2.1 数学、方程与比例
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（1）数学来源于人类的社会实践，包括工农业的劳动，商业、军事和科学技术研究等活动。
Mathematics comes from man’s social practice, for example, industrial and agricultural production, commercial activities, military operations and scientific and technological rearches.
（2）如果没有运用数学，任何一个科学技术分支都不可能正常地发展。
No modern scientific and technological branches could be regularly developed without the application of mathematics.
（3）符号在数学中起着非常重要的作用，它常用于表示概念和命题。
Notations are a special and powerful tool of mathematics and are ud to express conceptions and propositions very often.
（4）17 世纪之前，人们局限于初等数学，即几何、三角和代数，那时只考虑常数。
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Before 17th century, man confined himlf to the elementary mathematics, i. e. , geometry, trigonometry and algebra, in which only the constants were considered.
（5）方程与算数的等式不同在于它含有可以参加运算的未知量。
Equation is different from arithmetic identity in that it contains unknown quantity which can join operations.
英语五年级上册（6）方程又称为条件等式，因为其中的未知量通常只允许取某些特定的值。
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Equipment is called an equation of condition in that it is true only for certain values of unknown quantities in it. dostyle
（7）方程很有用，可以用它来解决许多实际应用问题。
Equations are of very great u. We can u equations in many mathematical problems.
（8）解方程时要进行一系列移项和同解变形，最后求出它的根，即未知量的值。
To solve the equation means to move and change the terms about without making the equation untrue, until the root of the equation is obtained, which is the value of unknown term.
2.2 几何与三角
（1）许多专家都认为数学是学习其他科学技术的必备基础和先决条件。 as far as
Many experts recognize that mathematics is the necessary foundation and prerequisite of studying other science technology.
（2）西方国家的专家认为几何起源于巴比伦和埃及人的土地测量技术，其实中国古代的数学 家对几何做了许多出色的研究。
The western experts think that geometry had its origin in the measurements by the Babylonians and Egyptians of their lands. Infect, the ancient Chine mathematicians made much remarkable study for geometry.
（3）几何的学习使学生在思考问题时更周密和审慎，他们将不会盲目接受任何结论。
In studying geometry, the student is taught to think clearly and critically and he is led away from the practice of blind acceptance of any conclusions.
（4）数学培养学生的分析问题的能力，使他们能应用毅力、创造性和逻辑推理来解决问题。
儿童动画故事Studying mathematics can develop the students’ ability to analyze problems and utilizing perverance, originality, and logical reasoning in solving the problem.
（5）几何主要不是研究数，而是形，例如三角形，平行四边形和圆，虽然它也与数有关。
Geometry mainly studies hot numbers but figures such as triangles, parallelograms and circles, though it is related with numbers.
（6）一个立体（图形）有长、宽和高；面（曲面或平面）有长和宽，但没有厚度；线（直线 或曲线）有长度，但既没有宽度，也没有厚度；点只有位置，却没有大小。
A solid (figure) has length, width and height. A surface (curved surface or plane surface) h
as length and width, but no thickness. A line (straight line or curved line) has length, but no width and thickness. A point has position, but no dimension.
（7）射线从某个点出发无限延伸；两条从同一点出发的射线构成了角。这两条射线称为这个 角的两边，当这两边位于同一直线上且方向相反时，所得的角是平角。
惧惮A ray starts from a point and extends infinitely far. Two rays starting from one point form an angle, which are called two edges of the angle. When two edges lie in the same line and have opposite direction named plane angle.
（8）平面上的闭曲线当其中每一点到一个固定点的距离均相等时叫做圆。这个固定点称为圆 心，经过圆心且其两个端点在圆周上的线段称为这个圆的直径，直径的一半叫做半径，这条 曲线的长度叫做周长。
A circle is a clod curve lying in one plane, all points of which are equidistant from a fixed point. The fixed point called the center. A diameter of a circle is a line gment through the center of the circle with endpoints on the circle. Half of the diameter is called radius. The length of the circle is called circumference.
2.3 集合论的基本概念
（1）由小于 10 且能被 3 整除的正整数组成的集是整数集的子集。
The t consisting of tho positive integers less than 10 which are divisible by 3 is a subt of the t of all integers.
（2）如果方便，我们通过在括号中列举元素的办法来表示集。
When convenient, we shall designate ts by displaying the elements in braces.
（3）用符号 表示集的包含关系，也就是说，式子 A B 表示 A 包含于 B。
The relation is referred to as t inclusion; AB means that A is contained in B.
（4）命题 A B 并不排除 B A 的可能性。
The statement AB does not rule out the possibility that BA.
（5）基础集可根据使用场合不同而改变。
The underlying t may vary from one application to another according to using occasions.
（6）为了避免逻辑上的困难，我们必须把元素 x 与仅含有元素 x 的集{x}区别开来。
To avoid logical difficulties, we must distinguish between the element x and the t {x} who only element is x.
（7）图解法有助于将集合之间的关系形象化。
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Diagrams often help using visualize relationship between ts.
（8）定理的证明仅仅依赖于概念和已知的结论，而不依赖于图形。
The proofs of theorems rely only on the definitions of the concepts and known result, not on the diagrams.
