1.Real Numbers 实数

1.1 Introduction 介绍

Real Number

n./ˈrɪəl ˈnʌmbə(r)/

Definition: A number that can be found on the number line. 实数是存在于数轴上的数。

Natations: The set of real number is denoted by (𝙍). 实数集用(𝙍)表示。

Properties:

Real Numbers Defined by Axioms

  1. Field Axioms

The real numbers form a field, satisfying algebraic properties:

  • Commutativity: a + b = b + a , a ⋅ b = b ⋅ a .
    Example: 2 + 3 = 3 + 2 , 4 ⋅ 5 = 5 ⋅ 4 .
  • Associativity: (a + b) + c = a + (b + c) , (a ⋅ b) ⋅ c = a ⋅ (b ⋅ c).
    Example: (1 + 2) + 3 = 1 + (2 + 3) .
  • Distributivity: a ⋅ (b + c) = a ⋅ b + a ⋅ c .
    Example: 3 ⋅ (4 + 5) = 3 ⋅ 4 + 3 ⋅ 5 .
  • Identity Elements:
    • Additive: a + 0 = a .
    • Multiplicative: a ⋅ 1 = a .
  • Inverses:
    • Additive: For every a , ∃ -a such that a + (-a) = 0 。
    • Multiplicative: For a≠0, ∃a−1∃${a}^{-1}$ such that a⋅${a}^{-1}$=1.
  1. 域公理

实数构成一个域，满足代数性质：

  • 交换律：( a + b = b + a ), ( a ⋅ b = b ⋅ a ).
    例：( 2 + 3 = 3 + 2 ), ( 4 ⋅ 5 = 5 ⋅ 4 )。
  • 结合律： (a + b) + c = a + (b + c) , (a ⋅ b) ⋅ c = a ⋅ (b ⋅ c)。
    例： (1 + 2) + 3 = 1 + (2 + 3)
  • 分配律：a ⋅ (b + c) = a ⋅ b + a ⋅ c。
    例：3 ⋅ (4 + 5) = 3 ⋅ 4 + 3 ⋅ 5。
  • 单位元：
    • 加法单位元：a + 0 = a 。
    • 乘法单位元： a ⋅ 1 = a。
  • 逆元：
    • 加法逆元：对任意 ( a )，存在 ( -a ) 使得 a + (-a) = 0 。
    • 乘法逆元：对任意非零 a ，存在${a}^{-1}$ 使得 a ⋅ ${a}^{-1}$=1。
  1. Order Axioms

𝙍 is a totally ordered field with:

  • Transitivity: If a < b and b < c , then a < c .
    Example: 1 < 2 < 3 ⟹ 1 < 3 .
  • Trichotomy: For any a, b , exactly one holds: a < b , a = b , or a > b .
  • Compatibility with Operations:
    • If a < b , then a + c < b + c .
    • If a < b and c > 0 , then a ⋅ c < b⋅ c .
      Example: 2 < 5 ⟹ 2 + 3 < 5 + 3 , 2 ⋅ 4 < 5 ⋅ 4 .
  1. 序公理

𝙍是一个全序域，满足：

  • 传递性：若 a < b 且 b < c ，则 a < c 。
    例： 1 < 2 < 3 ⟹ 1 < 3 。
  • 三分性：对任意 a, b ，有且仅有一种情况成立： a < b ， a = b ，或 a > b 。
  • 与运算兼容：
    • 若 a < b ，则 a + c < b + c 。
    • 若 a < b 且 c > 0 ，则 a ⋅ c < b ⋅ c 。
      例： 2 < 5 ⟹ 2 + 3 < 5 + 3 ， 2 ⋅ 4 < 5 ⋅ 4 。
  1. Completeness Axiom

The key property distinguishing 𝑹 from 𝑸:

  • Every non-empty subset of ℝ that is bounded above has a least upper bound (supremum) in ℝ.
    Example:
    Let S = { x ∈ 𝑸 | ${x}^{2}$ < 2 } . In 𝑸, 𝑺 has no supremum (since $\sqrt{x}$ $\notin$ 𝑸 ), but in 𝙍, $\sup$(S) = $\sqrt{2}$ .
  1. 完备性公理

𝑹区别于𝑸的关键性质：

  • 每个非空且有上界的𝑹子集在𝑹中都有最小上界（上确界）。
    例：
    设集合 S = { x ∈ 𝑸 | ${x}^{2}$ < 2 }。在𝑸中， S 无上确界（因 $\sqrt{x}$ $\notin$ 𝑸），但在𝑹中，$\sup$(S) = $\sqrt{2}$ 。

1.1.1 Positive,Negative,Zero 正数，负数，零

Positive 正(数)

adj./ ˈpɒzətɪv /

Definition: Greater than zero. 正数是大于0的数。

Notation: The set of positive numbers is denoted by ${R}^+$. 正数集用${R}^+$表示。

Properties:

  1. Positive numbers are always at the right side of 0 on the number line. 正数在数轴上0的右边。

  2. The union of the set of positive numbers and zero is also known as the set of nonnegative numbers. 正数集和零的并集是非负集。

Negative 负数

adj./ ˈneɡətɪv /

Definition: Less than zero. 负数是小于0的数。

Notation: The set of negative numbers is denoted by ${R}^-$. 负数集用${R}^-$表示。

Properties:

  1. Negative numbers are always at the left side of 0 on the number line. 负数在数轴上0的左边。

  2. The union of the set of negative numbers and zero is also known as the set of positive numbers. 负数集和零的并集是非正数集。

Zero 零
number / ˈzɪərəʊ /

Definition: Representation of a ’nothing-quantity’,also is a placeholder for rounding numbers. 零代表一个‘什么都没有’的数量，也是四舍五入后的占位符。

Notation: 0

Properties:

  1. The number 0 is neither positive nor negative. 零既不是正数，也不是负数。

  2. The product of 0 and any number is 0: a $\cdot$ 0 = 0,任何数和0的积均为0。

  3. The quotient of 0 and any nonzero number is 0: 0/a = 0,任何非零数和0的商均为0。