数学笔记.doc

数学

数值比较

比较的时候可以带数值，但may not be conclusive. 只有带出的值得出不一样的结论才可以确定选D (cannot be determined)。

表达
There is twice as much A as B = A is twice as much as B
A = 2B
The statement says that there is more A than B. Twice as much more. This means that more than 1 B will be needed to make 1 A. 
If there is twice as much A as B, then you need 2 B's to have the same amount of As.

There is half as much C as A
C = 0.5 A

硬币的说法
1 = penny
5 = nickel
10 = dime
25 = quarter
50 = half dollar

Integer 整数
Unit digit = one’s digit 个位数
Ten’s digit 十位数
Tenth’s digit 十分位
Perfect square 完全平方数
Perfect cube 完全立方数
Improper Fractions 假分数（大于或等于1的分数）
Simple Fraction 约分数
Rational Number 有理数
Irrational Number 无理数 （无限不循环小数）
Geometric Mean 为n个数的乘积开n次方

Mean
Median
Mode
三个的区别要看清啊

Range最大值与最小值的差

Factorial notation 阶乘 n!
Permutation 排列
Recombination 组合

Hypotenuse 直角三角形的斜边
Isosceles 等腰

任何一个大于2的偶数都可以表示为两个质数的和
奇数+奇数=偶数
如果a为质数，n为非负整数，则a的n次方的因数为n+1个，包括1和a的n次方本身
Least common composite 把两个数的因数写出来，把所有的因数相乘（如果有相同的就取幂比较高的那个。不用重复）
Largest common factor / divisible 把两个数的因数写出来，把相同的因数相乘（如果有相同的就取幂比较小的那个。）

1 mile = 5280 feet
1 pound (lbs) = 16 ounce (oz)

If r, s, and t are integers and rs=t, then r and s are factors, or divisors, of t; also,
t is a multiple of r (and of s) and t is divisible by r (and by s). The factors of an
integer include positive and negative integers. For example, –7 is a factor of
35, 8 is a factor of –40, and the integer 4 has six factors: –4, –2, –1, 1, 2, and 4.
The terms factor, divisor, and divisible are used only when r, s, and t are integers.
However, the term multiple can be used with any real numbers s and t provided r
is an integer. For example, 1.2 is a multiple of 0.4, and –2p is a multiple of p.

When an integer n is divided by a nonzero integer d resulting in a quotient q with
remainder r, then n=qd+r, where 0ent is 3 and the remainder is
0; and when –17 is divided by 7, the quotient is –3 and the remainder is 4.

A prime number is an integer greater than 1 that has only two positive divisors: 1
and itself. The first five prime numbers are 2, 3, 5, 7, and 11. A composite number
is an integer greater than 1 that is not a prime number. The first five composite
numbers are 4, 6, 8, 9, and 10.

Odd and even integers are not necessarily positive; for example, –7 is odd, and
–18 and 0 are even.

The order is as follows: parentheses; exponentiation; negation; multiplication and
division (from left to right); addition and subtraction (from left to right).

0!=1.

几何
Lines are assumed to be “straight” lines that extend in both directions without
end.

Geometric figures are not necessarily drawn to scale. That is, you should not
assume that quantities such as lengths and angle measures are as they appear in a
figure.

intersection of A and B 交集
union of A and B 并集
If all of the elements in A are also in B, then A is a subset of B.
By convention, the empty set is a subset of every set.
If A and B have no elements in common, they are called disjoint sets or mutually exclusive sets.

The terms data set and set of data are not sets in the mathematical sense given
above. Rather they refer to a list of data because there may be repetitions in the
data, and if there are repetitions, they would be relevant.

Sequences 数列 are lists that often have an infinite number of elements, or terms.

When mean is used in the context of data, it means arithmetic mean.

The median of an odd number of data is the middle number when the data are
listed in increasing order; the median of an even number of data is the arithmetic
mean of the two middle numbers when the data are listed in increasing order.

For a list of data, the mode of the data is the most frequently occurring number in
the list. Thus, there may be more than one mode 众数 for a list of data.

For data listed in increasing order, the first quartile, second quartile, and third
quartile of the data are three numbers that divide the data into four groups that
are roughly equal in size. The first group of numbers is from the least number up
to the first quartile. The second group is from the first quartile up to the second
quartile, which is also the median of the data. The third group is from the second
quartile up to the third quartile, and the fourth group is from the third quartile up
to the greatest number. Note that the four groups themselves are sometimes
referred to as quartiles—first quartile, second quartile, third quartile, and fourth
quartile. The latter usage is clarified by the word “in” as in the phrase “the cow’s
weight is in the third quartile of the herd.”

The 25th percentile equals the first quartile; the 50th percentile equals the second quartile, or median;
and the 75th percentile equals the third quartile.

方差 For a list of data, where the arithmetic mean is denoted by m, the standard
deviation of the data refers to the nonnegative square root of the mean of the
squared differences between m and each of the data. This statistic is also known
as the population standard deviation and is not to be confused with the sample
standard deviation. (sample standard deviation的最后一步-开方放前那个算差平方的平均数的时候-平均用的分母是n-1而不是n。)

For a list of data, the range of the data is the greatest number in the list minus the
least number. The interquartile range of the data is the third quartile minus the
first quartile.

Some questions display data in frequency distributions, where discrete data values
are repeated with various frequencies, or where preestablished intervals of
possible values are assigned frequencies corresponding to the numbers of data in
the intervals. For example, the lifetimes, rounded to the nearest hour, of 300
lightbulbs could be in the following 10 intervals: 501–550 hours, 551–600 hours,
601–650 hours, . . . , 951–1,000 hours; consequently, each of the intervals would
have a number, or frequency, of lifetimes, and the sum of the 10 frequencies is
300.

Questions may involve relative frequency distributions, where each frequency of a
frequency distribution is divided by the total number of data in the distribution,
resulting in a relative frequency. In the example above, the 10 frequencies of the
10 intervals would each be divided by 300, yielding 10 relative frequencies.

If E and F are two events in an experiment, then “E and F ” is an event, which is
the set of outcomes that are in the intersection of events E and F. Another event is
“E or F, ” which is the set of outcomes that are in the union of events E and F.

If E and F are two events and E and F are mutually exclusive, then P(E and F)=0.

If E and F are two events such that the occurrence of either event does not affect
the occurrence of the other, then E and F are said to be independent events. Events
E and F are independent if and only if P(E and F)=P(E)P(F).

Every value of a discrete random variable X, say X=a, has a probability denoted
by P(a). A histogram (or a table) showing all of the values of X and their
probabilities P(X) is called the probability distribution of X. The mean of the
random variable X is the sum of the products XP(X) for all values of X.

The mean of a random variable X is also called the expected value of X or the mean
of the probability distribution of X.

The mean of a continuous random variable X is the point m on the X-axis at which
region under the distribution curve would perfectly balance if a fulcrum were
placed at X_m. The median of X is the point M on the X-axis at which the line
X_M divides the region under the distribution curve into two regions of equal
area.

几何题的图不TO SCALE，但是数据题的图TO SCALE
Graphical data presentations, such as bar graphs, circles graphs, and line graphs,
are drawn to scale; therefore, you can read, estimate, or compare data values by
sight or by measurement.

When a multiple-choice question asks for an approximate quantity without
stipulating a degree of approximation, the correct answer is the choice that is
closest in value to the quantity that can be computed from the information given.

In questions involving real-life scenarios in which a variable is given to represent
a number of existing objects or another nonnegative amount, the context implies
that the variable is greater than 0. For example, “Jane sold x rugs and deposited
her profit of y dollars into her savings account” implies that x and y are greater
than 0.

1 dollar = 100 cents

In any question, there may be some information that is not needed for obtaining
the correct answer.

In many questions, mathematical expressions and words appear together in a
phrase. In such a phrase, each mathematical expression should be interpreted
separately from the words before it is interpreted along with the words. For
example, if n is an integer, then the phrase “the sum of the first two consecutive
integers greater than n_6” means (n_7)_(n_8); it does not mean “the sum
of the first two consecutive integers greater than n” plus 6, or
(n_1)_(n_2)_6. That is, the expression n_6 should be interpreted first,
separately from the words. However, in a phrase like “the function g is defined for
all x_0,” the phrase “for all x_0” is a mathematical shorthand for “for all
numbers x such that x_0.”