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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 11740 | . . . . 5 ⊢ 9 ∈ ℂ | |
2 | 10re 12120 | . . . . . . 7 ⊢ ;10 ∈ ℝ | |
3 | 2 | recni 10657 | . . . . . 6 ⊢ ;10 ∈ ℂ |
4 | nnnn0 11907 | . . . . . 6 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ0) | |
5 | expcl 13450 | . . . . . 6 ⊢ ((;10 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (;10↑𝑘) ∈ ℂ) | |
6 | 3, 4, 5 | sylancr 589 | . . . . 5 ⊢ (𝑘 ∈ ℕ → (;10↑𝑘) ∈ ℂ) |
7 | 3 | a1i 11 | . . . . . 6 ⊢ (𝑘 ∈ ℕ → ;10 ∈ ℂ) |
8 | 10pos 12118 | . . . . . . . 8 ⊢ 0 < ;10 | |
9 | 2, 8 | gt0ne0ii 11178 | . . . . . . 7 ⊢ ;10 ≠ 0 |
10 | 9 | a1i 11 | . . . . . 6 ⊢ (𝑘 ∈ ℕ → ;10 ≠ 0) |
11 | nnz 12007 | . . . . . 6 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℤ) | |
12 | 7, 10, 11 | expne0d 13519 | . . . . 5 ⊢ (𝑘 ∈ ℕ → (;10↑𝑘) ≠ 0) |
13 | divrec 11316 | . . . . 5 ⊢ ((9 ∈ ℂ ∧ (;10↑𝑘) ∈ ℂ ∧ (;10↑𝑘) ≠ 0) → (9 / (;10↑𝑘)) = (9 · (1 / (;10↑𝑘)))) | |
14 | 1, 6, 12, 13 | mp3an2i 1462 | . . . 4 ⊢ (𝑘 ∈ ℕ → (9 / (;10↑𝑘)) = (9 · (1 / (;10↑𝑘)))) |
15 | 7, 10, 11 | exprecd 13521 | . . . . 5 ⊢ (𝑘 ∈ ℕ → ((1 / ;10)↑𝑘) = (1 / (;10↑𝑘))) |
16 | 15 | oveq2d 7174 | . . . 4 ⊢ (𝑘 ∈ ℕ → (9 · ((1 / ;10)↑𝑘)) = (9 · (1 / (;10↑𝑘)))) |
17 | 14, 16 | eqtr4d 2861 | . . 3 ⊢ (𝑘 ∈ ℕ → (9 / (;10↑𝑘)) = (9 · ((1 / ;10)↑𝑘))) |
18 | 17 | sumeq2i 15058 | . 2 ⊢ Σ𝑘 ∈ ℕ (9 / (;10↑𝑘)) = Σ𝑘 ∈ ℕ (9 · ((1 / ;10)↑𝑘)) |
19 | 2, 9 | rereccli 11407 | . . . . 5 ⊢ (1 / ;10) ∈ ℝ |
20 | 19 | recni 10657 | . . . 4 ⊢ (1 / ;10) ∈ ℂ |
21 | 0re 10645 | . . . . . . 7 ⊢ 0 ∈ ℝ | |
22 | 2, 8 | recgt0ii 11548 | . . . . . . 7 ⊢ 0 < (1 / ;10) |
23 | 21, 19, 22 | ltleii 10765 | . . . . . 6 ⊢ 0 ≤ (1 / ;10) |
24 | 19 | absidi 14739 | . . . . . 6 ⊢ (0 ≤ (1 / ;10) → (abs‘(1 / ;10)) = (1 / ;10)) |
25 | 23, 24 | ax-mp 5 | . . . . 5 ⊢ (abs‘(1 / ;10)) = (1 / ;10) |
26 | 1lt10 12240 | . . . . . 6 ⊢ 1 < ;10 | |
27 | recgt1 11538 | . . . . . . 7 ⊢ ((;10 ∈ ℝ ∧ 0 < ;10) → (1 < ;10 ↔ (1 / ;10) < 1)) | |
28 | 2, 8, 27 | mp2an 690 | . . . . . 6 ⊢ (1 < ;10 ↔ (1 / ;10) < 1) |
29 | 26, 28 | mpbi 232 | . . . . 5 ⊢ (1 / ;10) < 1 |
30 | 25, 29 | eqbrtri 5089 | . . . 4 ⊢ (abs‘(1 / ;10)) < 1 |
31 | geoisum1c 15238 | . . . 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 1457 | . . 3 ⊢ Σ𝑘 ∈ ℕ (9 · ((1 / ;10)↑𝑘)) = ((9 · (1 / ;10)) / (1 − (1 / ;10))) |
33 | 1, 3, 9 | divreci 11387 | . . . 4 ⊢ (9 / ;10) = (9 · (1 / ;10)) |
34 | 1, 3, 9 | divcan2i 11385 | . . . . . 6 ⊢ (;10 · (9 / ;10)) = 9 |
35 | ax-1cn 10597 | . . . . . . . 8 ⊢ 1 ∈ ℂ | |
36 | 3, 35, 20 | subdii 11091 | . . . . . . 7 ⊢ (;10 · (1 − (1 / ;10))) = ((;10 · 1) − (;10 · (1 / ;10))) |
37 | 3 | mulid1i 10647 | . . . . . . . 8 ⊢ (;10 · 1) = ;10 |
38 | 3, 9 | recidi 11373 | . . . . . . . 8 ⊢ (;10 · (1 / ;10)) = 1 |
39 | 37, 38 | oveq12i 7170 | . . . . . . 7 ⊢ ((;10 · 1) − (;10 · (1 / ;10))) = (;10 − 1) |
40 | 10m1e9 12197 | . . . . . . 7 ⊢ (;10 − 1) = 9 | |
41 | 36, 39, 40 | 3eqtrri 2851 | . . . . . 6 ⊢ 9 = (;10 · (1 − (1 / ;10))) |
42 | 34, 41 | eqtri 2846 | . . . . 5 ⊢ (;10 · (9 / ;10)) = (;10 · (1 − (1 / ;10))) |
43 | 9re 11739 | . . . . . . . 8 ⊢ 9 ∈ ℝ | |
44 | 43, 2, 9 | redivcli 11409 | . . . . . . 7 ⊢ (9 / ;10) ∈ ℝ |
45 | 44 | recni 10657 | . . . . . 6 ⊢ (9 / ;10) ∈ ℂ |
46 | 35, 20 | subcli 10964 | . . . . . 6 ⊢ (1 − (1 / ;10)) ∈ ℂ |
47 | 45, 46, 3, 9 | mulcani 11281 | . . . . 5 ⊢ ((;10 · (9 / ;10)) = (;10 · (1 − (1 / ;10))) ↔ (9 / ;10) = (1 − (1 / ;10))) |
48 | 42, 47 | mpbi 232 | . . . 4 ⊢ (9 / ;10) = (1 − (1 / ;10)) |
49 | 33, 48 | oveq12i 7170 | . . 3 ⊢ ((9 / ;10) / (9 / ;10)) = ((9 · (1 / ;10)) / (1 − (1 / ;10))) |
50 | 9pos 11753 | . . . . . 6 ⊢ 0 < 9 | |
51 | 43, 2, 50, 8 | divgt0ii 11559 | . . . . 5 ⊢ 0 < (9 / ;10) |
52 | 44, 51 | gt0ne0ii 11178 | . . . 4 ⊢ (9 / ;10) ≠ 0 |
53 | 45, 52 | dividi 11375 | . . 3 ⊢ ((9 / ;10) / (9 / ;10)) = 1 |
54 | 32, 49, 53 | 3eqtr2i 2852 | . 2 ⊢ Σ𝑘 ∈ ℕ (9 · ((1 / ;10)↑𝑘)) = 1 |
55 | 18, 54 | eqtri 2846 | 1 ⊢ Σ𝑘 ∈ ℕ (9 / (;10↑𝑘)) = 1 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 class class class wbr 5068 ‘cfv 6357 (class class class)co 7158 ℂcc 10537 ℝcr 10538 0cc0 10539 1c1 10540 · cmul 10544 < clt 10677 ≤ cle 10678 − cmin 10872 / cdiv 11299 ℕcn 11640 9c9 11702 ℕ0cn0 11900 ;cdc 12101 ↑cexp 13432 abscabs 14595 Σcsu 15044 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-pm 8411 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-sup 8908 df-inf 8909 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-rp 12393 df-fz 12896 df-fzo 13037 df-fl 13165 df-seq 13373 df-exp 13433 df-hash 13694 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-clim 14847 df-rlim 14848 df-sum 15045 |
This theorem is referenced by: (None) |
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