159:< 1. The problem with this argument, however, is that it suggests that a statement S(n) that "a zero followed by a decimal point followed by n 9s represents a number less than 1", which is true for all integer n is also true when n is an infinite value. The problem here is that such a deduction is not generally valid (compare with a statement such as "n is finite", which by definition is true for any integer n but false if n is infinite). In fact, the behaviour of infinite properties when compared to their finite counterparts means it is actually possible to prove statements such as
453:, the infinite sum and the infinite limit of the partial sums are not the same (despite such a statement being a common definition to give the infinite sum a valid meaning) and so the numbers are equal. However, it is possible with a little effort to show that the two are in fact equal, if a few simple definitions are agreed upon.
120:
less than one". However, as many mathematics and science teachers are keen to point out, that which is "obvious" is not necessarily true, and such a statement is not particularly useful unless backed with some evidence that it can be proven.
451:
374:
215:
186:
91:
296:
249:
157:
114:
463:
379:
33:, not approximately but exactly, as well as some of the most common arguments that claim that 0.999… is less than 1, or that it is not exactly 1.
315:
298:
and the limit of partial sums that equals 1 both require an infinite number of steps, and thus cannot be "truly" calculated.
476:
by D. O. Tall & R. L. E. Schwarzenberger, University of
Warwick Published in Mathematics Teaching, 82, 44–49 (1978).
272:
477:
480:
191:
162:
67:
517:
278:
231:
139:
96:
507:
497:
228:
This argument claims that while it is possible to write an infinite decimal expansion such as
27:
534:
502:
512:
17:
529:
464:
Talk:Proof_that_0.999..._equals_1/Archive02#If_I_may_speak_to_the_article_itself...
446:{\displaystyle \lim _{n\rightarrow \infty }\sum _{i=1}^{n}{\frac {9}{10^{i}}}=1}
251:, this is only an approximation to the exact value and therefore cannot equal 1.
544:
61:
Should we include a counter-example for (each of) these claims? or an example?
124:
The argument may be strengthened by a suggestion along the following lines:
539:
369:{\displaystyle 0.999\ldots =\sum _{i=1}^{\infty }{\frac {9}{10^{i}}}}
264:
Infinite decimals and limits are the result of an infinite process.
26:
This article presents background and proofs of the fact that the
305:
An infinite sum is not the same as the limit of an infinite sum.
188:, and with greater rigor, in some cases, than it is to prove
275:
school of mathematics, in that both the full expression of
93:
works along the lines of "because it starts with zero,
382:
318:
281:
234:
194:
165:
142:
99:
70:
474:
Conflicts in the
Learning of Real Numbers and Limits
445:
368:
290:
243:
209:
180:
151:
108:
85:
64:Possibly the simplest argument offered as to why
57:Arguments that claim that 0.999… is not exactly 1
384:
261:Decimals and Limits are processes, not numbers.
8:
429:
420:
414:
403:
387:
381:
358:
349:
343:
332:
317:
280:
271:This may be considered a relation to the
233:
193:
164:
141:
98:
69:
312:This is an attempt to say that while
221:Infinite decimals are approximations.
7:
535:Ask A Scientist: Repeating Decimals
394:
344:
210:{\displaystyle 0.999\ldots \leq 1}
181:{\displaystyle 0.999\ldots \geq 1}
86:{\displaystyle 0.999\ldots \neq 1}
24:
391:
37:Definitions and justifications
1:
530:Why does 0.9999... = 1 ?
291:{\displaystyle 0.999\ldots }
244:{\displaystyle 0.999\ldots }
152:{\displaystyle 0.999\ldots }
136:and following this pattern,
109:{\displaystyle 0.999\ldots }
561:
258:Limits are approximations.
447:
419:
370:
348:
292:
245:
211:
182:
153:
110:
87:
448:
399:
371:
328:
293:
246:
212:
183:
154:
111:
88:
380:
316:
279:
232:
192:
163:
140:
97:
68:
518:Limit (mathematics)
443:
398:
366:
288:
241:
207:
178:
149:
106:
83:
508:Convergent series
498:Recurring decimal
435:
383:
364:
42:Elementary proofs
28:recurring decimal
552:
503:Geometric series
452:
450:
449:
444:
436:
434:
433:
421:
418:
413:
397:
375:
373:
372:
367:
365:
363:
362:
350:
347:
342:
297:
295:
294:
289:
250:
248:
247:
242:
216:
214:
213:
208:
187:
185:
184:
179:
158:
156:
155:
150:
115:
113:
112:
107:
92:
90:
89:
84:
31:0.9999… equals 1
560:
559:
555:
554:
553:
551:
550:
549:
540:Repeating Nines
526:
513:Infinite series
494:
489:
487:Popular culture
425:
378:
377:
354:
314:
313:
277:
276:
230:
229:
190:
189:
161:
160:
138:
137:
95:
94:
66:
65:
59:
54:
52:Generalizations
49:
47:Advanced proofs
44:
39:
22:
21:
20:
18:User:Jesushaces
12:
11:
5:
558:
556:
548:
547:
545:Is 0.999...=1?
542:
537:
532:
525:
524:External links
522:
521:
520:
515:
510:
505:
500:
493:
490:
488:
485:
484:
483:
467:
466:
457:
456:
455:
454:
442:
439:
432:
428:
424:
417:
412:
409:
406:
402:
396:
393:
390:
386:
361:
357:
353:
346:
341:
338:
335:
331:
327:
324:
321:
307:
306:
302:
301:
300:
299:
287:
284:
266:
265:
262:
259:
255:
254:
253:
252:
240:
237:
223:
222:
206:
203:
200:
197:
177:
174:
171:
168:
148:
145:
105:
102:
82:
79:
76:
73:
58:
55:
53:
50:
48:
45:
43:
40:
38:
35:
23:
15:
14:
13:
10:
9:
6:
4:
3:
2:
557:
546:
543:
541:
538:
536:
533:
531:
528:
527:
523:
519:
516:
514:
511:
509:
506:
504:
501:
499:
496:
495:
491:
486:
482:
479:
475:
472:
471:
470:
465:
462:
461:
460:
440:
437:
430:
426:
422:
415:
410:
407:
404:
400:
388:
359:
355:
351:
339:
336:
333:
329:
325:
322:
319:
311:
310:
309:
308:
304:
303:
285:
282:
274:
270:
269:
268:
267:
263:
260:
257:
256:
238:
235:
227:
226:
225:
224:
220:
219:
218:
204:
201:
198:
195:
175:
172:
169:
166:
146:
143:
134:
133:0.999 < 1
131:
128:
125:
122:
119:
103:
100:
80:
77:
74:
71:
62:
56:
51:
46:
41:
36:
34:
32:
29:
19:
473:
469:References:
468:
458:
273:constructive
135:
132:
130:0.99 < 1
129:
126:
123:
117:
63:
60:
30:
25:
127:0.9 < 1
459:See also:
401:∑
395:∞
392:→
345:∞
330:∑
323:…
286:…
239:…
202:≤
199:…
173:≥
170:…
147:…
118:obviously
104:…
78:≠
75:…
492:See also
481:webpage
320:0.999
283:0.999
236:0.999
196:0.999
167:0.999
144:0.999
101:0.999
72:0.999
16:<
376:and
478:pdf
385:lim
116:is
427:10
356:10
217:.
441:1
438:=
431:i
423:9
416:n
411:1
408:=
405:i
389:n
360:i
352:9
340:1
337:=
334:i
326:=
205:1
176:1
81:1
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.