Real numbers

Yeonsol Kim

May 31, 2026

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Definition 1.1.1. Let $x\in \mathbb R$.

1. $0$ is the number such that $x+0=x$ for all $x$.
2. $1$ is the nonzero number such that $x\cdot 1=x$ for all $x$.
3. We define the negative $x$, denoted by $-x$, as the number such that $x+(-x)=0$.
4. Let $x\ne0$. We define the reciprocal of $x$, denoted by $1/x$, as the number such that $x\cdot (1/x)=1$.

Definition 1.1.2. Let $x\in \mathbb R$. We define the absolute value of $x$ as follows:

\[\vert x\vert := \begin{cases} x, & x \ge 0,\\ -x, & x < 0. \end{cases}\]

Theorem 1.1.3. Let $x,y\in \mathbb R$. Then

1. $\vert x\vert \ge 0$
2. $\vert x\cdot y\vert = \vert x\vert \cdot \vert y\vert$
3. $\vert x+y\vert \le \vert x\vert +\vert y\vert$
4. $\vert x\vert ^2 = x^2$
5. $\vert x\vert = 0 \Rightarrow x = 0$

Definition 1.1.4 (Square Root). Let $x,y\in \mathbb R$ and $x\ge 0$.

\[\sqrt{x}=y :\Longleftrightarrow y \ge 0,\ y^2=x\]

Theorem 1.1.5. Let $x\in \mathbb R$. Then

1. $\sqrt{x^2}=\vert x\vert$
2. $(\sqrt{x})^2=x$

Definition 1.1.6.

1. (Harmonic Mean) For $x+y\ne 0$,

   \[HM(x,y):=\frac{2xy}{x+y}\]

2. (Geometric Mean) For $xy\ge 0$,

   \[GM(x,y):=\sqrt{xy}\]

3. (Arithmetic Mean)

   \[AM(x,y):=\frac{x+y}{2}\]

4. (Quadratic Mean)

   \[QM(x,y):=\sqrt{\frac{x^2+y^2}{2}}\]

5. (Weighted Arithmetic Mean) For $w_i\ge 0$ and $\sum_{i=1}^{n}w_i=1$,

   \[WAM(x_1,\ldots,x_n):=\sum_{i=1}^{n}w_i x_i\]

Theorem 1.1.7.

\[\min(x_1,\ldots,x_n) \le \operatorname{mean}(x_1,\ldots,x_n) \le \max(x_1,\ldots,x_n)\]

Theorem 1.1.8.

\[HM \le GM \le AM \le QM\]

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