GMAT數學算數知識精解

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GMAT數學算數知識精解

　　下面為大家整理了GMAT數學算數知識精解，供考生們參考，以下是詳細內容。

　　一.整數：integer，whole number 因子：factor or divisor

　　If x and y are integers and x0,x is a divisor 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=74, 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.倍數

　　2.商和余數：quotients and remainders

　　余數和商都可以為0

　　3.奇數和偶數：odd and even integers

　　奇數和偶數都可以是負數;零一定是偶數

　　4.質數和合數：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= , 81= , and 484= .

　　注:除了1和其本身外,還有其他因子的數叫合數。最小的質數為2，最小的合數為4，在討論質數和合數時，都指正數。1和0既不是質數，也不是合數。

　　5.整數中的重要概念：

　　 Perfect square耆?椒絞??釗? = 32

　　 Perfect cube　完全立方數，諸如8 = 23

　　 the greatest common divisor　最大公約數

　　幾個數所公有的最大因子稱最大公約數，諸如：48與36的公因子有1，2，3，4，6，12，其中12為最大公約數。

　　 the least common multiple最小公倍數

　　幾個數所公有的最小倍數稱最小公倍數，諸如：3，7和14的最小公倍數為42。

　　連續正整數的算術平均值也是首項和末項的算術平均值。

　　同理，連續奇數與連續偶數的算術平均值也是首項和末項的算術平均值。

　　 the properties of the number of factors因子個數的特性：

　　1)當一個正整數n有奇數個因子，則n必為一完全平方數。

　　2)除了n的平方根為其中一個因子外，小于n的平方根的因子與大于n的平方根的因子數相同。

　　3)當某一正整數n有偶數因子時，則n必不是完全平方數，且大于n的平方根的因子與小于其的因子數相同。

　　因子數的求解公式：將整數n分解為質因子相乘的形式，然后將每個質因子的冪分別加1之后連乘所得的結果就是n的因子的個數。

　　例：80的因子個數可以如下方式求得：80 = 2 45，則因子個數為= 10

　　整除特性：

　　能夠被2整除的數其個位一定是偶數。

　　能夠被3整除的數是各位數的和能夠被3整除。

　　能夠被4整除的數是最后兩位數能夠被4整除。

　　能夠被5整除的數的個位是0或5。

　　能夠被8整除的數是最后三位能夠被8整除。

　　能夠被9整除的數是各位數的和能夠被9整除。

　　能夠被11整除的數是其奇數位的和減去偶數位的和的差值可以被11整除。

　　記住：一個數要想被另一個數整除，該數需含有對方所具有的質數因子。

　　整數n次冪尾數特性：

　　尾數為2的數的冪的個位數一定以2，4，8，6循環

　　尾數為3的數的冪的個位數一定以3，9，7，1循環

　　尾數為4的數的冪的個位數一定以4，6循環

　　尾數為7的數的冪的個位數一定以7，9，3，1循環

　　尾數為8的數的冪的個位數一定以8，4，2，6循環

　　尾數為9的數的冪的個位數一定以9，1循環

　　例：7123　和3 的個位哪個大?

　　7和3冪的個位數均每4次循環一次，則將7123的冪指數1234余3，因此7123的個位數一定為3，同理將3 的冪指數3214余1，則3 的個位為3，則與7123的3 個位數相同。

　　二.分數：fractions

　　分子：numerator

　　分母：denominator

　　分數的加減乘除：addition，subtraction，multiplication and division of fractions

　　繁分數和假分數：mixed number and improper fraction

　　繁分數是指一個數由一個整數和一個分數構成。

　　假分數是指分子大于分母的分數。例如：7/3

　　三.小數：decimals

　　科學計數法：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 2.31102 and 0.0231 can be written as 2.3110-2. 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, 20.13103 is equal to 20,130 and 1.9110-4 is equal to 0.000191.

　　四舍五入：to the nearest

　　小數點：decimal point琾eriod

　　四.實數：real numbers

　　正數和負數：positive and negative numbers

　　絕對值：absolute value

　　五.比率與比例：ratio and proportion

　　一個比率ratio可以表示成許多方式，例如：the ratio of 2 to 3可以被表達為2 to 3，2：3，或者2/3。注意比率中的中項的順序是重要的，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是一個proportion。

　　六.百分比：percent

　　Percent means per hundred or number out of 100。

　　在考題中經常會問到從某一數量到另一數量百分比的增加或減少。首先算出增加或減少的量，然后除以原來的那個量，即from或than后面的量。

　　七.數的冪和根：powers and roots of numbers

　　x n意味著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.描述統計平均數

　　2.中數

　　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 /2 = 7. Note that the mean of these numbers is 7.5.

　　3.眾數：一組數中的眾數是指出現頻率最高的數。

　　例：the mode of 7，9，6，7，2，1 is 7。

　　4.值域：表明數的分布的量，其被定義為最大值減最小值的差。

　　例：the range of1，7，27，27，36 is 36-= 37。

　　5.標準方差：

　　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:

　　find the arithmetic mean ;

　　find the differences between the mean and each of the n numbers ;

　　square each of the differences ;

　　find the average of the squared differences ;

　　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.排列與組合

　　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 53 = 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.

　　階乘：factorial notation

　　假如一個大于1的整數n，計算n的階乘被表示為n!，被定義為從1至n所有整數的乘積，

　　例如：4! = 4321= 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!

　　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!/ !

　　例如：1, 2, 3, 4, 5這5個數字構成不同的5位數的總數為5! = 120

　　組合：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 0n . 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.

　　從n個元素中任選k個元素的數目為:

　　Ckn. = n!/ ! k!

　　例如：從5個不同元素中任選2個的組合為C25 = 5!/2! 3!= 10

　　排列組合的一些特性

　　加法原則：Rule of Addition

　　做某件事有x種方法，每種方法中又有各種不同的解決方法。例如第一種方法中有y1種方法，第二種方法有y2種方法，等等，第x種方法中又有yx種不同的方法，每一種均可完成這件事，即它們之間的關系用or表達，那么一般使用加法原則，即有：y1+ y2+。。。+ yx種方法。

　　乘法原則：Rule of Multiplication

　　完成一件事有x個步驟，第一步有y1種方法，第二步有y2種方法，。。。，第x步有yx種方法，完成這件事一共有y1 y2。。。yx種方法。

　　以上只是GMAT考題中經常涉及到的數學算術方面的問題，今后我們將陸續在新開辟的網上課堂中介紹代數、幾何以及系統的習題、講解，以幫助大家在GMAT數學考試中更好地發揮中國學生的優勢，拿到讓美國人瞠目結舌的成績!

　　以上就是GMAT數學算數知識精解的詳細內容，考生可針對文中介紹的方法進行有針對性的備考。最后預祝大家在GMAT考試中取得好成績！