Numbers and numerals:A number is an idea which answers the question "how many objects in a collection?"
数字是仅表示数字的单词或符号。
Natural numbers:The number 1, 2, 3,..... used for counting are called natural numbers.
Whole number:所有计数数字与“零”一起称为整数，并且
整数：The numbers 0, 1, - 1, 2, - 2, 3, - 3, .... are called integers, numbers 1, 2, 3, ..... are called positive integers and the numbers - 1, - 2, - 3, .... are called negative integers. Zero is called zero integers and is neither positive nor negative. The set of integers is denoted by 1 or Z.
有理数：A number of the form p/q where p and q are integers and q ¹ 0 is called a rational number.
Every integer is a rational number and every fraction is a rational number.
非理性数字：不是理性的数字称为非理性数字。因此，非理性数字不能以形式写
在哪里，p。问。i和q 0。Note:非理性数字是一个非终止和不重复的十进制。
实数：The union of the set of rational numbers and the set of irrational numbers is the set of real numbers. Its set is denoted by R.
Note:
(1) Completeness properties of number-line:实际数字集与数字行上的所有点集之间有一对一的对应关系。
(2) Absolute value:Absolute or modulus value of a real number x, denoted by |x|, is defined as
|x| = { x if x>= 0 or -x if x < 0 }
(3) R的属性
Laws of Addition:
（i）a + b -r，对于所有a，b-（封闭法）
（ii）a + b = b + a，全部，a，b -r（交换定律）
(iii) a + (b + c) = (a + b) + c; for all a, b, c Î R (Associative law)
（iv）There exists an identity element o in R such that a + 0 = 0 + a = a (Existence of additive identity)
(v) For every a Î R, there exists an additive inverse in R, denoted by -a, such that a + (-a) = (-a) + a = 0.
（添加词的存在）
乘法定律：
（i）ab - r，对于所有a，b- r（封闭法）
（ii）a×b = b×a，对于所有a，b -r（交换定律）
（iii）a×（b×c）=（a×b）×c：对于所有a，b，c- r（联想定律）
（iv）There exists a unity element in R, denoted by 1 such that a × 1 = 1 × a = a (Existence of unity element)
（v）对于每个a（¹0）îr，都存在r中的乘法逆，由1/2表示，使得a*1/a*1/a = 1 = 1
(μ的存在ltiplicative inverse).
分配法律：
（一世）a × (b + c) = a × b + a × c: for all a, b c Î R
(ii) (b + c) × a = b × a + c × a; for all a, b, c Î R.
(4) Order relations in R:R中的顺序关系的使用方式与合理数字的集合Q完全相同。
如果x，yîr，则必须满足以下条件之一。
（i）x = y（ii）x> y（iii）x （ii）如果x，y，z - r，则x x x (iii) If x, y, Z Î R and x < y, then x + Z < Y + Z and conversely.
（iv）If x, y, Z Î R and y < Z, then for x > 0, xy < xz.
多：A number which is an exact time of a given number is called the multiple.
Factor:确切地将给定数字划分的数字称为给定数字的因子。
Prime number:它是一个自然数，除1外，除1本身外，没有其他因素，例如2、3、5、7、11是质数。
复合数字：It is a natural number different from 1, which has at least one factor other than 1 and itself, e.g., 4, 6, 8, 9, 10, 12 are composite numbers.
Note:
零，一个既不是素数也不是复合数。
Division Algorithm:
股息=除数×商 +其余部分。
Tests of Divisibility:
（一世）A number is divisible by 2, if its last digit is divisible by 2.
（ii）如果数字之和可除以3，则数字可以排3。
（iii）一个数字可以排成4，如果其最后两个数字表示数字可除以4。
（iv）如果以5或0结束，则数字可除以5。
（v）一个数字可除以6的数字，如果可以除以2和3。
（vi）一个数字可以排出8，如果其最后三个数字表示一个数字可除以8。
(vii) A number is divisible by 9, if the sum of digits is divisible by 9.
(viii) A number is divisible by 10, if it ends with 0.
(ix) A number is divisible by 11, if the difference between the sum of the digits in odd places and the sum of the digits in even places is either divisible by 11 or is equal to zero.
独特的分解定理：
每个大于1的自然数量都可以分解为素数，除了因素顺序外，分解是唯一的。
H.C.F.:最高的两个或两个以上的共同因素(H.C.F.)given numbers is the greatest number which divides each of given numbers exactly.
L.C.M.两个给定数字的最低常见倍数（L.C.M.）是最低的数字，这是由每个给定数字完全排除的。
Note:两个数字的产物=其H.C.F和L.C.M.的乘积
Note:
（1）整数的绝对值是整数而不考虑其SIQN，将零的绝对值视为零。
(2) The integer 'zero' is less than every positive integer and greater than every negative integer.
乍看上去：
Representation of rational numbers as decimals:我们知道，以十进制形式表达时的每个理性数字在终止或重复小数形式中都可以表达。
Representation of a terminating decimal as a rational number:规则：将1放在分母中的小数点下，并在十进制点之后的数字数量和附件附件。现在，卸下小数点，并将分数以最简单的形式放置。
重复或重复小数：In such decimals a digit or a block of digits repeats itself again and again. We represent such decimals by putting a bar on repeated digit or digits.
（i）纯正的小数：十进制重复小数点后所有数字，称为纯回形小数。
(ii) Mixed Recurring Decimals:十进制重复小数点后至少一个数字，然后重复一个图形或一组图形，称为混合重复的小数。
Note:
If the denominator of a rational number (in standard form) contains no prime factors, other than 2 and 5, then this rational number can always be expressed in terminating decimal form. Otherwise, it is expressed as a non terminating repeating decimal.
特性：
（1）添加，乘法和减法的有理数集已关闭。两个有理数的添加（或乘法）是一个合理的数字。
(2) The sets of positive and negative rational numbers are closed for division.
(3) Division of two rational numbers is a rational number.
（4）有理数的集合可以用数字行上的点表示。
理性数字的属性：" a, b, C Î Q.
（一世）" a Î Q, 0 Î Q s. t. a + 0 = 0 + a = a [0 is known as additive identity]
(ii) " a Î Q, - a Î Q such that a + (-a) = (-a) + a = 0 [Additive inverse]
(iii) " a Î Q, 1 Î Q, such that 1 a = a. 1 = a [Multiplicative identity]
（v）“ a，b，c -q，
一个。（b + c）= a。b + a。C。
Also (a + b). c = a. c + b. c [Distributive law]
Order relation in Q.
（一世）三分法法：“ a，bcîq一，只有以下一个possibilities holds:a > b, a = b or a < b.
(ii)传播法：a > b and b > C Þ a > c.
(iii)Monotone law for addition:a > b Þ a + c > b + c " a, b, c Î Q.
（iv）Monotone law for multiplication:a> b，c>0ÞAC> bc“ a，b，c -q。
Q as ordered field:The set of rational numbers is an ordered field with respect to operations of addition and multiplication.
Archimedean property:Between any two distinct rational numbers a and b, there lies an infinite number of rational numbers.
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