Chapter 1 Numbers and Sets (数域与集合)

学堂在线：微积分-1 @陈酌

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微积分基本内容: 极限论, 函数微分、积分理论, 级数, 场论, 常微分方程 ...

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e.g.和i.e.是英语中两个容易混淆的拉丁语缩写:

• 前者是exempli gratia的缩写，意思是“for example” (举例)
• 后者是id est的缩写，意思是“that is” (即)

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Unit 1 Review of Real Numbers (回顾实数)

§1 Real Numbers (实数体系)

术语：

• R: real numbers (实数);
• Q: rational numbers (有理数),
  numbers that can be expressed as a/b,
  (a over b, called the quotient (商) of a and b),
  where a and b are integers and b ≠ 0,
• e.g. 2/3 (two-thirds), -5/4 (minus five over four, minus five fourths);
• Z: integers (整数),
• e.g. -3, -2, -1, 0, 1, 2, ...;
• Z+, +右上标: nonnegative integers (非负整数),
• e.g. 0, 1, 2, ...;
• N: natural numbers (自然数), also called positive integers (正整数),
• e.g. 1, 2, 3, 4, ...;
• R\Q=Qc, c右上标: Irrational numbers (无理数),
  numbers that cannot be expressed as a/b,
  where a and b are integers and b ≠ 0,
• e.g. 根号2 (square root of two), pi.
  [注] Q 以外的实数，用 R backslash Q 表示 Q 的余集

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§2 The R2 Plane (R2平面)

R2, 2右上标.

术语：

• x-axis (x轴): horizontal axis;
• y-axis (y 轴): vertical axis;
• (x, y): the coordinate (坐标) of a point;
• Origin (原点): the point of intersection of two axes, resented by (0, 0).

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§3 Operations With Real Numbers (实数的基本运算)

If a, b, c belong to R, we have:

(a plus parentheses b plus c equals parentheses a plus b plus c)

这个叫做 单位律. ↑

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§4 Inequalities (不等式)

描述两个数大小的时候, 建立不等式.

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§5 Absolute Values of Real Numbers (实数的绝对值)

• |a|: the absolute value (绝对值) of a.
• Two commonly used inequalities:

(the absolute value of a plus b is always less than or equal to the absolute value of a plus the absolute value of b)

(the absolute value of the difference of the absolute value of a and the absolute value of b is always less than or equal to the absolute value of the difference of a and b)

附录:

运算结果	运算符	运算法
Sum 和	plus +, add 加上	addition 加法
Difference 差	minus -, subtract 减去	subtraction 减法
Product 积	times *, multiplied by 乘以	multiplication 乘法
Quotient 商	over /, divided by 除以	division 除法

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Unit 1 Summary: 实数, 运算, 两个重要的不等式, 英文术语

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Unit 2 Exponentiations, Logarithms and Sets(指数、对数与集合)

§1 Exponentiations and Roots (指数与方根)