整数的读法

和中文四位一节不同，英文是三位一节(period)，加逗号隔开：

4,321 四千三百二十一 four thousand, three hundred twenty-one

This is a four-digit number (四位数), where the digit 1 is in the ones place (个位), standing for 1 one (1个一) the digit 2 is in the tens place (十位), standing for 2 tens (2个十), the digit 3 is in the hundreds place (百位), standing for 3 hundreds (3个百), and the digit 4 is in the thousands place (千位), standing for 4 thousands (4个千).

注：十位和个位之间加连字符；美式英语不会在百位和十位中间加 and.

不像中文会在一节中的四位中间出现0时把0读出来：

101 一百零一 one hundred one

4,012 四千零十二 four thousand, twelve

不会像中文在读十/百/千位本身不为0，后面各位都为0的数字时，可以省略“十/百/千”：

250 二百五(二百五十) two hundred fifty, but not two hundred five (which is 205).

3,800 三千八(三千八百) three thousand, eight hundred, but not three thousand, eight (which is 3,008)

每个逗号之间都按三位数的读法，从右往左的第一个逗号读作 thousand (千)，第二个逗号读作 million (百万，一千个千)，第三个逗号读作 billion (十亿，一千个百万)，第四个逗号读作 trillion (万亿，一千个十亿), etc.

9,876,543,210 九十八亿七千六百五十四万三千二百一十 nine billion, eight hundred seventy-six million, five hundred forty-three thousand, two hundred ten.

This is a 10-digit number (十位数), where 4 is in the ten thousands place (十千位，也就是万位), standing for 4 ten thousands, 5 is in the hundred thousands place (百千位，也就是十万位), 6 is in the millions place (百万位), standing for 6 millions, 7 is in the ten millions place (十百万位，也就是千万位), standing for 7 ten millions, 8 is in the hundred millions place (百百万位，也就是亿位), standing for 8 hundred millions, 9 is in the billions place (十亿位), standing for 9 billions。

刚才讨论的是正数 positive numbers 的读法，为强调可加上 positive 或者 plus

负数 negative numbers 的读法须在前面加上 negative 或者 minus

-375 is read as negative/minus three hundred seventy-five

分数的读法

fraction n. 分数 fractional adj. 分数的

numerator 分子 the number of parts that are taken.

denominator 分母 the number of parts into which the whole is equally divided.

总的来说：用中文读分数，先读分母，再读分子；用英文读分数，先读分子，再读分母。分子大于1，或者等于0时，分母序数词用复数(+s).

1. 分母为2 (when the denominator is 2)

frac{1}{2} 二分之一 one-half, frac{2}{2} 二分之二 two-halves

frac{3}{2} 二分之三 three-halves, frac{0}{2} 二分之零 zero-halves

2. 分母是大于2的整数 (when the denominator is any whole number other than 2)

先读分子，分子用基数词，再读分母，分母用序数词，分子大于1，分母用复数。在书写时分子和分母之间加连字符：

frac{1}{3} 三分之一 one-third, frac{2}{3} 三分之二 two-thirds

frac{1}{4} 四分之一 one-fourth, or a quarter, frac{3}{4} 四分之三 three-fourths, or three quarters

frac{1}{21} 二十一分之一 one twenty-first

frac{3}{22} 二十二分之三 three twenty-seconds

frac{5}{23} 二十三分之五 five twenty-thirds

3. 当分母很大，或者不是正整数时 (when the denominator is too big or not a whole number).

分子和分母都用基数词，先读分子，中间加over（因为分子在分数线fraction line/bar的上方，分母在分数线的下方），再读分母。对于小分母也可以这么读。

frac{355}{113} 一百一十三分之三百五十五 three hundred fifty-five over one hundred thirteen

frac{3}{-2} 负二分之三 three over negative two; frac{1}{x} one over x;

frac{-bpm sqrt{b^2-4ac}}{2a} the opposite of b plus minus the square root of b squared minus 4ac, the quantity over 2a.

注：中文不管分母是什么，都可以读成“分母”分之“分子”，但英文却要分情况而定，这正如次方的读法，中文不管指数n是什么，都可以读成“底数”的n次方，但英文也要视情况而定，读成nth power (n是正整数), 或者 to the (power of) n (n不是正整数).

理解英文读法的关键在于 unit fraction 单位分数 (fraction with numerator one)，序数词表示单位分数。

英文分数读法的逻辑在于：先说明占多少份，然后说每份是多少。这就类似于我们读任何测量值时先读数值，再读单位。单位分数就是把1等分之后得到的新的单位。

比如 frac{2}{5} 五分之二用英文来读就是 two-fifths, 两个五分之一。

这种分数读法的好处在于能够很容易地看出分数和除法的关系：

Three-fifths or three over five is just three divided by five.

三个五分之一（五分之三）正好等于把三个单位1平均分为五份每一份的大小，也就是单位1分为五分每份大小（五分之一）的三倍。

用英文就很容易读出 frac{a}{b}=adiv b , a over b equals a divided by b (所以分数线也经常读成 divided by).

我们知道（分子分母为自然数的）分数分为真分数 (proper fraction, in which the numerator is less than the denominator) 和假分数 (improper fraction, in which the numerator is greater than or equal to the denominator).

真分数是大于等于0小于1的分数，可以用"分子" +out of "分母" 的读法。

frac{0}{2} zero out of two, frac{1}{2} one out of two, frac{2}{2} two out of two.

假分数可以通过带余除法转化成带分数 (mixed number). 英文的带分数先读整数部分 (whole-number part) 中间加 and，再读分数部分 (fractional part)，比如

2frac{1}{3} 二又三分之一，读作 two and one-third.

小数的读法

decimal 这个单词本身是指十进制小数。

3.1415926… 正像中文读小数点 (decimal point) 后的数字是一位一位直接读一样，英语也是 three point one four one five nine two six… 但是对于位数比较少的有限小数 (terminating decimal)，小学教科书一开始教的读法是按照分数的方式来读，比如

3.1 three and one tenth (三又十分之一), 1 is in the tenths place (十分位).

注：在分数的分子和分母之间为了避免歧义不再加连字符。

3.14 three and fourteen hundredths (三又百分之十四), 4 is in the hundredths place (百分位).

3.141 three and one hundred forty-one thousandths (三又千分之一百四十一), the last digit 1 is in the thousandths place (千分位).

3.1415 three and one thousand, four hundred fifteen ten-thousandths (三又万分之一千四百一十五), the last digit 5 is in the ten-thousandths place (万分位).

3.14159 three and fourteen thousand, one hundred fifty-nine hundred-thousandths (三又十万分之一万四千一百五十九), the last digit 9 is in the hundred-thousandths place. (十万分位)

3.141592 three and one hundred forty-one thousand, five hundred ninety-two millionths (三又百万分之十四万一千五百九十二), the last digit 2 is in the millionths place (百万分位).

3.1415926 three and one million, four hundred fifteen thousand, nine hundred twenty-six ten-millionths (三又千万分之一百四十一万五千九百二十六), the last digit 6 is in the ten-millionths place (千万分位).

这种读法要确定小数点后最后一位不为0的数字所在数位，所以对于循环小数 (repeating decimal) 或者无限不循环小数 (non-repeating decimal) 自然就不适用了。

循环小数的记法 (notation): dfrac{1}{3}=0.overline{3}, dfrac{1}{6}=0.1overline{6}, dfrac{1}{7}=0.overline{142857} 是在循环节 repetend 上用上划线表示。

1.2overline{34} may be read "one point two repeating three four", "one point two repeated three four", "one point two recurring three four", "one point two repetend three four" or "one point two into infinity three four". (From Wikipedia.)

四舍五入到万位 round to the nearest ten thousand

四舍五入到亿分位 round to the nearest hundred-millionth

保留三位小数 correct to 3 decimal places

保留四位有效数字 correct to 4 significant digits/figures

数系 number system

1. natural number (counting number) 自然数（不包括0），正整数

prime number/prime n. 质数/素数，prime adj. relatively prime 互质的

A prime number is a natural nature that has exactly two distinct factors: 1 and the number itself.

composite number 合数

A composite number is a natural number that has more than two factors, or equivalently, has at least one factor other than 1 and the number itself.

2. whole number (AmE) 0和自然数的统称，但也有地方用作整数的同义词。

whole number part 整数部分，whole number ratio (正)整数比，whole number multiple (正)整数倍.

3. integer n. 整数 integral adj. 整数的 integral domain 整环 integral/integer coefficient 整系数

Natural number, 0, and their opposites are called integers. The set mathbb{Z} , standing for the German word Zahlen ([ˈtsaːlən], "numbers"), consists of all integers.

odd number 奇数

An odd number is an integer not divisible by 2 (leaving a remainder of 1 when divided by 2).

even number 偶数

An even number is an integer divisible by 2 (leaving no remainder when divided by 2).

4. rational number 有理数 rational adj. et n. 有理数和理性没有关系，而跟ratio有关

A rational number is a number that can be expressed as a ratio of two integers. Represented as a decimal, it is either terminating or repeating. All the rationals constitute the set mathbb{Q} , it was thus denoted in 1895 by Peano after quoziente, Italian for "quotient".

5. irrational number 无理数 irrational adj. et n.

An irrational number is a number that is real but not rational. Represented as a decimal, it is non-repeating, e.g. π, e and sqrt{2} .

6. real number 实数 real adj. et n.

Real number system mathbb{R} is the completion of rationals mathbb{Q} .

7. complex number 复数 complex adj. et n.

A complex number that has a real part x and imaginary part y is written as z=x+yi, where i=sqrt{-1} is the imaginary unit.