【正文】 馬數(shù)字 pencilandpaper 紙和筆的regardless of , 不顧 contraction n. 收縮，縮寫式，緊縮gross n. 羅，為一計(jì)數(shù)單位， 1羅 =12打 positional notation n. 位置記數(shù)法 The Decimal SystemOur present system of numbers has 10 separate symbols 0, 1, 2, 3,…, 9, wich are called Arabic numerals. We would be forced to stop at 9 or to invent more symbols if it were not for the use of positional notation. An example of earlier types of notation can be found in Roman numeral, which are essentially additive: III=I+I+I, XXV=X+X+V. New symbols (X, C, M, etc.) were used as the numbers increased in value. Thus V rather than IIIII=5. The only importance of position in Roman numerals lies in whether a symbol precedes or follows another symbol (IV=4 and VI=6).我們當(dāng)前的數(shù)字系統(tǒng)有 0、 3….9 十個(gè)單獨(dú)的符號(hào)，稱之為阿拉伯?dāng)?shù)字。如果不使用位置符號(hào)，我們數(shù)到 9就被迫停下來，或發(fā)明更多的符號(hào)。在羅馬數(shù)字里可以找到早期符號(hào)類型的例子，他們基本上是加法的： Ⅲ =Ⅰ +Ⅰ +Ⅰ ，XXV=X+X+V。當(dāng)數(shù)值增加時(shí)采用新符號(hào)（ X、 C、 M等）。這樣 V就不是 IIIII=５。羅馬數(shù)字中位的唯一重要性在于這個(gè)符號(hào)處于另一個(gè)符號(hào)之前或之后（Ⅳ =4和 Ⅵ =6）。 The Decimal SystemThe clumsiness of this system can easily be seen if we try to multiply XII by XIV. Calculating with Roman numerals was so difficult that early mathematicians were forced to perform arithmetic operations almost entirely on abaci, or counting boards, translating their results back into Romannumber form. Pencilandpaper putations are unbelievably intricate and difficult in such systems. In fact, the ability to perform such operations as addition and multiplication was considered a great acplishment in earlier civilization. 如果你要用 XIV乘 XII ，很容易看出這個(gè)數(shù)字系統(tǒng)是笨拙的。用羅馬數(shù)字計(jì)算太難了，以至于早期的數(shù)字家?guī)缀跬耆黄仍谒惚P或演算板完成算術(shù)運(yùn)算，然后再把結(jié)果翻譯成羅馬數(shù)字形式。在這樣的數(shù)字系統(tǒng)中，紙和筆運(yùn)算達(dá)到以難置信的復(fù)雜和困難程度。事實(shí)上，在早期文明中能進(jìn)行這樣的加法和乘法運(yùn)算被看作是一項(xiàng)偉大的成就。 The Decimal System The Decimal SystemThe great beauty and simplicity of our number system can now be seen. It is necessary to learn only the basic numerals and the positional notation system in order to count to any desired figure. After memorizing the addition and multiplication tables and learning a few simple rules, it is possible to perform all arithmetic operations. Notice the simplicity of multiplying 1214 using the present system.現(xiàn)在可以看到我們的數(shù)字系統(tǒng)的巨大優(yōu)勢(shì)和簡(jiǎn)單明了，為了要數(shù)到任意想到的數(shù)字，只需要學(xué)會(huì)基本數(shù)字和進(jìn)位符號(hào)，再記住加法和乘法表及學(xué)會(huì)一些簡(jiǎn)單規(guī)則，就可能完成所有的算術(shù)運(yùn)算?？匆幌掠矛F(xiàn)在數(shù)制計(jì)算 1214的簡(jiǎn)單性。 The Decimal SystemThe actual meaning of the number 168 can be seen more clearly if we notice that it is spoken as “one hundred and sixtyeight”. Basically, the number is a contraction of (1100)+(610)+8. The important point is that the value of each digit is determined by its position. For example, the 2 in 2,000 has a different value than the 2 in 20. We show this verbally by saying “two thousand” and “twenty”. Different verbal representations have been invented for numbers from 10 to 20 (eleven, twelve), but from 20 upward we break only at powers of 10 (hundreds, thousands, millions, billions). Written numbers are always contracted, however, and only the basic 10 numerals are used regardless of the size of the integer written. The general rule for representing numbers in the decimal system using positional notation is as follows.如果我們注意到說 ‘一百六十八 ’時(shí)，數(shù)字 168的實(shí)際意義就能更清楚地看出來?；旧希@個(gè)數(shù)字是（ 1100） +610） +8的緊縮形式。更重要的是每個(gè)數(shù)字的值由它的位置來決定。例如 2022中的 2和 20中的２的值是不同的。我說 ‘二千 ’和 ‘二十 ’來口頭表達(dá)這些。從 10到 20我們發(fā)明出不同的口頭表示方式。但是從 20往上起，我們只在 10的權(quán)位上斷開。書寫出的數(shù)字總緊湊的，不論寫出的整數(shù)大小，只用 10個(gè)基本數(shù)字。十進(jìn)制使用進(jìn)位符號(hào)表示數(shù)字的通則是 ： The Decimal SystemThe integer digit in different position is expressed as a