In this post, we will study several important examples of set. The elements of these sets are all numbers. Notations are:

$\mathbb{N}$: Natural numbers.

$\mathbb{Z}$: Integers.

$\mathbb{Q}$: Rational numbers.

$\mathbb{R}$: Real numbers.

$\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}$.

Also, assume the addition (加法) and multiplication (乘法) of integers, i.e., $1+1=2, 3-5=-2, 4 \cdot (-2)=-8$. Now we can start to develop these number systems (sets).

1. Natural numbers

$\mathbb{N}$ is another basic concept for which we don’t have a mathematical definition. People understand $\mathbb{N}$ as ‘courting number’ (计数). For example, there are 6 apples on the table; This is my 3rd year here. Informally, $\mathbb{N}=\{1,2,3,4, \ldots \}$. Another traditional definition includes $0$ into $\mathbb{N}$, namely $\mathbb{N}=\{0,1,2,3,\ldots\}$. There is not much difference between these two in application.

There are a lot of ‘good’ properties of $\mathbb{N}$, most of which are quite obvious. But it is still worth bringing them up so that we could use them comfortably in following text.

1. We can define some operations (运算) on $\mathbb{N}$, such as addition ($+$) and multiplication ($\times\ \text{or}\ \cdot$). That satisfies commutativity (交换律), associativity (结合律), and distributivity (分配率). We can also define substraction (减法) and division (除法), but these two operation are not closed (封闭的) in ﻿﻿﻿﻿﻿﻿﻿﻿﻿$\mathbb{N}$. For example, $1 \in \mathbb{N}, 2 \in \mathbb{N}$, but $1-2=-1 \notin \mathbb{N}, 1 \div 2=0.5 \notin \mathbb{N}$.
2. There is a zero element, $0$, such that $\forall\ n \in \mathbb{N}, 0+n=n$. The zero element is unique (唯一的).
3. There is an identity element, $1$, such that $\forall\ n \in \mathbb{N}, 1 \cdot n=n$. The identity element is unique.
4. Natural numbers are comparable (可比的), i.e., $\forall\ m, n \in \mathbb{N}$, we have either $m or $m>n$.
5. $\forall\ n \in \mathbb{N}$, we call $n+1$ and $n-1$ the successor (后继) and predecessor (前继) of $n$, respectively. Apparently, any element in $\mathbb{N}$ has a successor, and any element, except $0$, has a predecessor.
6. If a set $A$ satisfies: i)$0 \in A$, ii) $n \in A \Rightarrow n+1 \in A$, then $A=\mathbb{N}$.

Remark 1: In mathematical logic (数理逻辑), the Dedekind-Peano Axioms are used to present $\mathbb{N}$. Notice that the set satisfying these axioms (功力) is not unique (think about one example). However, if there is a set which satisfies the Dedekind-Peano Axioms besides $\mathbb{N}$, we, algebraically, view this set the ‘same’ as $\mathbb{N}$, which means there is an isomorphism (同构) between this set and $\mathbb{N}$. (We will explain more on this in the chapter of Abstract Algebra (抽象代数).)

Remark 2: The 6th properties is also call the Mathematical Induction (数学归纳法). We can prove a statement is true for all ‘steps’, if we can prove it is true for the first step, and if any step later than the first step is true, then the next step is also true.

A useful property is as follows:

 Theorem 1: $\forall\ a, b \in \mathbb{N}$ such that $b=0$, $\exists\ q, r \in \mathbb{N}$ with  $a=q \cdot b +r$. Additionally, if $r, then $q\ \text{and}\ r$ are uniquely determined.

2. Integers

As one can see, $\mathbb{N}$ is not closed under the subtraction since the ‘negative’ number may come up. Therefore, we need to extend it to make subtraction a reasonable operation. Define $-\mathbb{N}=\{(-1) \cdot n|n \in \mathbb{N}\}$, then $\mathbb{Z}=\mathbb{N} \cup -\mathbb{N}$.

$\mathbb{Z}$ shares properties 1~5 from $\mathbb{N}$ except $0$ also has an predecessor  in $\mathbb{Z}$. Here we don’t need to add substraction, since $-1 \in \mathbb{Z}$. Therefore, $m-n=m+(-1) \cdot n$. Several more properties are as follows:

1. $\forall\ n \in \mathbb{Z}$, there is an inverse element $-n$ defined as $(-n)+n=0$.
2. Define $\mathbb{Z^+}=\mathbb{N}-\{0\}, \mathbb{Z^-}=-\mathbb{N}-\{0\}$. $\forall\ n \in \mathbb{Z}$, either $n \in \mathbb{Z^+}, n \in \mathbb{Z^-}$, or $n=0$.

3. Rational Numbers

$\mathbb{Z}$ is perfect for addition, subtraction, and multiplication. However, as math evolved, division (除法) started to become important. Assuming the division, then $\mathbb{Q}=\left\{\dfrac{p}{q} | p, q \in \mathbb{Z}, q \neq 0\right\}$. Especially, $\dfrac{p}{1}=p$. Sometimes we write $\dfrac{p}{q}$ as $p/q$.

• Addition: $\dfrac{a}{b}+\dfrac{c}{d}=\dfrac{ad+bc}{bd}$.
• Multiplication: $\dfrac{a}{b} \cdot \dfrac{c}{d}=dfrac{ac}{bd}$.
• Additive inverse (加法逆元): $-\left(\dfrac{p}{q} \right)=\dfrac{-p}{q}=\dfrac{p}{-q}$.
• Multiplicative inverse (乘法逆元): $\left(\dfrac{p}{q} \right)^{-1}=\dfrac{q}{p}$, if $p \neq 0$.
• Division: $p \div q=\dfrac{p}{q}; \dfrac{a}{b} \div \dfrac{c}{d}=\dfrac{a/b}{c/d}$, if $c \neq 0$. (You can prove the division between two rational numbers (fractions) are well defined.)

$\mathbb{Q}$ shares properties 1~4 from $\mathbb{N}$ and properties 1~2 from $\mathbb{Z}$. However, property 5 from $\mathbb{N}$ doesn’t hold here. Given an arbitrary rational number, there is neither a successor nor a predecessor of it. More specifically, we have the following theorem.

 Theorem 2: $\forall\ a, b \in \mathbb{Q}$ s.t.  $a, then $\exists\ c \in \mathbb{Q}$ s.t. $a

4. Real numbers

It seems the number system has been able to satisfy all tasks one can meet with, but there was at least one man who was not satisfied, Pythagoras (毕达哥拉斯). He proved the following statement.

 Proposition 1: There is no rational number for which the square (二次方) is 2.

To solve this problem, man introduced irrational numbers (无理数), thus real numbers $\mathbb{R}$. There are several ways to construct $\mathbb{R}$ from $\mathbb{Q}$, such as Cauchy sequence (柯西序列) and Dedekind cut (戴德金分割). You can find more details about the construction (构造) and other methods here.

One can understand real numbers from the real line (实数轴). Each point on the real line corresponds (对应) to a real number, and vice versa (反之亦然). One can also refer to the decimal system. Each real number can be written by a decimal representation (小数). If it’s a rational number, then the representation will be either finite (有限) or repeating infinite (无限循环); otherwise, the representation will be infinite without repeating (无限不循环).

One of the most important properties of $\mathbb{R}$ is the Archimedean Property (阿基米德性).

 Archimedean Property (of $\mathbb{R}$): $\forall\ p, q \in \mathbb{R}$ positive such that  $p < q$, $\exists\ n \in \mathbb{N}$ satisfies $np >q$.

$\mathbb{R}$ shares all of the good properties above. It turns out to be the core part of modern analysis.

From the construction or the decimal representation, it’s easy to see that $\mathbb{Q}$ is dense in $\mathbb{R}$ in the following sense:

 Density of $\mathbb{Q}$ in $\mathbb{R}$: $\forall\ p, q \in \mathbb{R}$ such that  $p < q$, $\exists\ r \in \mathbb{Q}$ satisfies $p.

Notice that $p\ \text{and}\ q$ can be either rational numbers or irrational numbers.

 Corollary 1: (Density of the irrational numbers in $\mathbb{R}$) $\forall\ p, q \in \mathbb{R}$ such that  $p < q$, $\exists$ an irrational number $r$ satisfies $p.

5. Others

$\mathbb{R}$ is not a perfect set. For example, it’s not an algebraic closed (代数封闭) set/field, i.e., Of the polynomial (多项式) equation:

$a_n x^n+a_{n-1}x^{n-1}+ \cdots a_1x+a_0 =0$,

we choose all of the coefficient (系数) $a_n, \ldots, a_0 \in \mathbb{R}$, the solution may not be in $\mathbb{R}$. For example, $x^2+1=0$ has no root in $\mathbb{R}$. Therefore, we still need to extend number sets. In the above example, the equation has roots in the complex numbers set $\mathbb{C}$. More examples are Algebraic numbers $\bar{\mathbb{Q}}$ (代数数), Quaternions $\mathbb{H}$ (四分数), Octonions $\mathbb{O}$ (八分数), Sedenions $\mathbb{S}$ (十六分数), etc. You can find more discussion about these sets in the Abstract Algebra.

Reviewed by Rachel Reed and Karthick Alagarsamy.