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Mirrors > Home > MPE Home > Th. List > 0.999... | Structured version Visualization version GIF version |
Description: The recurring decimal 0.999..., which is defined as the infinite sum 0.9 + 0.09 + 0.009 + ... i.e. 9 / 10↑1 + 9 / 10↑2 + 9 / 10↑3 + ..., is exactly equal to 1, according to ZF set theory. Interestingly, about 40% of the people responding to a poll at http://forum.physorg.com/index.php?showtopic=13177 disagree. (Contributed by NM, 2-Nov-2007.) (Revised by AV, 8-Sep-2021.) |
Ref | Expression |
---|---|
0.999... | ⊢ Σ𝑘 ∈ ℕ (9 / (;10↑𝑘)) = 1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 9cn 11108 | . . . . 5 ⊢ 9 ∈ ℂ | |
2 | 10re 11517 | . . . . . . 7 ⊢ ;10 ∈ ℝ | |
3 | 2 | recni 10052 | . . . . . 6 ⊢ ;10 ∈ ℂ |
4 | nnnn0 11299 | . . . . . 6 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ0) | |
5 | expcl 12878 | . . . . . 6 ⊢ ((;10 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (;10↑𝑘) ∈ ℂ) | |
6 | 3, 4, 5 | sylancr 695 | . . . . 5 ⊢ (𝑘 ∈ ℕ → (;10↑𝑘) ∈ ℂ) |
7 | 3 | a1i 11 | . . . . . 6 ⊢ (𝑘 ∈ ℕ → ;10 ∈ ℂ) |
8 | 10pos 11515 | . . . . . . . 8 ⊢ 0 < ;10 | |
9 | 2, 8 | gt0ne0ii 10564 | . . . . . . 7 ⊢ ;10 ≠ 0 |
10 | 9 | a1i 11 | . . . . . 6 ⊢ (𝑘 ∈ ℕ → ;10 ≠ 0) |
11 | nnz 11399 | . . . . . 6 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℤ) | |
12 | 7, 10, 11 | expne0d 13014 | . . . . 5 ⊢ (𝑘 ∈ ℕ → (;10↑𝑘) ≠ 0) |
13 | divrec 10701 | . . . . 5 ⊢ ((9 ∈ ℂ ∧ (;10↑𝑘) ∈ ℂ ∧ (;10↑𝑘) ≠ 0) → (9 / (;10↑𝑘)) = (9 · (1 / (;10↑𝑘)))) | |
14 | 1, 6, 12, 13 | mp3an2i 1429 | . . . 4 ⊢ (𝑘 ∈ ℕ → (9 / (;10↑𝑘)) = (9 · (1 / (;10↑𝑘)))) |
15 | 7, 10, 11 | exprecd 13016 | . . . . 5 ⊢ (𝑘 ∈ ℕ → ((1 / ;10)↑𝑘) = (1 / (;10↑𝑘))) |
16 | 15 | oveq2d 6666 | . . . 4 ⊢ (𝑘 ∈ ℕ → (9 · ((1 / ;10)↑𝑘)) = (9 · (1 / (;10↑𝑘)))) |
17 | 14, 16 | eqtr4d 2659 | . . 3 ⊢ (𝑘 ∈ ℕ → (9 / (;10↑𝑘)) = (9 · ((1 / ;10)↑𝑘))) |
18 | 17 | sumeq2i 14429 | . 2 ⊢ Σ𝑘 ∈ ℕ (9 / (;10↑𝑘)) = Σ𝑘 ∈ ℕ (9 · ((1 / ;10)↑𝑘)) |
19 | 2, 9 | rereccli 10790 | . . . . 5 ⊢ (1 / ;10) ∈ ℝ |
20 | 19 | recni 10052 | . . . 4 ⊢ (1 / ;10) ∈ ℂ |
21 | 0re 10040 | . . . . . . 7 ⊢ 0 ∈ ℝ | |
22 | 2, 8 | recgt0ii 10929 | . . . . . . 7 ⊢ 0 < (1 / ;10) |
23 | 21, 19, 22 | ltleii 10160 | . . . . . 6 ⊢ 0 ≤ (1 / ;10) |
24 | 19 | absidi 14117 | . . . . . 6 ⊢ (0 ≤ (1 / ;10) → (abs‘(1 / ;10)) = (1 / ;10)) |
25 | 23, 24 | ax-mp 5 | . . . . 5 ⊢ (abs‘(1 / ;10)) = (1 / ;10) |
26 | 1lt10 11681 | . . . . . 6 ⊢ 1 < ;10 | |
27 | recgt1 10919 | . . . . . . 7 ⊢ ((;10 ∈ ℝ ∧ 0 < ;10) → (1 < ;10 ↔ (1 / ;10) < 1)) | |
28 | 2, 8, 27 | mp2an 708 | . . . . . 6 ⊢ (1 < ;10 ↔ (1 / ;10) < 1) |
29 | 26, 28 | mpbi 220 | . . . . 5 ⊢ (1 / ;10) < 1 |
30 | 25, 29 | eqbrtri 4674 | . . . 4 ⊢ (abs‘(1 / ;10)) < 1 |
31 | geoisum1c 14611 | . . . 4 ⊢ ((9 ∈ ℂ ∧ (1 / ;10) ∈ ℂ ∧ (abs‘(1 / ;10)) < 1) → Σ𝑘 ∈ ℕ (9 · ((1 / ;10)↑𝑘)) = ((9 · (1 / ;10)) / (1 − (1 / ;10)))) | |
32 | 1, 20, 30, 31 | mp3an 1424 | . . 3 ⊢ Σ𝑘 ∈ ℕ (9 · ((1 / ;10)↑𝑘)) = ((9 · (1 / ;10)) / (1 − (1 / ;10))) |
33 | 1, 3, 9 | divreci 10770 | . . . 4 ⊢ (9 / ;10) = (9 · (1 / ;10)) |
34 | 1, 3, 9 | divcan2i 10768 | . . . . . 6 ⊢ (;10 · (9 / ;10)) = 9 |
35 | ax-1cn 9994 | . . . . . . . 8 ⊢ 1 ∈ ℂ | |
36 | 3, 35, 20 | subdii 10479 | . . . . . . 7 ⊢ (;10 · (1 − (1 / ;10))) = ((;10 · 1) − (;10 · (1 / ;10))) |
37 | 3 | mulid1i 10042 | . . . . . . . 8 ⊢ (;10 · 1) = ;10 |
38 | 3, 9 | recidi 10756 | . . . . . . . 8 ⊢ (;10 · (1 / ;10)) = 1 |
39 | 37, 38 | oveq12i 6662 | . . . . . . 7 ⊢ ((;10 · 1) − (;10 · (1 / ;10))) = (;10 − 1) |
40 | 10m1e9 11630 | . . . . . . 7 ⊢ (;10 − 1) = 9 | |
41 | 36, 39, 40 | 3eqtrri 2649 | . . . . . 6 ⊢ 9 = (;10 · (1 − (1 / ;10))) |
42 | 34, 41 | eqtri 2644 | . . . . 5 ⊢ (;10 · (9 / ;10)) = (;10 · (1 − (1 / ;10))) |
43 | 9re 11107 | . . . . . . . 8 ⊢ 9 ∈ ℝ | |
44 | 43, 2, 9 | redivcli 10792 | . . . . . . 7 ⊢ (9 / ;10) ∈ ℝ |
45 | 44 | recni 10052 | . . . . . 6 ⊢ (9 / ;10) ∈ ℂ |
46 | 35, 20 | subcli 10357 | . . . . . 6 ⊢ (1 − (1 / ;10)) ∈ ℂ |
47 | 45, 46, 3, 9 | mulcani 10666 | . . . . 5 ⊢ ((;10 · (9 / ;10)) = (;10 · (1 − (1 / ;10))) ↔ (9 / ;10) = (1 − (1 / ;10))) |
48 | 42, 47 | mpbi 220 | . . . 4 ⊢ (9 / ;10) = (1 − (1 / ;10)) |
49 | 33, 48 | oveq12i 6662 | . . 3 ⊢ ((9 / ;10) / (9 / ;10)) = ((9 · (1 / ;10)) / (1 − (1 / ;10))) |
50 | 9pos 11122 | . . . . . 6 ⊢ 0 < 9 | |
51 | 43, 2, 50, 8 | divgt0ii 10941 | . . . . 5 ⊢ 0 < (9 / ;10) |
52 | 44, 51 | gt0ne0ii 10564 | . . . 4 ⊢ (9 / ;10) ≠ 0 |
53 | 45, 52 | dividi 10758 | . . 3 ⊢ ((9 / ;10) / (9 / ;10)) = 1 |
54 | 32, 49, 53 | 3eqtr2i 2650 | . 2 ⊢ Σ𝑘 ∈ ℕ (9 · ((1 / ;10)↑𝑘)) = 1 |
55 | 18, 54 | eqtri 2644 | 1 ⊢ Σ𝑘 ∈ ℕ (9 / (;10↑𝑘)) = 1 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 ℝcr 9935 0cc0 9936 1c1 9937 · cmul 9941 < clt 10074 ≤ cle 10075 − cmin 10266 / cdiv 10684 ℕcn 11020 9c9 11077 ℕ0cn0 11292 ;cdc 11493 ↑cexp 12860 abscabs 13974 Σcsu 14416 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-rp 11833 df-fz 12327 df-fzo 12466 df-fl 12593 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-rlim 14220 df-sum 14417 |
This theorem is referenced by: (None) |
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