2.2 Absolute Value and the Real Line

<< 2.1 The Algebraic and Order Properties of R | 2.3 The Completeness Property of R >>

2.2.1 Definition: Absolute value

Definition

The absolute value $a\in \mathbb{R}$, denoted by $|a|$, is defined by $ |a| := \begin{cases} -a, &a<0\ 0, &a=0\ a, &a>0 \end{cases} $$

We see from the definition of absolute value that, $\forall a\in \mathbb{R}$,

• $|a|\ge 0$
• $|-a|=|a|$
• $|a|=0⟺a=0$

2.2.2 Theorem: Additional property of absolute values

Theorem

$\forall a,b,c\in \mathbb{R},c\ge 0:$

1. $|ab|=|a||b|$
2. $|a{|}^2=a^2$
3. $|a|\le c⟺-c\le a\le c$
4. $-|a|\le a\le |a|$

2.2.3 Theorem: Triangle inequality

Theorem

$ |a+b|\leq |a| + |b|,\quad\forall a,b\in \mathbb{R}$$

2.2.4 Corollary: From 2.2.3

Corollary 1

There are multiple variations of Triangle Inequality, which is:

$\forall a,b\in \mathbb{R}:$

1. $||a|-|b||\le |a-b|$
2. $|a-b|\le |a|+|b|$

2.2.5 Corollary: From 2.2.3

Corollary 2

$ |a_1+\dots+a_n|\leq |a_1|+\dots+|a_n|,\quad \forall a_{1},\dots,a_{n}\in \mathbb{R} $$

2.2.7 Definition: Neighborhood

Definition

Let $a\in \mathbb{R}$ and $\epsilon >0$. Then the $\epsilon$-neighborhood of $a$ is the set $ V_\varepsilon(a):=\set{ x\in\mathbb R : |x-a|<\varepsilon } $$

Remark

This essentially means that $x$ is “close” to $a$. $V_{\epsilon }(a)$ is the set of all real number within $\epsilon$ unit “distance” of $a$.

For example 2-neighborhood of 3 means numbers within 2 unit distance of 3, which ranges from 1 to 5.

2.2.8 Theorem: Uniqueness of a point in neighborhoods

Theorem

Let $a\in \mathbb{R}$.

Then $ x\in V_{\varepsilon}(a)\implies x=a,\quad \forall \varepsilon>0 $$

Remark

We can make sense of this theorem by imagining even if $\epsilon$ (or the distance between $x$ and $a$) is infinitesimally small, but $x$ is still in $V_{\epsilon }(a)$.

This can only happen if the distance between $x$ and $a$ is $0$, which means $x=a$.