:57,559 –> 00:01:59,339 So we can conclude when we
72 00:01:59,349 –> 00:02:01,099 form our equivalence classes
73 00:02:01,110 –> 00:02:03,069 soon, these two sequences
74 00:02:03,080 –> 00:02:04,779 here should land in one
75 00:02:04,790 –> 00:02:06,639 single box simply
76 00:02:06,650 –> 00:02:08,059 because they have the same
77 00:02:08,070 –> 00:02:08,538 limit.
78 00:02:09,020 –> 00:02:10,699 However, you already know
79 00:02:10,710 –> 00:02:12,139 we cannot just look at the
80 00:02:12,149 –> 00:02:13,740 limits because there are
81 00:02:14,050 –> 00:02:15,880 Cauchy sequences in Q
82 00:02:15,889 –> 00:02:16,860 without limits.
83 00:02:17,589 –> 00:02:19,100 Indeed, for this, we will
84 00:02:19,110 –> 00:02:20,550 find a simple solution.
85 00:02:21,000 –> 00:02:21,419 OK.
86 00:02:21,429 –> 00:02:23,020 The first step is to define
87 00:02:23,029 –> 00:02:24,660 the set of all the Cauchy
88 00:02:24,669 –> 00:02:26,250 sequences we consider.
89 00:02:26,880 –> 00:02:28,729 So here XN stands for a
90 00:02:28,740 –> 00:02:30,259 sequence which means an
91 00:02:30,270 –> 00:02:31,580 infinite list with the
92 00:02:31,589 –> 00:02:33,449 properties that
93 00:02:33,460 –> 00:02:35,000 for all natural numbers
94 00:02:35,009 –> 00:02:36,990 NXN is
95 00:02:37,000 –> 00:02:38,050 a rational number
96 00:02:38,729 –> 00:02:40,130 and that the sequence
97 00:02:40,139 –> 00:02:41,779 XN is indeed a Cauchy
98 00:02:42,119 –> 00:02:42,779 sequence.
99 00:02:43,570 –> 00:02:45,509 And this nice set that contains
100 00:02:45,520 –> 00:02:47,270 all the Koshy sequences
101 00:02:47,279 –> 00:02:48,750 is denoted by C.
102 00:02:49,539 –> 00:02:51,320 Now for two elements from
103 00:02:51,330 –> 00:02:52,789 the set, let’s call them
104 00:02:52,800 –> 00:02:54,479 A and BN.
105 00:02:54,990 –> 00:02:56,240 So for two Cauchy
106 00:02:56,250 –> 00:02:58,160 sequences, we define
107 00:02:58,330 –> 00:02:59,559 an equivalence relation
108 00:03:00,250 –> 00:03:00,729 OK.
109 00:03:00,740 –> 00:03:02,250 At this point we want that
110 00:03:02,259 –> 00:03:03,990 these two sequences here
111 00:03:04,000 –> 00:03:05,990 are equivalent, but we don’t
112 00:03:06,000 –> 00:03:07,419 want to use the explicit
113 00:03:07,429 –> 00:03:07,940 limit.
114 00:03:08,789 –> 00:03:09,839 Therefore, the question is
115 00:03:09,850 –> 00:03:11,500 here, how can we do that
116 00:03:11,509 –> 00:03:13,350 in a simple way for
117 00:03:13,360 –> 00:03:13,770 this?
118 00:03:13,779 –> 00:03:15,490 Let’s imagine that we see
119 00:03:15,500 –> 00:03:16,880 all the sequence members
120 00:03:16,889 –> 00:03:18,250 here on the number line.
121 00:03:19,009 –> 00:03:20,110 In the case that the two
122 00:03:20,119 –> 00:03:22,009 Cauchy sequences, we present
123 00:03:22,020 –> 00:03:23,289 the same point, they
124 00:03:23,300 –> 00:03:24,770 accumulate around this
125 00:03:24,779 –> 00:03:25,300 number.
126 00:03:25,899 –> 00:03:27,350 So the sequence members get
127 00:03:27,360 –> 00:03:29,000 closer and closer to this
128 00:03:29,009 –> 00:03:29,449 point.
129 00:03:30,130 –> 00:03:31,389 Therefore, the difference
130 00:03:31,399 –> 00:03:32,979 between two members from
131 00:03:32,990 –> 00:03:34,509 the two sequences gets
132 00:03:34,520 –> 00:03:35,910 smaller and smaller.
133 00:03:36,630 –> 00:03:38,360 Hence, this difference is
134 00:03:38,369 –> 00:03:39,779 the new sequence we should
135 00:03:39,789 –> 00:03:40,380 consider.
136 00:03:41,139 –> 00:03:42,639 And now we already discussed
137 00:03:42,649 –> 00:03:44,259 that it should be a convergence
138 00:03:44,270 –> 00:03:46,080 sequence with limit zero.
139 00:03:46,970 –> 00:03:48,550 In other words, for any point
140 00:03:48,559 –> 00:03:50,009 on the number line, we just
141 00:03:50,020 –> 00:03:51,889 shift the problem to zero.
142 00:03:52,649 –> 00:03:53,110 OK.
143 00:03:53,119 –> 00:03:54,309 Now we have everything we
144 00:03:54,320 –> 00:03:56,029 need because we can show
145 00:03:56,039 –> 00:03:57,750 that this defines indeed
146 00:03:57,759 –> 00:03:58,990 an equivalence relation.
147 00:03:59,529 –> 00:04:00,860 This means that we can show
148 00:04:00,869 –> 00:04:02,050 the three properties.
149 00:04:02,059 –> 00:04:03,850 It’s reflexive symmetric
150 00:04:03,860 –> 00:04:04,750 and transitive.
151 00:04:05,449 –> 00:04:06,830 Now, as always, when we have
152 00:04:06,839 –> 00:04:08,250 such a nice equivalence
153 00:04:08,259 –> 00:04:09,990 relation, we can go over
154 00:04:10,000 –> 00:04:11,830 to the boxes where we put
155 00:04:11,839 –> 00:04:13,190 in the equivalent Cauchy
156 00:04:13,199 –> 00:04:13,960 sequences.
157 00:04:14,399 –> 00:04:16,089 And we call these boxes
158 00:04:16,100 –> 00:04:17,350 equivalence classes
159 00:04:18,100 –> 00:04:19,559 by definition, the equivalence
160 00:04:19,570 –> 00:04:21,279 class of X and is
161 00:04:21,290 –> 00:04:23,000 just a set of all Cauchy
162 00:04:23,010 –> 00:04:24,380 sequences that are
163 00:04:24,390 –> 00:04:25,690 equivalent to XN.
164 00:04:26,279 –> 00:04:28,079 And therefore, this equivalence
165 00:04:28,089 –> 00:04:29,440 class now uniquely
166 00:04:29,450 –> 00:04:31,119 represents one point on the
167 00:04:31,130 –> 00:04:31,920 number line.
168 00:04:32,760 –> 00:04:33,299 To put it.
169 00:04:33,309 –> 00:04:34,910 In other words, we now can
170 00:04:34,920 –> 00:04:36,540 actually define the real
171 00:04:36,549 –> 00:04:38,429 numbers, the set
172 00:04:38,440 –> 00:04:40,359 R is just given by the
173 00:04:40,369 –> 00:04:42,200 set of all equivalence
174 00:04:42,209 –> 00:04:42,760 classes.
175 00:04:43,500 –> 00:04:45,279 So the set of these boxes
176 00:04:45,290 –> 00:04:47,119 defines the complete number
177 00:04:47,130 –> 00:04:47,480 line.
178 00:04:48,279 –> 00:04:49,459 Now, the only things that
179 00:04:49,470 –> 00:04:50,660 are still missing are the
180 00:04:50,670 –> 00:04:52,160 two operations, addition
181 00:04:52,170 –> 00:04:53,700 and multiplication and the
182 00:04:53,709 –> 00:04:54,239 ordering.
183 00:04:54,880 –> 00:04:55,859 Therefore, for the first
184 00:04:55,869 –> 00:04:57,500 part, let’s define how we
185 00:04:57,510 –> 00:04:59,410 can add two equivalence classes
186 00:04:59,420 –> 00:04:59,779 here.
187 00:05:00,359 –> 00:05:01,510 Indeed, that’s not hard at
188 00:05:01,519 –> 00:05:01,899 all.
189 00:05:01,910 –> 00:05:03,529 We can just use the addition
190 00:05:03,540 –> 00:05:04,720 we have for the rational
191 00:05:04,730 –> 00:05:06,579 numbers for all the members
192 00:05:06,589 –> 00:05:07,829 of the two sequences.
193 00:05:08,399 –> 00:05:10,019 Of course, as always, then
194 00:05:10,029 –> 00:05:11,410 we need to check that this
195 00:05:11,420 –> 00:05:13,179 definition does not depend
196 00:05:13,190 –> 00:05:14,700 on the chosen representation
197 00:05:14,709 –> 00:05:16,170 of the two occurrence classes.
198 00:05:16,890 –> 00:05:18,179 In this case, you just have
199 00:05:18,190 –> 00:05:19,190 to calculate a little bit
200 00:05:19,200 –> 00:05:20,309 with the sequences.
201 00:05:20,320 –> 00:05:21,529 And then we get out.
202 00:05:21,540 –> 00:05:23,040 Yes, it’s well defined.
203 00:05:23,750 –> 00:05:24,890 And afterwards, you will
204 00:05:24,899 –> 00:05:26,299 believe me that everything
205 00:05:26,309 –> 00:05:27,190 works the same.
206 00:05:27,200 –> 00:05:28,970 When we define the multiplication,
207 00:05:29,720 –> 00:05:31,339 we just use again the well
208 00:05:31,350 –> 00:05:32,690 known multiplication for
209 00:05:32,700 –> 00:05:33,579 rational numbers.
210 00:05:34,390 –> 00:05:34,850 OK.
211 00:05:34,859 –> 00:05:36,570 Now looking back at the axioms
212 00:05:36,579 –> 00:05:38,279 we want to fulfill, you see
213 00:05:38,290 –> 00:05:39,630 that the last thing we need
214 00:05:39,640 –> 00:05:41,480 to define is an ordering.
215 00:05:41,940 –> 00:05:43,649 So we want to define in which
216 00:05:43,660 –> 00:05:45,619 cases one equivalence class
217 00:05:45,630 –> 00:05:47,070 is greater than another.
218 00:05:47,079 –> 00:05:47,410 One.
219 00:05:47,619 –> 00:05:48,529 Of course.
220 00:05:48,540 –> 00:05:49,869 Here we need to use the
221 00:05:49,880 –> 00:05:51,299 ordering we have for the
222 00:05:51,309 –> 00:05:52,450 rational numbers again.
223 00:05:53,250 –> 00:05:54,880 However, claiming this for
224 00:05:54,890 –> 00:05:56,679 all sequence members is
225 00:05:56,690 –> 00:05:58,100 much too restrictive for
226 00:05:58,109 –> 00:05:59,799 us to see this.
227 00:05:59,809 –> 00:06:01,260 Please keep the number line
228 00:06:01,269 –> 00:06:01,799 in mind.
229 00:06:02,529 –> 00:06:04,320 For example, our B and C
230 00:06:04,329 –> 00:06:06,040 could accumulate around
231 00:06:06,049 –> 00:06:07,950 one third and
232 00:06:07,959 –> 00:06:09,700 maybe the sequence with AM
233 00:06:09,709 –> 00:06:11,350 has zero as the limit.
234 00:06:12,049 –> 00:06:13,929 But of course, we have infinitely
235 00:06:13,940 –> 00:06:15,510 many sequence members here.
236 00:06:16,290 –> 00:06:17,880 Therefore, for example, a
237 00:06:17,890 –> 00:06:19,739 four could lie over here,
238 00:06:20,619 –> 00:06:22,450 but still the limit would
239 00:06:22,459 –> 00:06:23,109 be zero.
240 00:06:23,670 –> 00:06:25,399 Hence, this inequality
241 00:06:25,410 –> 00:06:27,149 here should be satisfied
242 00:06:27,160 –> 00:06:28,470 only eventually.
243 00:06:29,230 –> 00:06:30,399 In other words, we can just
244 00:06:30,410 –> 00:06:31,959 ignore finally many
245 00:06:31,970 –> 00:06:32,600 members.
246 00:06:33,269 –> 00:06:34,609 Therefore, we would write
247 00:06:34,619 –> 00:06:36,369 there exists a capital N
248 00:06:37,109 –> 00:06:38,940 such that for all indices
249 00:06:38,950 –> 00:06:40,820 greater than capital N, we
250 00:06:40,829 –> 00:06:42,579 have A N is less
251 00:06:42,589 –> 00:06:43,459 than BN.
252 00:06:44,140 –> 00:06:44,559 OK.
253 00:06:44,570 –> 00:06:46,239 This now looks better, but
254 00:06:46,250 –> 00:06:47,940 it is still not correct.
255 00:06:48,549 –> 00:06:50,100 In order to see this, just
256 00:06:50,109 –> 00:06:51,679 imagine one point on a number
257 00:06:51,690 –> 00:06:53,489 line where the sequence BN
258 00:06:53,500 –> 00:06:55,239 comes from above and the
259 00:06:55,250 –> 00:06:56,970 sequence A comes from
260 00:06:56,980 –> 00:06:57,559 below.
261 00:06:58,190 –> 00:06:59,500 So they should describe the
262 00:06:59,510 –> 00:07:01,179 same point but this
263 00:07:01,190 –> 00:07:02,720 one is still fulfilled.
264 00:07:03,420 –> 00:07:04,799 Therefore, maybe it’s better
265 00:07:04,809 –> 00:07:06,329 to think in distances.
266 00:07:06,339 –> 00:07:07,720 So let’s bring this one to
267 00:07:07,730 –> 00:07:09,720 the other side, then it
268 00:07:09,730 –> 00:07:11,500 simply reads that BN
269 00:07:11,510 –> 00:07:13,440 minus A is greater than
270 00:07:13,450 –> 00:07:14,000 zero.
271 00:07:14,799 –> 00:07:16,010 Therefore, this means that
272 00:07:16,019 –> 00:07:17,440 the distance here could be
273 00:07:17,450 –> 00:07:18,640 arbitrary small.
274 00:07:19,279 –> 00:07:20,239 And when you look back at
275 00:07:20,250 –> 00:07:21,929 the first example here, that’s
276 00:07:21,940 –> 00:07:22,850 not what we want.
277 00:07:23,559 –> 00:07:25,239 So there should be an actual
278 00:07:25,250 –> 00:07:26,529 minimal distance between
279 00:07:26,540 –> 00:07:27,269 the two points.
280 00:07:28,040 –> 00:07:29,500 And maybe we just call this
281 00:07:29,510 –> 00:07:30,899 distance simply
282 00:07:30,910 –> 00:07:31,540 delta.
283 00:07:32,350 –> 00:07:33,809 So we simply have to add,
284 00:07:33,820 –> 00:07:35,500 there exists a distance
285 00:07:35,510 –> 00:07:37,369 delta greater than zero.
286 00:07:38,269 –> 00:07:39,649 Of course, the distance delta
287 00:07:39,660 –> 00:07:40,690 could be very small.
288 00:07:40,750 –> 00:07:42,369 The point is it holds then
289 00:07:42,380 –> 00:07:43,369 for all end
290 00:07:44,369 –> 00:07:44,910 OK.
291 00:07:44,940 –> 00:07:46,350 There we have it, these are
292 00:07:46,359 –> 00:07:47,929 all the operations we need
293 00:07:47,940 –> 00:07:49,029 for the real numbers.
294 00:07:49,809 –> 00:07:51,230 Now, in the last step, one
295 00:07:51,239 –> 00:07:52,890 can just calculate that all
296 00:07:52,899 –> 00:07:54,470 the properties we want, the
297 00:07:54,480 –> 00:07:56,170 axioms are actually
298 00:07:56,179 –> 00:07:56,910 satisfied.
299 00:07:57,660 –> 00:07:59,390 For example, we can define
300 00:07:59,399 –> 00:08:00,720 the zero element and the
301 00:08:00,730 –> 00:08:02,279 one element and show that
302 00:08:02,290 –> 00:08:03,440 we have a field again.
303 00:08:04,220 –> 00:08:05,519 However, the important part
304 00:08:05,529 –> 00:08:07,320 we have now by construction
305 00:08:07,329 –> 00:08:09,010 is the completeness axiom.
306 00:08:09,019 –> 00:08:10,959 Every Cauchy sequence is now
307 00:08:10,970 –> 00:08:12,429 actually convergent
308 00:08:13,059 –> 00:08:14,489 with this, we can close this
309 00:08:14,500 –> 00:08:15,929 video and I hope that you
310 00:08:15,940 –> 00:08:17,700 now know a lot about real
311 00:08:17,709 –> 00:08:18,359 numbers.
312 00:08:18,929 –> 00:08:20,359 And in the next videos we
313 00:08:20,369 –> 00:08:21,859 will talk about the complex
314 00:08:21,869 –> 00:08:22,519 numbers.
315 00:08:23,029 –> 00:08:24,230 Therefore, I hope I see you
316 00:08:24,239 –> 00:08:25,720 there and have a nice day.
317 00:08:25,769 –> 00:08:26,489 Bye.
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Quiz Content
Q1: This is how the quizzes for the topics look like. You should start with the next video :)
A1: Yes!
A2: No!
A3: Never!
A4: Why???
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Last update: 2024-10
