PinYin and BoPoMoFo ZhuYin Equivalence


There is a straight one-to-one correspondence between ZhuYin (BoPoMoFo) and PinYin (Chinese Romanization), with the few exceptions listed at the end of the next table. Despite the usually emotionally charged arguments (usually by those who do not bother to take the time to understand the differences across the Taiwan Strait), it is just a matter of choice of symbols to represent the many sounds in Chinese. In the following table, BoPoMoFo are coded in Big5 (because BoPoMoFo is not used in mainland China; thus, the corresponding GB coding does not contain these symbols). The table below lists the 37 BoPoMoFo phonetic symbols, along with the corresponding romanized equivalents. The first half are the consonants and the second half are the vowels. Note that 8, 9, and : can be both, and the corresponding PinYin are different.

Consonants
 b p m f
 d t n l
 g k h
! j" q# x
¥ zh% ch& sh' r
( z) c* s
8 y9 w: yu
Vowels
8 i9 u: u, u:
+ a, o- e. e, e^
/ ai0 ei1 au2 ou
3 an4 en5 ang6 eng
7 er, r

Exceptions for some vowels (but not consonants):

One more exception to the rule for ZhuYin symbols that can stand alone: The above are all the rules and exceptions to the rules. Period. Lastly, we indicate the tone by appending 1, 2, 3, 4, or 0 (e.g., wang2 nan2 xin1, which is my name).

The following table contains a complete set of all valid sounds in Mandarin, and BoPoMoFo, pinyin, etc. are the different ways of expressing the various sounds. The Big5 and GB characters in the table are only representative characters with that pronunciation. Many Chinese characters share an identical pronunciation. Conversely, there are sometimes multiple pronunciations for the same character. Note that many combinations of the consonants and vowels (e.g., "bui") are missing in Mandarin Chinese, unless you speak with a very strange accent or unless you are imitating the sounds of animals. One single font will not display the next table correctly; you need to switch fonts to view this page in Big5, GB (present setting), and HZ codes (which you accomplish, for example, in Internet Explorer through the menu |View|Encoding|.

Big5GBHZZhuYinPinYinWadeYale
_ A A A
~{0'~}Ai Ai Ai
~{02~}An An An
~{09~}Ang Ang Ang
_ Ao Ao Ao
~{0K~}+ Ba Pa Ba
~{0W~}/ Bai Pai Bai
ф~{0b~}3 Ban Pan Ban
ü~{0n~}5 Bang Pang Bang
~{0|~}1 Bao Pao Bau
~{10~}0 Bei Pei Bei
~{1<~}4 Ben Pen Ben
~{1@~}6 Beng Peng Beng
~{1F~}8 Bi Pi Bi
~{1`~}83Bian Pien Byan
~{1k~}81Biao Piao Byau
~{1n~}8.Bie Pieh Bye
~{1r~}84Bin Pin Bin
~{1y~}86Bing Ping Bing
~{2(~}, Bo Po Bwo
~{2;~}9 Bu Pu Bu
~{2A~})+ Ca Ts'a Tsa
~{2B~})/ Cai Ts'ai Tsai
~{2N~})3 Can Ts'an Tsan
~{2V~})5 Cang Ts'ang Tsang
~{2Y~})1 Cao Ts'ao Tsau
~{2a~})- Ce Ts'e Tse
~{a/~})4 Cen Ts'en Tsen
~{Tx~})6 Ceng Ts'eng Tseng
~{2f~}%+ Cha Ch'a Cha
~{2p~}%/ Chai Ch'ai Chai
~{2t~}%3 Chan Ch'an Chan
~{2}~}%5 Chang Ch'ang Chang
~{33~}%1 Chao Ch'ao Chau
ó~{35~}%- Che Ch'e Che
~{h!~}%4 Chen Ch'en Chen
~{3F~}%6 Cheng Ch'eng Cheng
~{3T~}Chi Ch'ih Chr
~{3d~}%96Chong Ch'ung Chung
~{3i~}%2 Chou Ch'ou Chou
~{3v~}%9 Chu Ch'u Chu
~{4'~}%9/Chuai Ch'uai Chwai
~{4(~}%93Chuan Ch'uan Chwan
~{44~}%95ChuangCh'uangChwang
~{45~}%90Chui Ch'ui Chwei
~{4:~}%94Chun Ch'un Chwun
~{4B~}%9,Chuo Ch'o Chwo
~{4C~}Ci Tz'u Tsz
~{4R~})96Cong Ts'ung Tsung
~{4V~})9 Cu Ts'u Tsu
~{4\~})93Cuan Ts'uan Tswan
~{4^~})90Cui Ts'ui Tswei
~{4e~})94Cun Ts'un Tswun
~{4i~})9,Cuo Ts'o Tswo
~{4p~}+ Da Ta Da
~{4t~}/ Dai Tai Dai
~{5$~}3 Dan Tan Dan
~{51~}5 Dang Tang Dang
~{56~}1 Dao Tao Dau
~{5C~}- De Te De
~{5C~}0 Dei Tei Dei
~{5G~}6 Deng Teng Deng
~{5M~}8 Di Ti Di
~{5a~}83Dian Tien Dyan
~{5s~}81Diao Tiao Dyau
~{5y~}8.Die Tieh Dye
~{6!~}86Ding Ting Ding
~{6*~}82Diu Tiu Dyou
~{6,~}96Dong Tung Dung
~{65~}2 Dou Tou Dou
~{6=~}9 Du Tu Du
~{6K~}93Duan Tuan Dwan
~{6Q~}90Dui Tui Dwei
~{6X~}94Dun Tun Dwun
~{6`~}9,Duo To Dwo
_ E E E
~{6p~}E E E
_ Ei Ei Ei
~{6w~}En En En
_ Eng Eng Eng
~{6y~}Er Erh Er
~{7%~}+ Fa Fa Fa
~{74~}3 Fan Fan Fan
~{7=~}5 Fang Fang Fang
~{ez~}0 Fei Fei Fei
~{7V~}4 Fen Fen Fen
~{7a~}6 Feng Feng Feng
~{7p~}, Fo Fo Fwo
~{8!~}2 Fou Fou Fou
~{FM~}9 Fu Fu Fu
Ъ~{^N~}+ Ga Ka Ga
~{[r~}/ Gai Kai Gai
~{8I~}3 Gan Kan Gan
~{?:~}5 Gang Kang Gang
~{8_~}1 Gao Kao Gau
~{8j~}- Ge Ke Ge
~{8x~}0 Gei Kei Gei
~{8y~}4 Gen Ken Gen
~{8{~}6 Geng Keng Geng
~{9$~}96Gong Kung Gung
~{94~}2 Gou Kou Gou
~{9@~}9 Gu Ku Gu
~{9O~}9+Gua Kua Gwa
~{9T~}9/Guai Kuai Gwai
~{9Y~}93Guan Kuan Gwan
~{9b~}95Guang Kuang Gwang
~{9g~}90Gui Kuei Gwei
~{9v~}94Gun Kun Gwun
~{9y~}9,Guo Kuo Gwo
~{9~~}+ Ha Ha Ha
~{:"~}/ Hai Hai Hai
~{:(~}3 Han Han Han
~{:;~}5 Hang Hang Hang
~{:>~}1 Hao Hao Hau
~{:G~}- He He He
~{:Z~}0 Hei Hei Hei
~{:[~}4 Hen Hen Hen
~{:`~}6 Heng Heng Heng
~{:e~}96Hong Hung Hung
~{:n~}2 Hou Hou Hou
~{:u~}9 Hu Hu Hu
~{;/~}9+Hua Hua Hwa
~{;2~}9/Huai Huai Hwai
~{;6~}93Huan Huan Hwan
~{kA~}95Huang Huang Hwang
η~{;R~}90Hui Hui Hwei
~{;h~}94Hun Hun Hwun
~{;m~}9,Huo Huo Hwo
~{<8~}!8 Ji Chi Ji
~{!8+Jia Chia Jya
~{!83Jian Chien Jyan
~{=-~}!85Jiang Chiang Jyang
~{=;~}!81Jiao Chiao Jyau
~{=T~}!8.Jie Chieh Jye
~{=m~}!84Jin Chin Jin
~{>)~}!86Jing Ching Jing
~{>=~}!:6Jiong Chiung Jyung
~{>E~}!82Jiu Chiu Jyou
~{>S~}!: Ju Chu: Jyu
~{>j~}!:3Juan Chuan: Jywan
~{f^~}!:.Jue Chueh: Jywe
~{>}~}!:4Jun Chun: Jyun
~{?(~}+ Ka K'a Ka
~{?*~}/ Kai K'ai Kai
~{?/~}3 Kan K'an Kan
~{?5~}5 Kang K'ang Kang
σ~{?<~}1 Kao K'ao Kau
~{?L~}- Ke K'e Ke
~{?O~}4 Ken K'en Ken
~{?S~}6 Keng K'eng Keng
~{?U~}96Kong K'ung Kung
~{?Z~}2 Kou K'ou Kou
~{?]~}9 Ku K'u Ku
~{?d~}9+Kua K'ua Kwa
е~{?l~}9/Kuai K'uai Kwai
~{?m~}93Kuan K'uan Kwan
~{?o~}95Kuang K'uang Kwang
~{?x~}90Kui K'uei Kwei
~{@$~}94Kun K'un Kwun
~{@(~}9,Kuo K'uo Kwo
~{@-~}+ La La La
~{@4~}/ Lai Lai Lai
~{@7~}3 Lan Lan Lan
~{@I~}5 Lang Lang Lang
~{@L~}1 Lao Lao Lau
~{@U~}- Le Le Le
~{@a~}0 Lei Lei Lei
~{c6~}6 Leng Leng Leng
~{A(~}8 Li Li Li
~{A)~}8+Lia Lia Lya
~{A,~}83Lian Lien Lyan
~{A<~}85Liang Liang Lyang
~{AC~}81Liao Liao Lyau
~{_V~}8.Lie Lieh Lye
~{AV~}84Lin Lin Lin
~{An~}86Ling Ling Ling
~{Ao~}82Liu Liu Lyou
~{B!~}96Long Lung Lung
~{B'~}2 Lou Lou Lou
~{B,~}9 Lu Lu Lu
~{BM~}93Luan Luan Lwan
~{BS~}:.Lue: Lueh: Lywe
~{BU~}94Lun Lun Lwun
~{B^~}9,Luo Luo Lwo
~{Bh~}+ Ma Ma Ma
~{Bq~}/ Mai Mai Mai
~{r)~}3 Man Man Man
Γ~{C&~}5 Mang Mang Mang
~{C(~}1 Mao Mao Mau
~{C;~}0 Mei Mei Mei
~{CF~}4 Men Men Men
~{CI~}6 Meng Meng Meng
~{_d~}8 Mi Mi Mi
~{C_~}83Mian Mien Myan
~{_w~}81Miao Miao Myau
~{XB~}8.Mie Mieh Mye
~{Cq~}84Min Min Min
~{C{~}86Ming Ming Ming
~{C}~}82Miu Miu Myou
~{C~~}, Mo Mo Mwo
θ~{D2~}2 Mou Mou Mou
~{D>~}9 Mu Mu Mu
ê~{DG~}+ Na Na Na
~{DK~}/ Nai Nai Nai
~{DP~}3 Nan Nan Nan
~{DR~}5 Nang Nang Nang
~{DU~}1 Nao Nao Nau
~{DZ~}0 Nei Nei Nei
~{D\~}6 Neng Neng Neng
~{Da~}8 Ni Ni Ni
~{Dj~}83Nian Nien Nyan
~{Do~}85Niang Niang Nyang
~{Dq~}81Niao Niao Nyau
~{Ds~}8.Nie Nieh Nye
~{Dz~}84Nin Nin Nin
~{D~~}86Ning Ning Ning
~{E#~}82Niu Niu Nyou
~{E)~}96Nong Nung Nung
~{qq~}92Nou Nou Nou
~{E+~}9 Nu Nu Nu
~{E.~}: Nu: Nu: Nyu
~{E/~}93Nuan Nuan Nwan
~{E0~}:.Nue: Nueh: Nywe
~{E2~}9,Nuo No Nwo
~{E<~}Ou Ou Ou
~{0E~}+ Pa P'a Pa
~{ED~}/ Pai P'ai Pai
~{EK~}3 Pan P'an Pan
~{ER~}5 Pang P'ang Pang
~{EW~}1 Pao P'ao Pau
Д~{;5~}0 Pei P'ei Pei
~{Eg~}4 Pen P'en Pen
~{Es~}6 Peng P'eng Peng
~{F%~}8 Pi P'i Pi
~{F,~}83Pian P'ien Pyan
~{F1~}81Piao P'iao Pyau
~{F2~}8.Pie P'ieh Pye
~{^U~}84Pin P'in Pin
~{F9~}86Ping P'ing Ping
~{FB~}, Po P'o Pwo
~{FJ~}2 Pou P'ou Pou
~{FM~}9 Pu P'u Pu
~{F_~}"8 Qi Ch'i Chi
~{F~~}"8+Qia Ch'ia Chya
~{G'~}"83Qian Ch'ien Chyan
~{G<~}"85Qiang Ch'iangChyang
~{GD~}"81Qiao Ch'iao Chyau
~{GP~}"8.Qie Ch'ieh Chye
~{GV~}"84Qin Ch'in Chin
~{G`~}"86Qing Ch'ing Ching
~{q7~}":6Qiong Ch'iungChyung
~{Gp~}"82Qiu Ch'iu Chyou
Ρ~{Gz~}": Qu Ch'u: Chyu
~{H&~}":3Quan Ch'uan:Chywan
~{H1~}":.Que Ch'ueh:Chywe
~{H:~}":4Qun Ch'un: Chyun
~{H;~}'3 Ran Jan Ran
~{HB~}'5 Rang Jang Rang
~{f,~}'1 Rao Jao Rau
~{HH~}'- Re Je Re
~{HK~}'4 Ren Jen Ren
~{HS~}'6 Reng Jeng Reng
~{HU~}Ri Jih R
Ε~{HV~}'96Rong Jung Rung
~{Ha~}'2 Rou Jou Rou
~{Hg~}'9 Ru Ju Ru
ǹ~{Hn~}'93Ruan Juan Rwan
~{Hp~}'90Rui Jui Rwei
~{Hr~}'94Run Jun Rwun
~{Ht~}'9,Ruo Jo Rwo
~{X&~}*+ Sa Sa Sa
~{H{~}*/ Sai Sai Sai
~{H}~}*3 San San San
~{I#~}*5 Sang Sang Sang
~{I&~}*1 Sao Sao Sau
~{I+~}*- Se Se Se
~{I-~}*4 Sen Sen Sen
~{I.~}*6 Seng Seng Seng
~{I3~}&+ Sha Sha Sha
~{I9~}&/ Shai Shai Shai
~{I=~}&3 Shan Shan Shan
~{IL~}&5 Shang Shang Shang
~{IS~}&1 Shao Shao Shau
~{I]~}&- She She She
~{K-~}&0 Shei Shei Shei
~{Ij~}&4 Shen Shen Shen
~{I}~}&6 Sheng Sheng Sheng
~{J,~}Shi Shih Shr
Μ~{JU~}&2 Shou Shou Shou
~{Ji~}&9 Shu Shu Shu
~{K"~}&9+Shua Shua Shwa
~{K%~}&9/Shuai Shuai Shwai
~{K)~}&93Shuan Shuan Shwan
~{K*~}&95ShuangShuang Shwang
~{K-~}&90Shui Shui Shwei
~{K1~}&94Shun Shun Shwun
~{K5~}&9,Shuo Shuo Shwo
~{K>~}Si Ssu Sz
~{bl~}*96Song Sung Sung
~{[E~}*2 Sou Sou Sou
~{Jh~}*9 Su Su Su
~{Ka~}*93Suan Suan Swan
~{Kd~}*90Sui Sui Swei
~{Ko~}*94Sun Sun Swun
~{Kt~}*9,Suo So Swo
~{K{~}+ Ta T'a Ta
~{L(~}/ Tai T'ai Tai
~{L.~}3 Tan T'an Tan
~{L@~}5 Tang T'ang Tang
~{LR~}1 Tao T'ao Tau
~{LX~}- Te T'e Te
~{LZ~}6 Teng T'eng Teng
~{L^~}8 Ti T'i Ti
~{Ll~}83Tian T'ien Tyan
~{Y,~}81Tiao T'iao Tyau
~{L{~}8.Tie T'ieh Tye
~{L}~}86Ting T'ing Ting
~{M,~}96Tong T'ung Tung
~{M5~}2 Tou T'ou Tou
~{M:~}9 Tu T'u Tu
~{ME~}93Tuan T'uan Twan
~{MF~}90Tui T'ui Twei
~{ML~}94Tun T'un Twun
~{Y"~}9,Tuo T'o Two
~{M^~}9+ Wa Wa Wa
~{Ma~}9/ Wai Wai Wai
~{X`~}93 Wan Wan Wan
~{Mv~}95 Wang Wang Wang
~{N/~}90 Wei Wei Wei
~{ND~}94 Wen Wen Wen
~{NL~}96 Weng Weng Weng
~{YA~}9, Wo O O
~{[X~}Wu Wu Wu
~{Yb~}#8 Xi Hsi Xyi
~{O;~}#8+Xia Hsia Sya
~{OI~}#83Xian Hsien Syan
~{O`~}#85Xiang Hsiang Syang
~{Ow~}#81Xiao Hsiao Syau
~{P)~}#8.Xie Hsieh Sye
~{PD~}#84Xin Hsin Syin
~{PG~}#86Xing Hsin Sying
~{PW~}#:6Xiong Hsiung Syung
~{P]~}#82Xiu Hsiu Syou
~{Pl~}#: Xu Hsu: Syu
~{P{~}#:3Xuan Hsuan: Sywan
~{Q%~}#:.Xue Hsueh: Sywe
~{Q+~}#:4Xun Hsun: Syun
~{Q>~}8+ Ya Ya Ya
~{QY~}83 Yan Yan Yan
~{Qk~}85 Yang Yang Yang
~{R"~}81 Yao Yao Yau
~{R9~}8. Ye Yeh Ye
~{R;~}Yi I Yi
~{Rr~}84 Yin Yin Yin
~{S"~}86 Ying Ying Ying
~{g_~}:6 Yong Yung Yung
я~{X|~}82 You Yu You
ū~{SX~}Yu Yu: Yu
~{T*~}:3 Yuan Yuan: Ywan
~{TB~}:. Yue Yueh: Ywe
~{TN~}:4 Yun Yun: Yun
~{TQ~}(+ Za Tsa Dza
~{TV~}(/ Zai Tsai Dzai
~{t"~}(3 Zan Tsan Dzan
~{j0~}(5 Zang Tsang Dzang
~{Tb~}(1 Zao Tsao Dzau
~{Tr~}(- Ze Tse Dze
~{Tt~}(0 Zei Tsei Dzei
~{Tu~}(4 Zen Tsen Dzen
~{Tx~}(6 Zeng Tseng Dzeng
~{Tz~}¥+ Zha Cha Ja
~{U*~}¥/ Zhai Chai Jai
~{U<~}¥3 Zhan Chan Jan
~{UE~}¥5 Zhang Chang Jang
~{UY~}¥1 Zhao Chao Jau
~{UZ~}¥- Zhe Che Je
~{Ud~}¥4 Zhen Chen Jen
~{U}~}¥6 Zheng Cheng Jeng
~{V.~}Zhi Chih Jr
~{VP~}¥96Zhong Chung Jung
~{V]~}¥2 Zhou Chou Jou
Χ~{Vl~}¥9 Zhu Chu Ju
ъ~{W%~}¥9+Zhua Chua Jwa
~{W'~}¥9/Zhuai Chuai Jwai
~{W(~}¥93Zhuan Chuan Jwan
З~{W1~}¥95ZhuangChuang Jwang
~{v?~}¥90Zhui Chui Jwei
~{W<~}¥94Zhun Chun Jwun
~{W?~}¥9,Zhuo Cho Jwo
~{WS~}Zi Tzu Dz
~{WZ~}(96Zong Tsung Dzung
~{W^~}(2 Zou Tsou Dzou
~{Wb~}(9 Zu Tsu Dzu
~{Wj~}(93Zuan Tsuan Dzwan
~{Wl~}(90Zui Tsui Dzwei
~{Wp~}(94Zun Tsun Dzwun
~{Ww~}(9,Zuo Tso Dzwo

Additional Points

The above table deals with the phonetic symbols employed to represent the Chinese pronunciation of a character. This is analogous to the many different schemes used in English dictionaries to represent pronunciation (long vowels, short consonants, etc). As we can see from the above table, there are many methods (and many more are not listed above). The most common ones are pinyin (China) and BoPoMoFo (Taiwan). The purpose of this web page is to demonstrate the nearly one-to-one correspondence between these two methods. Personally, I prefer the pinyin method, because I already know my alphabets. Why waste time learning another set of phonetic symbols, especially when they are so nearly equivalent?

Big5 vs. GB Because of the political division of China, there are two major coding schemes: Big5 (mainly in Taiwan) and GB (mainly in mainland China). Do not confuse Big5/GB with the BoPoMoFo/pinyin phonetic representation. The coding scheme is simply a way to represent a character with a number, much like how we represent an alphabet with a number with the ASCII scheme, e.g., 65 for the letter "A". Both coding schemes are based on 4 bytes ( 32 bits) per character. For example, the character of my last name is represented with different numerical values in different coding schemes: Big5, GB (and, for that matter, Japanese, or Korean).

Big5|GB vs. Unicode|HZ To make the matter worse, there exist many coding schemes other than just Big5 and GB. At one time, every Chinese software company seemed to advocate its own proprietary coding scheme. Another fairly common coding scheme, although not as popular as Big5 or GB, is HZ. Then there is Unicode, which is used to express characters/letters of not just Chinese Mandarin, but all sorts of languages, Asian and European alike; thus, the Unicode coding system contains Chinese characters along with many non-Chinsese characters.

Traditional vs. Simplified Style. A completely separate idea (not to be confused with phonetic representation or coding schemes) is the traditional style (which has more strokes for certain characters, favored in Taiwan) versus the simplified style (favored in China). The styles are analogous to different fonts in the US, much like how my wingdings font and my courier font display two different symbols for the same alphabet "A".

Input Method. Another yet completely different concept is the input method. Because the English-based keyboard is what we normally see in the US, we need to adopt that to Chinese. There are many input methods for picking out a specific Chinese character -- I am aware of at least 15 such methods. One of these is pinyin, where you simply type in the alphabet (which I think is the simplest because I am comfortable with typing English with a US keyboard). For example, I would type "w" "a" "n" "g"; about 15 characters pop up on my screen; and I pick out the character that corresponds to my last name. The BoPoMoFo method is similar in that each key maps into a phonetic symbol. There are many mappings for BoPoMoFo, one such mapping is shown here. There are sufficient number of keys on a US keyboard for each of the 37 BoPoMoFo phonetic symbols. Other input methods are based on pronunciation in other dialects (such as Cantonese, rather than Mandarin), radicals (bu4shou3), number of strokes, the type of strokes at four corners of the character (si4jiao3hao4ma3), the shape of the character (cang1jie2), ..., even English equivalent. For example, my last name translates to "king" in English; thus, when I employ the last input method, I type "king" on my US keyboard; a number of Chinese characters or phrases that correspond to "king" in meaning pop up on my screen; and I choose the correct one. As for me -- well, I hardly ever type in Chinese -- but when I do, I do most of my input with pinyin. Without exception, everyone I know who masters both BoPoMoFo and pinyin prefers pinyin, just like myself. I occasionally switch to other methods, when I, for example, do not know the pronunciation of the character that I am trying to type.

Directionality. Western languages are written in horizontal lines left-to-right. However, certain languages (such as Hebrew and Arabic) are written in horizontal lines right-to-left. Chinese languages (and those with a strong Chinese influence such as Japanese and Korean) are traditionally written in vertical lines (top-to-bottom) arranged right-to-left. However, this tradition is not commonly practiced today. China has long changed the directionality to be in line with the Western practice (i.e., horizontal lines left-to-right). However, many Taiwanese publications seem to follow no rules. It is typical to see top-to-bottom, right-to-left, and left-to-right conventions all coexist on the same page in a newspaper. This can cause serious confusions, especially when the phrases are sufficiently short such that both directions make perfect sense, but convey completely different meanings. For example, with my name read in an opposite direction, I become another equally plausible person: a new Southern King, which of course I am not.

There are many permutations of the above concepts. To make the matter worse, there are people who mix already messy Chinese schemes with other languages that also employ Chinese characters (such as Japanese and Korean). Such is the sad state of things when a country that should be united remains divided, and ordinary folks like I pay for the resulting chaos. I wrote this web page in an attempt to reduce such misunderstandings.


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PinYin and BoPoMoFo ZhuYin Equivalence

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Nam Sun Wang
Department of Chemical Engineering
University of Maryland
College Park, MD 20742-2111
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e-mail: nsw@eng.umd.edu ©1999 by Nam Sun Wang
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