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SAT 数学考试：SAT Numbers
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SATII 考试有别于 SATI，
它是一种专项考试，
考察学生某一科目的能力，
故又被称为 SAT SUBJECT TESTS。
目前 SATII 分为数学、科学、语言、英语、历史与社会科学五大类。其中数学类有数学一和数学二两门，
科学类有物理、化学、生物三门，语言类有法语、德语、西班牙语、中文、日语、韩语、希伯莱语、拉丁
语等十二门，英语类只有英语文学一门;历史与社会科学类有美国历史和世界历史两门。以上加起来共计二
十门考试。 以下就数学部分做专项讲解分析。
Before you take the Math IC, you should know the common types of numbers. Of these types,
the most important ones to understand are probably integers and real numbers. They can be spotted
in nearly every question on the test and will be explicitly mentioned at times.
Whole Numbers. The set of counting numbers, including zero {0, 1, 2, 3, . . .}.
Natural Numbers. The set of all whole numbers except zero {1, 2, 3, 4, 5, . . .}.
Integers. The set of all positive and negative whole numbers, including zero. Fractions
and decimals are not included {. . . , –3, –2, –1, 0, 1, 2, 3, . . .}.
Rational Numbers. The set of all numbers that can be expressed as a quotient of integers.
That is, any number that can be expressed in the form m?n , where m and n are integers. The
set of rational numbers includes all integers and all fractions that can be created using integers
in the numerator and denominator.
Irrational Numbers. The set of all numbers that cannot be expressed as a quotient of integers.
Examples include π, , 1.01001000100001000001 . . . . The sets of irrational numbers and rational
numbers are mutually exclusive. Any given number must be either rational or irrational; no number
can be both.
Real Numbers. Every number on the number line. The set of real numbers includes all rational
and irrational numbers.
Imaginary Numbers. See the “Miscellaneous Math” chapter later in this book.
On the Math IC, integers and real numbers will appear far more often than any of the other
number types.
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Even and Odd Numbers
Even numbers are those numbers that are divisible by two with no remainder.
Only integers can be even or odd, meaning decimals and fractions are not included. Zero,
however, is an integer and thus a member of the set.
. . . , –6, –4, –2, 0, 2, 4, 6, . . .
Odd numbers are those numbers not evenly divisible by two.
. . . , –5, –3, –1, 1, 3, 5, . . .
The set of even numbers and the set of odd numbers are mutually exclusive.
A more rigorous definition of even and odd numbers appears below:
Even numbers are numbers that can be written in the form 2n, where n is an integer. Odd
numbers are the numbers that can be written in the form 2n + 1, where n is an integer.
This definition is nothing more than a technical repetition of the fact that even numbers
are divisible by two, and odd numbers are not. It may come in handy, though, when you need to
represent an even or odd number with a variable.
Operations of Odd and Even Numbers
There are a few basic rules regarding the operations of odd and even numbers that you should
know well. If you grasp the principles behind the two types of signed numbers, these rules should
all come easily.
Addition:
even + even = even
odd + odd = even
even + odd = odd
Subtraction:
even – even = even
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odd – odd = even
even – odd = odd
Multiplication and Division:
eveneven = even
oddodd = odd
evenodd = even
Positive and Negative Numbers
Positive and negative numbers are governed by rules similar to those that have to do with
even and odd numbers. First, for their quick definitions:
Positive numbers are numbers that are greater than zero. Negative numbers are numbers that
are less than zero. The number zero is neither positive nor negative.
Operations of Positive and Negative Numbers
The following rules define how positive and negative numbers operate under various operations.
Addition and Subtraction:
When adding and subtracting negative numbers, it helps to remember the following:
Adding a negative number is the same as subtracting its opposite. For example:
Subtracting a negative number is the same as adding its opposite. For example:
Multiplication:
positivepositive = positive
negativenegative = positive
positivenegative = negative
Division:
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positivepositive = positive
negativenegative = positive
positivenegative = negative
The rules for multiplication and division are exactly the same since any division operation
can be written as a form of multiplication: ab = a/b = a 1/b.
Absolute Value
The absolute value of a number is the distance on a number line between that number and
zero. Or, you could think of it as the positive “version” of every number. The absolute value
of a positive number is that same number, and the absolute value of a negative number is the
opposite of that number.
The absolute value of x is symbolized by |x|.
Solving an equation with an absolute value in it can be particularly tricky. As you will
see, the answer is often ambiguous. Take a look at the following equation:
We can simplify the equation in order to isolate |x|:
Knowing that |x| = 2 means that x = 2 and x = –2 are both possible solutions to the problem.
Keep this in mind; we’ll deal more with absolute values in equations later on in the Algebra
chapter.
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