数学基础

一.整数：integer，whole number

　　1.因子：factor or divisor

　 　If x and y are integers and x≠0,x is a divisor (factor) of y provided that y=xn for some integer n. In this case y is also said to be divisible by x or to be a multiple of x. For example, 7 is a divisor or factor of 28 since 28=7•4, but 8 is not a divisor of 28 since there is no integer n such that 28=8n.Divisible adj.可以被整除的　　multiple　n.倍数

　　2.商和余数：quotients and remainders

　　余数和商都可以为0

　　3.奇数和偶数：odd and even integers

　　奇数和偶数都可以是负数；零一定是偶数

　　4.质数和合数：prime numbers and composite numbers

　 　A prime number is a positive integer that has exactly two different positive divisors，1 and itself. For example, 2,3,5,7,11, and 13 are prime numbers, but 15 is not, since 15 has four different positive divisors, 1, 3, 5, and 15. The number 1 is not a prime number, since it has only one positive divisor. Every integer greater than 1 is either prime or can be uniquely expressed as a product of prime factors. For example, 14= (2) (7), 81= (3) (3) (3) (3), and 484= (2) (2) (11) (11).

　　注:除了1和其本身外,还有其他因子的数叫合数。最小的质数为2，最小的合数为4，在讨论质数和合数时，都指正数。1和0既不是质数，也不是合数。

　　5.整数中的重要概念：

　　* Perfect square完全平方数，诸如9 = 32

　　* Perfect cube　完全立方数，诸如8 = 23

　　* the greatest common divisor　最大公约数

　　几个数所公有的最大因子称最大公约数，诸如：48与36的公因子有1，2，3，4，6，12，其中12为最大公约数。

　　* the least common multiple最小公倍数

　　几个数所公有的最小倍数称最小公倍数，诸如：3，7和14的最小公倍数为42。

　　*连续正整数的算术平均值也是首项和末项的算术平均值。

　　同理，连续奇数与连续偶数的算术平均值也是首项和末项的算术平均值。

　　* the properties of the number of factors因子个数的特性：

　　1)当一个正整数n有奇数个因子，则n必为一完全平方数。

　　2)除了n的平方根为其中一个因子外，小于n的平方根的因子与大于n的平方根的因子数相同。

　　3)当某一正整数n有偶数因子时，则n必不是完全平方数，且大于n的平方根的因子与小于其的因子数相同。

　　*因子数的求解公式：将整数n分解为质因子相乘的形式，然后将每个质因子的幂分别加1之后连乘所得的结果就是n的因子的个数。

　　例：80的因子个数可以如下方式求得：80 = 2 4•5，则因子个数为(4+1)(1+1)= 10

　　　　　

　　*整除特性：

　　能够被2整除的数其个位一定是偶数。

　　能够被3整除的数是各位数的和能够被3整除。

　　能够被4整除的数是最后两位数能够被4整除。

　　能够被5整除的数的个位是0或5。

　　能够被8整除的数是最后三位能够被8整除。

　　能够被9整除的数是各位数的和能够被9整除。

　　能够被11整除的数是其奇数位的和减去偶数位的和的差值可以被11整除。

　　记住：一个数要想被另一个数整除，该数需含有对方所具有的质数因子。

　　　　　

　　*整数n次幂尾数特性：

　　尾数为2的数的幂的个位数一定以2，4，8，6循环

　　尾数为3的数的幂的个位数一定以3，9，7，1循环

　　尾数为4的数的幂的个位数一定以4，6循环

　　尾数为7的数的幂的个位数一定以7，9，3，1循环

　　尾数为8的数的幂的个位数一定以8，4，2，6循环

　　尾数为9的数的幂的个位数一定以9，1循环

　　例：7123　和3 (321)的个位哪个大？

　　7和3幂的个位数均每4次循环一次，则将7123的幂指数123÷4余3，因此7123的个位数一定为3，同理将3 (321)的幂指数321÷4余1，则3 (321)的个位为3，则与7123的3 (321)个位数相同。

二.分数：fractions

　　分子：numerator

　　分母：denominator

　　分数的加减乘除：addition，subtraction，multiplication and division of fractions

　　繁分数和假分数：mixed number and improper fraction

　　繁分数是指一个数由一个整数和一个分数构成。

　　假分数是指分子大于分母的分数。例如：7/3

三.小数：decimals

　　科学计数法：scientific notation

　 　Sometimes decimals are expressed as the product of a number with only one digit to the left of the decimal point and a power of 10. This is called scientific notation. For example, 231 can be written as 2.31×102 and 0.0231 can be written as 2.31×10-2. When a number is expressed in scientific notation, the exponent of the

　　　10 indicates the number of places that the decimal point is to be moved in the number that is to be multiplied by a power of 10 in order to obtain the product. The decimal point is moved to the right if the exponent is positive and to the left if the exponent is negative. For example, 20.13×103 is equal to 20,130 and 1.91×10-4 is equal to 0.000191.

　　四舍五入：to the nearest

　　小数点：decimal point，period

四.实数：real numbers

　　正数和负数：positive and negative numbers

　　绝对值：absolute value

五.比率与比例：ratio and proportion

　　一个比率ratio可以表示成许多方式，例如：the ratio of 2 to 3可以被表达为2 to 3，2：3，或者2/3。注意比率中的中项的顺序是重要的，即2 to 3和3 to 2不同。A proportion is　a

　　statement that two ratios are equal。例如：2/3=8/12是一个proportion。

六.百分比：percent

　　Percent means per hundred or number out of 100。

　　在考题中经常会问到从某一数量到另一数量百分比的增加或减少。首先算出增加或减少的量，然后除以原来的那个量，即“from”或“than”后面的量。

七.数的幂和根：powers and roots of numbers

　　x n意味着the nth power of x。例如：64 is the 6th power of 2。2 is a 6th root of 64。

　　立方根是指cube root。

八.描述统计(descriptive statistics)

　　1.平均数(average or arithmetic mean)

　　2.中数(median)

　 　To calculate the median of n numbers，first order the numbers from least to greatest；if n is odd，the median is defined as the middle number，while if n is even，the median is defined as the average of the two middle numbers. For the data 6, 4, 7, 10, 4, the numbers, in order, are 4, 4, 6, 7, 10, and the median is 6, the middle number. For the numbers 4, 6, 6, 8, 9, 12, the median is (6+8 )/2 = 7. Note that the mean of these numbers is 7.5.

　　3.众数(mode)：一组数中的众数是指出现频率最高的数。

　　例：the mode of 7，9，6，7，2，1 is 7。

　　4.值域(range)：表明数的分布的量，其被定义为最大值减最小值的差。

　　例：the range of–1，7，27，27，36 is 36-(-1)= 37。

　　5.标准方差(standard deviation)：

　 　One of the most common measures of dispersion is the standard deviation. Generally speaking, the greater the data are spread away from the mean, the greater the standard deviation. The standard deviation of n numbers can be calculated as follows:

　　(1)find the arithmetic mean ;

　　(2)find the differences between the mean and each of the n numbers ;

　　(3)square each of the differences ;

　　(4)find the average of the squared differences ;

　　(5)take the nonnegative square root of this average.

　 　Notice that the standard deviation depends on every data value, although it depends most on values that are farthest from the mean. This is why a distribution with data grouped closely around the mean will have a smaller standard deviation than data spread far from the mean.

　　6.排列与组合

　　There are some useful methods for counting objects and sets of objects without actually listing the elements to be counted. The following principle of Multiplication is fundamental to these methods.

　　If a first object may be chosen in m ways and a second object may be chosen in n ways, then there are mn ways of choosing both objects.

　　As an example, suppose the objects are items on a menu. If a meal consists of one entree and one dessert and there are 5 entrees and 3 desserts on the menu, then 5×3 = 15 different meals can be ordered from the menu. As another example, each time a coin is flipped, there are two possible outcomes, heads and tails. If an experiment consists of 8 consecutive coin flips, the experiment has 28 possible outcomes, where each of these outcomes is a list of heads and tails in some order.

　　☆阶乘：factorial notation

　　假如一个大于1的整数n，计算n的阶乘被表示为n!，被定义为从1至n所有整数的乘积，

　　例如：4! = 4×3×2×1= 24

　　注意：0! = 1! = 1

　　☆排列：permutations

　 　The factorial is useful for counting the number of ways that a set of objects can be ordered. If a set of n objects is to be ordered from 1st to nth, there are n choices for the 1st object, n-1 choices for the 2nd object, n-2 choices for the 3rd object, and so on, until there is only 1 choice for the nth object. Thus, by the multiplication principle, the number of ways of ordering the n objects is

　　　　　　　　　　n (n-1) (n-2)…( 3) (2) (1) = n!

　　For example, the number of ways of ordering the letters A, B, and C is 3!, or 6：ABC, ACB, BAC, BCA, CAB, and CBA.

　　These orderings are called the permutations of the letters A, B, and C.也可以用P 33表示.

　　　　　　　　　　　　　　　　　　　　　　　　　　Pkn = n!/ (n-k)!

　　例如：1, 2, 3, 4, 5这5个数字构成不同的5位数的总数为5! = 120

　　　　　　

　　☆组合：combination

　 　A permutation can be thought of as a selection process in which objects are selected one by one in a certain order. If the order of selection is not relevant and only k objects are to be selected from a larger set of n objects, a different counting method is employed.

　 　Specially consider a set of n objects from which a complete selection of k objects is to be made without regard to order, where 0≤k≤n . Then the number of possible complete selections of k objects is called the number of combinations of n objects taken k at a time and is Ckn.

　　从n个元素中任选k个元素的数目为:

　　　　　　　　　　　　　　　　　　　Ckn. = n!/ (n-k)! k!

　　例如：从5个不同元素中任选2个的组合为C25 = 5!/2! 3!= 10

　　排列组合的一些特性(properties of permutation and combination)

　　☆加法原则：Rule of Addition

　 　做某件事有x种方法，每种方法中又有各种不同的解决方法。例如第一种方法中有y1种方法，第二种方法有y2种方法，等等，第x种方法中又有yx种不同的 方法，每一种均可完成这件事，即它们之间的关系用“or”表达，那么一般使用加法原则，即有：y1+ y2+。。。+ yx种方法。

　　　　　

　　☆乘法原则：Rule of Multiplication

　　完成一件事有x个步骤，第一步有y1种方法，第二步有y2种方法，。。。，第x步有yx种方法，完成这件事一共有y1• y2•。。。•yx种方法。