㘆 ঌ ⒵ ኗ
㙣┋ዣ 0DKƗYƯUD ⒍ύᓴߨ߉ 800 枼צ׆⒉ᨐ⿻ᓴࣾ
တㄧ⒫ᵺ ώ⒍࣯ᓴߨ߉ 870 ώᓄฎ ώ
㘆 ঌ ⒵ ኗ
㔉 ت ᅂ
ᭇॠ⌋ጦⲣ᷹⊣
㙣┋ዣ 0DKƗYƯUD ⒍ύᓴߨ߉ 800 枼צ׆⒉ᨐ⿻ᓴࣾ
တㄧ⒫ᵺ ώ⒍࣯ᓴߨ߉ 870 ώᓄฎ ώ
㛦▂ፕᖺ६ႃॠㆤ┡ᷱڂ 栾⛔ㆽᔢᔢᄡ 栾ᙄாፕḚፕႇ
ộᙙ5ƗV
· W
·UDNnjW
·D ⃞ຢჂⴂῳ᪃ㅻᄚ㒆تᥧᗆ㒙 ц┃ᕧࡕ࠶ 850
ၣ 栾ڒᒪ໎٦ъⷆ␡ⓙ⥩ы*DQ
·LWD VƗUD VDPթJUDKDتᙀ ц⸋ᙀா
६ႃॠᙄⳝႵᨼ܃Ό 栾॒تغ⓿ⳝ⻌ᩍℛத⸩ ц٩॒غ⓿ ࢩ 栾ⳝ㊟ᕟ⃘Ἢ ц1912 ၣ 栾ா㛦ᄹፕᕞ⻌⢿ᕄ᷹ цъⷆ
␡ⓙ⥩ыᖺ६ႃ⏜تᙥࢩ࡛Ἢڜჽ⃞ᔷᔢ⌋ᙀ 栾Ἢڌᔷᔢᙿ ف⃞تٲ⹐㙃ਘ╁ᠫா࡚فထমࢲ ц
㛦▂ፕ⃞ဉۦ༝ᕧ┓ᔷ㘥ఝ 栾ೳᕄક㙃ၬْᨈᙌ᪵
Ễ цㅚ⠛ڒ⃞ࣆミݜᖺ㘤سী⃞ цாতڜ६ႃ 栾ᔷວت⠴ٜ
ᖺೳᕄວ ц
㛦▂ፕ⃞ъⷆ␡ⓙ⥩ыࡘٚ⎮ 桌1 ⲣ⸩ 桍2 ␡ⲣㅹ␡ 桍 3 ⠛࢛ᔷᙌ㓊⃞ㅹ␡ 桍4 ᙌহ⍁Ḛ㦼⃞ㅹ␡ 桍5 ⠛د࢛ᄝ
ᄝ ᙌ㓊⃞ㅹ␡ 桍6 ᬈ়ㅹ␡ 桍7 㖭⍕ⷆ␡ 桍8 ⠛ᎎᏢᙌ㓊
⃞ⷆ␡ 桍9 ⠛ᄉᙌ㓊⃞ⷆ␡ ц
㛦▂ፕᔌㅪਘᏰႵ٦ڒ⃞ࣆミݜ⃞೬╁ᚶ 栾࡚فᙆᙌḚ⡥
⃞⇩⍭भ፩ 桌㕦⃞ㅹ␡ х٩ᤄᕥ⌫ хࢯổⷆ␡ хᔳᔷᆄ⽶ хᏞ ࢧ┺় х࢛ᔷᨷࣁ · · · ⏯⏯ ц
1 㒇—⍖
ъⷆ␡ⓙ⥩ыفᔡㄼ٦㕦⃞␡ᨷ 桌³تݗᔷٗ㕦ᄣ㕦 栾تݗᔷ
ࣺ㕦 хᬕ㕦ዜ㔀ڞ㕦 栾ㅚᔷㇺس⻝´ цㅚⲯᖣ 栾‧ᗆᙣ⸟⺼ࢲ 㕦س❫ۦ㔀ᔷ ц
2 ղޅֳᡂᒥ≢य़ջොᒥ≢
ாㅚᕥ㖭 栾ᙀᕧ┃ࡕ࠶تغ⓿ᗆ⃞فமъٚ⎮␡ⲣыထᙌ ダ೬⃞ᚶ цࡕ࠶دغ⓿ᗆ 栾သ⟷ᔷວؾ‡ழ 'LRSKDQWXV ⦰ ъ␡ⲣыتᙀ 栾ٜⶫ᧳٦س٩ᤄᕥ⌫ хسຖᕥ⌫ક㙃 栾ےڒ سጥ⸟⼻ᔷ⃞়ἱᆄ ц㛦▂ፕڞࣆ⃞६ႃᔷວسᕣ⇩⍭
٦ㅚٲᕥ⌫⃞ⶫᨷ 栾㓨⛴ᩂ೬ ƖU\DEKDW
·D , Ⴣ⎧٦تᤄ▤ᆄس ຖᕥ⌫ᤳᔳᔷㅛⶫ⃞ᨷࣁ 栾९ႇ॥ .XWWDND цථ♭ᑵ⏄೬
%UDKPDJXSWD ═٦ت࠶٩ᤄᕥ⌫⃞تݗࡕ ц㛦▂ፕ
ٜⷉ⹐٦ᄚ೬ㅚᕥ㖭⃞ક㙃 цഫ 桌
³تᛱ⃞ 1
12 ٗڞ 1
30 ᨈᕧᧃف 栾ࣔر⃞ 1
20 ٗڞ 3 16 ఔ
ாᩄⴂ 栾ㆧᙌ ➝㖔ாᧃ㖭ذ 栾ⶏሬ⃞ᙎঙ 栾⹈કㅚᛱ
ᙌ೬㞖 桑´
㛦▂ፕ═ᕥ⌫
x − x2 12 × 30
− 1
20 · 3 16
x − x2 360
2
= 20 栾 ؿ٦⃞ᙌἱⶫ ц
ாদتٲક㙃ف 栾ڒㆧ═٦ჽഫ x − (bx + c√
x + a) = 0
⃞ᕥ⌫⃞ⶫ ц
㛦▂ፕႇ॥ٜۦ٦تٲᔌㅪ цڒாݝঔᕥ⌫⃞ⶫᗆ ℯ‟٦Ό⏜تݗએᔷক⠛ㅹ␡⃞تᤵ 栾تݗᙣ⇉ᔷᖺΌڜࡏᕥ⌫ᨷ
ᄣ⃞ цےڒ◮ᖺゐ㒍⦰س⻣ロヷ㔀ᨷ⃞㚣ᔷ 栾ㅚ້࡚ᖺسᅃ
⃞ ц
3 ³⟃Ởᑸ´
ࡒᔳᔷ℩ٗ 栾⢵࡚ٗ⍕⃞ᔷ৭فᅂ⍄ 栾㛦▂ፕܨ⍄ِ
³⢔ᾦᔷ´ 栾ഫ 桌ф
14287143 × 7 = 100010001 桍 142857143 × 7 = 1000000001 桍 12345679 × 9 = 111111111 桍
333333666667 × 33 = 11000011000011 桍 11011011 × 91 = 1002002001 桍
27994681 × 441 = 12345654321 ц
ڒㅚ⍁Ṋی⢔ᾦ⃞Ḛ᥉ᔳᔷ⃞ᠫᄝㅪⲟ٦⇩⍭ ц
4 ጠ߶⑭आ
তڜ⛔ㆽᔢ⒗فᙌتٲ⑹⃞Ꮮࢧ┺়ક㙃 栾㛦▂ፕ═
ࡕ
Crn = n(n − 1)(n − 2)· · ·(n − r + 1) 1 · 2 · 3· · ·r
ц
5 ਝߪᑸࠐ
࢛ᔷᖺ࢛ 1 ⃞࢛ᔷ цতঘᔷວ㓨ᝆᕞ $KPHV ᙄㅡ
ጮ࢛ᔷ 2
2n + 1 ⲯ⊡࢛ᔷਘ⃞ⲯ (n = 1 梐 49) 栾࡚উἱᄚম
❫ᖺ╲㝨ᆄ⃞ ц㛦▂ፕ⇩⍭تダຌᔳ⃞࢛ᔷⲯ⊡ᨷ 桌 1 ጮ 1 ⲯ⊡ n ݗ࢛ᔷ⃞ਘ
1 = 1
2 + 1
3 + 1
32 + 1
33 + · · · + 1
3n−2 + 1
2 · 3n−2 ц
2 ጮ 1 ⲯ⊡ 2n − 1 ݗ࢛ᔷ⃞ਘ
1 = 1
2 · 3 · 1 2
+ 1
3 · 4 · 1 2
+ · · ·
+ 1
(2n − 1) · 2n · 1 2
+ 1
2n · 1 2
ц
3 ጮ═ຖ⃞࢛ᔷⲯ⊡ r ݗ࢛࢛ࢭ a1 хa2 х· · · хar
⃞࢛ᔷِਘ 1
n = a1
n(n + a1) + a2
(n + a1)(n + a1 + a2) + · · ·
+ ar−1
(n + a1 + a2 + · · · + ar−2)(n + a1 + a2 + · · · + ar−1)
+ ar
ar(n + a1 + · · · + ar−1) ц 4 ጮک࢛۟ᔷⲯ⊡࢛ᔷ⃞ਘ
⢵ p
q ᖺ═ຖ⃞࢛ᔷ p < q 栾㆛ᒸ i ܃ᄣ q + i
p تݗᔳ
ᔷ 栾ഫ⸲ r 栾ࣁ p
q = 1
r + i
rq ц i
1 − q ㊟ⴜᤴㅻ⌫ 栾ங
i < q 栾ᤴㅻ⌫╲ᙌ㓳ᤵܨޒᤲ٦ ц 5 ጮتݗ࢛ᔷⲯ⊡ࡒݗ࢛ᔷ⃞ਘ ᕥᨷت 桌 1
n = 1
pn + 1 pn p − 1
栾࡚ف p ⃞㆛ᒸ܃ p − 1 ❫ⳝ n
ᔳ㔀 桍
ᕥᨷ٩ 桌 1
ab = 1
a(a + b) + 1
b(a + b) (b ≥ 1) ц
6 ጮک࢛۟ᔷⲯ⊡ᙌ═ຖ࢛⃞ࡒ࢛ᔷِਘ m
n = a
ap + b m · n
p
+ b
ap + b m · n
栾
࡚ف p মᔳ㔀 n хm মᔳ㔀 ap + b ц
╁় 1 ਘ 6 栾ک࢛۟ᔷㇺমⲯ⊡ 2n ݗ့═ຖ࢛⃞࢛ᔷ
ِਘ ц
ாၬ۟ᕥ㖭 栾㛦▂ፕ㊟ᕟ⇩⍭٦ථ♭ᑵ⏄೬㓊ᕧㆬᙌἱ ᔷ⃞லࡐᏮகㆬჽ⃞ۦழ цߡථ♭ᑵ⏄೬تᡥ 栾㛦▂ፕٜᨈᙌ ໂࢲㅚ㙒கㆬჽᅃ㘙ᖺࡐᏮᕧல⃞ ц㛦▂ፕዷⷉ⹐ㅻ⃞ၬ۟
ۦழક㙃ᄚ೬ 栾ഫ 桌1 ═ຖتㆬ 栾ۦتݗ࡚ࡒㆬᙌ ἱᔷ⃞Ωدჽ 桍2 ═ຖتᕏㆬ c 栾ۦتΩدჽ 栾܃
٩Ωㆬᙌἱᔷ 桍3 ۦتݗدㆬ℩⏯⃞ᝦჽ 桍4 ۦت ᙌ═ຖ㖭⍕⃞லࡐᏮகㆬჽ 桍5 ۦتᙌ═ຖৼ㒆⃞லࡐᏮகㆬ ჽ 桍6 ۦت㒆ᕥჽ 栾܃࡚㖭⍕ாᔷ㊡ذᖺ࡚ৼ㒆ዜ▤㒆
⃞ݙᔷ 栾ዜ⛓ت⠴ 栾ᖺ࡚ㆬ㒆⠛▤㒆⃞▤ᆄ┺় 桍7 ۦ ࡒݗ㒆ᕥჽ 桌1 ࡚ৼ㒆℩⏯ 栾ے࡚فتݗ⃞㖭⍕ᖺদتݗ⃞ 2 ݙ 桍2 ࡚㖭⍕℩⏯ 栾ے࡚فتݗ⃞ৼ㒆ᖺদتݗ⃞ 2 ݙ 桍3 ت ݗ㒆ᕥჽ⃞ৼ㒆ᖺদتݗ⃞ 2 ݙ 栾࡚㖭⍕ᖺদتݗ⃞ 1
2 цᙆᄞㅚ ݗۦழક㙃 栾ت⠴ኩ⦰ᕥ⌫
m(x + y) = n(u + v) 栾 pxy = quv 栾
࡚ف (x, y) х(u, v) ࢛ࢭࡒݗ㒆ᕥჽ⃞ㆬ 栾m хn хp хq ᖺ═
ຖ⃞ᔷ ц㛦▂ፕᄣࢲ٦ࡒݗⶫ ц
㛦▂ፕㆧ⇩⍭ㅻᢐலਘ࡚ၬ۟ழჽ цڒ═ᢐல⃞㖭⍕
ৼ㒆 × 1
2 ख़⇎ザ 栾ㅚݗ㎟⸭⃞╁ᚶম❫ᖺᄨல㖭⍕ πd × d
4 ࡚
ف d ல⃞Ωᄠ 㙒Ᏸ⛘܊⃞ цᕧბჽ㖭⍕ 栾ڒΌ٦ㄩیࡕ
S = 1
2 (b + h)h ࡚ف b ბჽ⃞ო㒆 栾h ბჽ⃞㞖 栾ㅚتࡕ
ᙆᖆἪாفமতڜᔷু⦰ъٚ⎮␡ⲣыف ц
ᒄ Ḃ
ࣕತᒄḂ
[1] 0DKƗYƯUD *DQ
·LWDVƗUDVDQթJUDKD 0DGUDV 1912 ц
℠⊤ᒄḂ
[2] % 'DWWD DQG $1 6LQJK +LVWRU\ RI +LQGX PDWKHPDWLFV $VLD 3XEOLVKLQJ +RXVH %RPED\ 1938 ц
[3] $. %DJ 0DWKHPDWLFDO LQ DQFLHQW DQG PHGLHYDO ,QGLD
&KDXNKDPEKD 2ULHQWDOLD 9DUDQDVL 1979 ц
[4] 'DYLG 3LQJUHH 0DKƗYƯUD 'LFWLRQDU\ RI VFLHQWLILF ELRJUDSK\
9RO 9 1981 22 ц
[5] '( 6PLWK 7KH *DQLWD6DUD6DQJUDKD RI 0DKƗYƯUƗFƗU\D %LE OLRWKHFD 0DWKHPDWLFD 31908 ± 1909 9 106 ± 110 ц
[6] (7 %HOO 0DKDYLUD¶V 'LRSKDQWLQH V\VWHP %XOOHWLQ RI WKH &DO FXWWD 0DWKHPDWLFDO 6RFLHW\ 381946 121 ± 122 ц
[7] Ⱥ ɉ ɘɲɤɟɜɢɱ ɂɫɬɨɪɢɹ ɦɚɬɟɦɚɬɢɤɢ ɜ ɫɪɟɞɧɢɟ ɜɟɤɚ Ɇɨɫɤɜɚ 1961 ц
[8] Ⱥ ɂ ȼɨɥɨɞɚɪɫɤɢɣ Ɉɱɟɪɤɢ ɢɫɬɨɪɢɢ ɫɪɟɞɧɟɜɟɤɨɜɨɣ ɢɧɞɢɣɫɤɨɣ ɦɚɬɟɦɚɬɢɤɢ ɂɷɞɚɬɟɥɶɫɬɜɨ ъɇɚɍɤɚыɆɨɫɤɜɚ 1977 ц
