The Letters of Wolfgang Amadeus Mozart — Volume 01 by Wolfgang Amadeus Mozart - Free Ebook
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The Letters of Wolfgang Amadeus Mozart — Volume 01
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Mar 1, 2004
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Enter your search terms separated by spaces,
then press &Enter&.
Avoid punctuation except as indicated below:
exact match
Always put spaces around these.
this query
shakespeare hamlet
"Hamlet" by Shakespeare
"qui", not "Quixote"
love stories
love stories
a.shakespeare
by Shakespeare
s.shakespeare
about Shakespeare
ebook no. 74
juvenile l.german
juvenile lit in German
verne ( l.fr | l.it )
by Verne in French or Italian
love stories ! austen
love stories not by Austen
jane austen cat.audio
audio books by Jane Austen数字~年份~月份的标准读法_百度知道
数字~年份~月份的标准读法
1.数字中and的用法，在数字中and一般用与百与十之间作连接憨骇封较莩记凤席脯芦，但如果没有百位树，要不要and开连接呢？如：101 读作one hundred and one
1001读作one thousand (and) one 括号中的and到底要不要啊，我在网上问了好几次了，有的说要，有的又说不要，把我搞糊涂了2.年的读法在英语中的标准是怎样的如：2006年读作 1.two thousand (and) six
2.ten hundred
(and) six上面写法到底哪个是对的，另外请用英语写出1973年的表达方法3.用完整的英语写出以下几个日期日日PS：请各位高手帮忙举例解释，菜鸟自重，自己都没把握的不要来误人子弟小弟感激不敬.
回答的好我会加分的请自重~不诚心的不要来浪费时间~
提问者采纳
1。可要可不要。但尽量不要，因为and其实应该是用在小数点上的。列：101.1读作one 憨骇封较莩记凤席脯芦hundred one AND one tenths2。这里需要and因为年数没有小数点，所以应该是第一个。it's year TWO THOUSAND AND SIX.3。year nineteen ninety eight, June 26th (twenty sixth), year two thousand seven, December 30th (thirtyeth).
提问者评价
谢谢你~加20吧
其他类似问题
其他2条回答
1001读作one thousand (and) one2006年读作 1.two thousand (and) six 1973nineteen seventy-three日 june twenty-sixth, nineteen ninty憨骇封较莩记凤席脯芦-eight日 december thirtith, two thouand and seven
1.Cardinals and Ordinals 基数和序数
(1)100以内的数词
Roman Arabic
Numerals Numerals
罗马数字 阿拉伯数字 读法 数码 读法
I 1 one 1st the first
II 2 two 2nd the second
III 3 three 3rd the third
IV 4 four 4th the fourth
V 5 five 5th the fifth
VI 6 six 6th the sixth
VII 7 seven 7th the seventh
VIII 8 eight 8th the eighth
IX 9 nine 9th the ninth
X 10 ten 10th the tenth
XI 11 eleven 11th the eleventh
XII 12 twelve 12th the twelfth
XIII 13 thirteen 13th the thirteenth
XIV 14 fourteen 14th the fourteenth
XV 15 fifteen 15th the fifteenth
XVI 16 sixteen 16th the sixteenth
XVII 17 seventeen 17th the seventeenth
XVIII 18 eighteen 18th the eighteenth
XIX 19 nineteen 19th the nineteenth
XX 20 twenty 20th the twentieth
XXI 21 twenty-one 21st the twenty-first
XXV 25 twenty-five 25th the twenty-fifth
XXX 30 thirty 30th the thirtieth
XL 40 forty 40th the fortieth
L 50 fifty 50th the fiftieth
LX 60 sixty 60th the sixtieth
LXX 70 seventy 70th the seventieth
LXXX 80 eighty 80th the eightieth
XC 90 ninety 90th the ninetieth
IC 99 ninety-nine 99th the ninety-ninth
C 100 one hundred 100th the hundredth
CII 102 a hundred 102nd the (one) hundred
and two and second
246 two hundred and forty-six
751 seven hundred and fifty-one
(2)1000以上的数词
1,000 = one thousand 一千
10,000 = ten thousand 一万
100,000 = one hundred thousand 十万
1,000,000 = one million 一百万
10,000,000 = ten million 一千万
100,000,000=one hundred million 一亿
(3)十亿以上的大数，英美有不同的读法：
十亿 one thousand million = one billion
百亿 ten thousand million = ten billion
千亿 one hundred thousand million = one hundred billion
万亿 one billion = one trillion
2.Fractions 分数
通常将分子读为基数，将分母读为序数。
1/2 = a (or one) half
1/3 = a (or one) third
1/4 = a quarter or one fouth
1/5 = a (or one) fifth
2/3 = two thirds
9/10 = nine tenths
53/4 = five and three quarters
15/64= fifteen over (or by) sixty-four
15% = fifteen per cent
4‰ = four per mill
3.Decimals 小数
0.4 = zero (or nought) point four
.01 = point (or decimal) nought one
12.34 = twelve point three four
567.809 = five hundred and sixty-seven point eight nought nine
30.45 = thirty point four five, five recurring
0.3% = decimal three percent
4.Mathematic Forms 数学式
(1) Addition 加法
1+2=3 One and two are three.
2+3=5 Two plus three equals five.
4+0=4 Four and nought is equal to four.
45+70+152=267 45,70 and 152 added are (or make) 267
the sum (or total) is 267.
演算时的读法： Thr I write(or I write down,or I put down)a nought
and carry one. Four and one that I carry are five, and eight make thirteen,and three I write down six and carry one. One and one (that I carry) I put down
two.- The sum (or total, or the result of the addition) is two hundred and sixty. 37,80
and 143 added together, and(or make) 260.
(2)Subtraction 减法
9-4=5 Nine minus four equals (or is equal to) five.
15-7=8 Seven from fifteen leaves eight.
23,654-8,175=15,479 8,175 (take or subtracted) from 23,654 leaves 15,479. The difference
(or The remainder) is 15,479.
Nine from five won`t go.
演算时的读法：Nought from nought (leaves) nought. One from one leaves nought (or nothing).
Two from three (leaves) one. I can`t take (or subtract) I
five from fourteen leaves nine.- The difference (or The remainder) is nine thousand one
hundred. 5210 (take or substracted) from 14,310 leaves 9,100.
(3)Multiplication 乘法
1×0=0 One multiplied by nought equals nought.
1×1=1 Once one is one.
2×1=2 Twice one is two.
3×5=15 Three times five is fifteen
6×0=0 Multiply six by nothing, and the result is nothing.
演算时的读法: Five times nine (or Nine multiplied by five) are forty- I put down five
and carry four. Five times seven are thirty-five and four (that I carry) are thirty- I write
down nine and carry three. Five times six make thirty and three (that I carry) thirty- I put
down thirty-three.
Eight times nine (or Eight nine) I write two and carry seven. Eight sevens make
fifty-six and seven are sixty-three. I put down three and carry six. Eight sixes make forty-eight
and six fifty- I write down fifty-four.
I now add the partial results (or products) Five. Two and nine are eleven. Three and three are
six and one are seven. Four and three make seven. Five.
(4)Division 除法
9÷3=3 Nine divided by three maks (or is equal to) three.
20÷5=4 Five into twenty goes four times.
余13 23 into 4567 goes 198 times, and 13 remainder.
The quctient is 198, and 13 remainder.
演算时的读法: Fifteen into thirteen won` fifteen into one hundred and thirty-eight goes nine nine times fifteen are one hundred and threty- one hundred and thirty-five from one hundred
and thirty- I fifteen into thirty- twice fifteen thirty from thirty-seven leaves seven.
The (exact) quotient is ninety-two, 15 into 1387 goes 92 times, and 7 remainder.
5.Time 时间
(1)Hours 钟点
2h.5’8& = two hours five minutes eight seconds 2小时5分8秒
6.18 = six eighteen 6时18分
8.30 a.m.= eight thirty a.m.[’ei ’em] 上午8时30分
the 6.05 p.m.train = the six (nought) five p.m.[’pi:’em] train 下午6时零5分列车
又二十四小时混合制的写法和读法如下：
0900 = 0 nine hunderd (上午)9时
0910 = 0 nine ten (上午)9时10分
1300 = thirteen hundred 13时(下午1时)
1525 = fifteen twenty-five 15时25分(下午3时25分)
2000 = twenty hundred 20时(下午8时)
at 5 o’c =at five o’clock 五点钟
(2)Date 日期
Oct.1 =October first 10月1日
Oct.1st = October the first 10月1日
Ist Oct. 1949 = teh first of October, nineteen forty-nine 日
3/5 = [英]May (the) third 5月3日；[美]March fifth 3月5日
[附注]联系日期前置词用on.
(3)Year 年份
684 B.C. =Six eighty-four B.C. [’bi:’si:] 公元前684年
1960 = nineteen hundred and sixty
19- nineteen something
1950’s nineteen fifties 二十世纪五十年代
[附注]联系年份的前置词用in.6.Numbers 号码
(1)Telephones Numbers 电话号码
1023 = one O ten twenty-three
1227 = one double two (or two two) seven
0386 = O three eight six
0096 = double O(or O O )nine six
7000 = seven O double O = seven thousand
No.26= Number 26 第26号
Room 201 = Room two O one 第201房间
10 Changan Street = Ten Changan Street 长安街10号
(2)Writings 书籍作品
Vol.I = Volume one (or the first volume) 卷一
Chap.II= Chapter two (or the second chapter) 第2章
Page 3 = page Three (or the third page) 第3页
See pp.5-10 = See Pages five to ten 见第5-10页
Act V = Act five (or the fifth act) 第5幕
Hamlet III 1:56 = Hamlet Act Three, Secene One.Line fifty-six 《哈姆雷特》第3幕第一场第56行
Matt.7:12 = Matthew, Chapter Seven, Verse Twelve 《马可福音》第七章第12节
Beethoven Op.49 = Beethoven Opus forty-nine 贝多芬作品第49号
4to = quarto 四本开 8vo = octavo 八本开
(3)Other 其他
World War I = World War One 第一次世界大战
World War II = World War Two 第二次世界大战
Charles I= Charles the First 查理一世
Henry V = Henry the Fifth 亨利五世
Mr.- = Mr. D Mr. So-and-so 某某先生
Mr.B - = Mr. B
-Esq.,of -= the town of Blank Esquire of Blank University 某大学某某先生
the town ōf = the town of Blank 某某城
The result of the game was 3-0 比赛结果是三比零。7.Money 货币
(1)British currency 英币
6d. = six pence 六便士
1/2d. = a halfpenny 半便士
11/2d. = a peney halfpenny or three halfpeny or three halfpence一便士半
1/4d. = a farthing 一个铜元
33/4d.= three pence three farghings 三便士三铜元
1s.6d.(or 1/6) = one (shilling) and six (pence) 一先令六便士
￡1.18 = (or￡1.18s.)= one pound eighteen (shillings) 一镑十八先令
￡1.3s.6d. = one pound, three shillings (and) sixpence
4/51/2(or4s.51/2d.) = four (shillings) and five pence halfpenny
(2)U.S. currency
?1.20 dollar (and) twenty (cents) 美金一元二角
?4.25 = four dollars twenty-five cents 美金四元二角五分
(3)Soviet currency 前苏联币
6 Rb. 15 = six roubles fifteen kopecks 六卢布十五戈比
(4) German currency 德币
1 m. 60 = one mark sixty pfennig -马克六十芬尼
(5)French currency 法国币
1 fr.30 = one franc. thirty (centimes) -法郎三十生丁
0 fr.15 = fifteen centimes 十五生丁
(6) Chinese People’s Currency 人民币
?1.50 = one yuan and a half 一元五角
JMP 10.35 = JMP ten yuan thirty-five (fen) 人民币十元三角五分
附注：yuan单复数没有变化：?也可作为日本本位币“圆”的符号。8. Weight and Measures 度量衡
(1)length,area, and volume 长度、面积和容积
3 in = three inches 三英寸
15 ft.5 in = fifteen foot five (inches) 十五英尺五英寸
[附注] 尤其在inches省略时, ft.读作如果inches也念出来，ft可以读作fett.
18’6 5/1&= eighteen foot six and a fifth (inches)
10×8feet= ten by eight feet 十英尺长，八英尺宽
5&×4×31/2 = five inches by four by three and a half 长五英寸，宽四英寸，高三英寸半
[附注] 以上二例中，乘号×表示面积或容积。
(2)Weight 重量
12 dr. 23 gr.= Twelve drams twenty-three grains 十二打兰二十三喱
10 oz. 4 dr. = Ten ounces four drams 十盎司打兰
(3)Capacity 容量
3 gi. = three gills 三及耳
1 qt. 1 pt = one quart one pint 一夸脱一品脱
[附注] qt.,pt. 的复数是qts.,pts.,也可以不加s.
20 gal. 5 qt. =Twenty gallons five quarts 二十加仑五夸脱
5 bu. 3 pk. = Five bushels three pecks 五蒲式耳三配克一、一般符号对应的英文单词
． period 句号
， comma 逗号
： colon 冒号
； semicolon 分号
！ exclamation 惊叹号
？ question mark 问号
— hyphen 连字符
’ apostrophe 省略号；所有格符号
- dash 破折号
‘’single quotation marks 单引号
“”double quotation marks 双引号
( ) parentheses 圆括号
[ ] square brackets 方括号
《 》French quotes 法文引号；书名号
... ellipsis 省略号
¨ tandem colon 双点号
& ditto 同上
‖ parallel 平行
／ virgule 斜线号
＆ ampersand = and
～ swung dash 代字号
§ division 分节号
→ arrow 箭号；参见号
＋ plus 加号；正号
－ minus 减号；负号
± plus or minus 正负号
× is multiplied by or cross 叉乘
÷ is divided by 除号
＝ is equal to 等于号
≠ is not equal to 不等于号
≡ is equivalent to 全等于号
≌ is equal to or approximately equal to 等于或约等于号
≈ is approximately equal to 约等于号
＜ is less than 小于号
＞ is more than （is greater than在数学中更常用） 大于号
≮ is not less than 不小于号
≯ is not more than 不大于号
≤ is less than or equal to 小于或等于号
≥ is more than or equal to 大于或等于号
％ per cent 百分之…
‰ per mill 千分之…
∞ infinity 无限大号
∝ varies as 与…成比例
√ (square) root 平方根
∵ because 因为
∴ hence 所以
∷ equals, as (proportion) 等于，成比例
∠ angle 角
⌒ semicircle 半圆
⊙ circle 圆
○ circumference 圆周
π pi 圆周率
△ triangle 三角形
⊥ perpendicular to 垂直于；另外normal to，right to也都有垂直的意思。
∪ union of 并，合集
∩ intersection of 交，通集
∫ the integral of …的积分
∑ (sigma) summation of 总和
° degree 度
′ minute 分
〃 second 秒
＃ pound …号
. dot (点乘就是centered dot)
f’ f prime f撇
A上面一个横杠：A bar
A上面一个星星*： A asterisk
A上面一个波浪线～：A tilde
A的厄米共轭（注意不是加号，那个竖比横长）：A dagger（dagger：短剑，匕首）
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出门在外也不愁From Wikipedia, the free encyclopedia
This article is about the arithmetic concept.
For the group theory concept, see .
It has been suggested that
into this article. () Proposed since November 2014.
In , the hyperoperation sequence is an infinite
of arithmetic operations (called hyperoperations) that starts with the
(n = 0), then continues with the
(n = 2), and
(n = 3), after which the sequence proceeds with further binary operations extending beyond exponentiation, using . For the operations beyond exponentiation, the nth member of this sequence is named by
of n suffixed with -ation (such as
(n = 5), hexation (n = 6), etc.) and can be written as using n - 2 arrows in
(when n ≥ 3). Each hyperoperation may be understood
in terms of the previous one by:
It may also be defined according to the recursion rule part of the definition, as in Knuth's up-arrow version of the :
This can be used to easily show numbers much larger than those which
can, such as
and , but there are some numbers which even they cannot easily show, such as
This recursion rule is common to many variants of hyperoperations (see ).
The hyperoperation sequence is the
as follows:
(Note that for n = 0, the
essentially reduces to a
() by ignoring the first argument.)
For n = 0, 1, 2, 3, this definition reproduces the basic arithmetic operations of
(which is a unary operation), , , and , respectively, as
and for n ≥ 4 it extends these basic operations beyond exponentiation to what can be written in
Knuth's notation could be extended to negative indices ≥ -2 in such a way as to agree with the entire hyperoperation sequence, except for the lag in the indexing:
The hyperoperations can thus be seen as an answer to the question "what's next" in the : , , , , and so on. Noting that
the relationship between basic arithmetic operations is illustrated, allowing the higher operations to be defined naturally as above. The parameters of the hyperoperation hierarchy are sometimes referred to by their analogous so a is the base, b is the exponent (or hyperexponent), and n is the rank (or grade).
In common terms, the hyperoperations are ways of compounding numbers that increase in growth based on the iteration of the previous hyperoperation. The concepts of successor, addition, multiplication and exponentiation are the successor operation (producing x+1 from x) is the most primitive, the addition operator specifies the number of times 1 is to be added to itself to produce a final value, multiplication specifies the number of times a number is to be added to itself, and exponentiation refers to the number of times a number is to be multiplied by itself.
This is a list of the first seven (0th to 6th) hyperoperations. (Notice that in this article, we define 00 as 1)
(Hn(a, b))
Definition
hyper0, increment, , zeration
b arbitrary
a & 0, b real, or a ≥ 0, b nonnegative, or a non-zero, b an integer, with some multivalued extensions to
a ≥ 0 or an integer, b an integer ≥ -1 (with some proposed extensions)
a, b integers ≥ -1
hyper6, hexation
a, b integers ≥ -1
Hn(0, b) =
0, when n = 2, or n = 3, b ≥ 1, or n ≥ 4, b odd (≥ -1)
1, when n = 3, b = 0, or n ≥ 4, b even (≥ 0)
b, when n = 1
b + 1, when n = 0
Hn(a, 0) =
0, when n = 2
1, when n = 0, or n ≥ 3
a, when n = 1
Hn(a, -1) =
0, when n = 0, or n ≥ 4
a - 1, when n = 1
-a, when n = 2
1/a , when n = 3
One of the earliest discussions of hyperoperations was that of Albert Bennett in 1914, who developed some of the theory of commutative hyperoperations (see ). About 12 years later,
defined the function
which somewhat resembles the hyperoperation sequence.
In his 1947 paper,
introduced the specific sequence of operations that are now called hyperoperations, and also suggested the Greek names , pentation, etc., for the extended operations beyond exponentiation (because they correspond to the indices 4, 5, etc.). As a three-argument function, e.g., , the hyperoperation sequence as a whole is seen to be a version of the original
— as modified by Goodstein to incorporate the primitive
together with the other three basic operations of arithmetic (, , ), and to make a more seamless extension of these beyond exponentiation.
The original three-argument
uses the same recursion rule as does Goodstein's version of it (i.e., the hyperoperation sequence), but differs from it in two ways. First,
defines a sequence of operations starting from addition (n = 0) rather than the , then multiplication (n = 1), exponentiation (n = 2), etc. Secondly, the initial conditions for
result in , thus differing from the hyperoperations beyond exponentiation. The significance of the b + 1 in the previous expression is that
= , where b counts the number of operators (exponentiations), rather than counting the number of operands ("a"s) as does the b in , and so on for the higher-level operations. (See the
article for details.)
This is a list of notations that have been used for hyperoperations.
Notation equivalent to
(for n ≥ 3), and found in several reference books.
Goodstein's notation
(for n ≥ 1)
This corresponds to hyperoperations for base 2 (a = 2)
Nambiar's notation
Used by Nambiar (for n ≥ 1)
Box notation
Used by Rubtsov and Romerio.
Superscript notation
Subscript notation (for lower hyperoperations)
Used for lower hyperoperations by Robert Munafo.
Operator notation (for "extended operations")
Used for lower hyperoperations by
Square bracket notation
Used i convenient for .
(for n ≥ 3)
(for n ≥ 3)
Main article:
defined a 3-argument function
which gradually evolved into a 2-argument function known as the . The original Ackermann function
was less similar to modern hyperoperations, because his initial conditions start with
for all n & 2. Also he assigned addition to n = 0, multiplication to n = 1 and exponentiation to n = 2, so the initial conditions produce very different operations for tetration and beyond.
An offset form of . The iteration of this operation is different than the
of tetration.
Not to be confused with .
Another initial condition that has been used is
(where the base is constant ), due to Rózsa Péter, which does not form a hyperoperation hierarchy.
In 1984, C. W. Clenshaw and F. W. J. Olver began the discussion of using hyperoperations to prevent computer
overflows. Since then, many other authors have renewed interest in the application of hyperoperations to
representation. (Since Hn(a, b) are all defined for b = -1) While discussing , Clenshaw et al. assumed the initial condition , which makes yet another hyperoperation hierarchy. Just like in the previous variant, the fourth operation is very similar to , but offset by one.
An offset form of . The iteration of this operation is much different than the
Not to be confused with .
An alternative for these hyperoperations is obtained by evaluation from left to right. Since
define (with ° or subscript)
This was extended to ordinal numbers by Donner and Tarski,[Definition 1] by :
It follows from Definition 1(i), Corollary 2(ii), and Theorem 9, that, for a ≥ 2 and b ≥ 1, that[]
But this suffers a kind of collapse, failing to form the "power tower" traditionally expected of hyperoperators:[Theorem 3(iii)]
If α ≥ 2 and γ ≥ 2, [Corollary 33(i)]
increment, successor, zeration
Not to be confused with .
Not to be confused with .
Similar to .
Commutative hyperoperations were considered by Albert Bennett as early as 1914, which is possibly the earliest remark about any hyperoperation sequence. Commutative hyperoperations are defined by the recursion rule
which is symmetric in a and b, meaning all hyperoperations are commutative. This sequence does not contain , and so does not form a hyperoperation hierarchy.
This is due to the .
Not to be confused with .
Sequences similar to the hyperoperation sequence have historically been referred to by many names, including: the
(3-argument), the Ackermann hierarchy, the
(which is more general), Goodstein's version of the Ackermann function, operation of the nth grade, z-fold iterated exponentiation of x with y,
operations, reihenalgebra and hyper-n.
a [n] (-1) = 0 for all real number a and all integer n ≥ 4, because if so, than a [n] 0 = a [n - 1] (a [n] (-1)) = a [n - 1] 0 = 1 (because of n ≥ 4), this is keeping with the define: a [n] 0 = 1 for all real number a.
Ordinal additio see
for more information
Daniel Geisler (2003). .
Harvey M. Friedman (Jul 2001). . Journal of Combinatorial Theory, Series A 95 (1): 102–144. :.
Manuel Lameiras Campagnola and
and José Félix Costa (Dec 2002). . Journal of Complexity 18 (4): 977–1000. :.
Marc Wirz (1999). . CiteSeer.
R. L. Goodstein (Dec 1947). "Transfinite Ordinals in Recursive Number Theory". Journal of Symbolic Logic 12 (4): 123–129. :.  .
Albert A. Bennett (Dec 1915). "Note on an Operation of the Third Grade". Annals of Mathematics. Second Series 17 (2): 74–75. :.  .
Paul E. Black (). . . U.S. National Institute of Standards and Technology (NIST).
J. E. Littlewood (Jul 1948). "Large Numbers". Mathematical Gazette 32 (300): 163–171. :.  .
Markus Müller (1993).
Robert Munafo (November 1999). . Large Numbers at MROB.
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