Intervals in Real Numbers

Bounded intervals

We will call interval the set of numbers included between two given limits.

If $a$ and $b$ are two real numbers such that $a\leq b$, the interval of endpoints $a$ and $b$ is the segment $\overline{ab}$, or, also, the set of numbers included between $a$ and $b$.

If we consider that the endpoints $a$ and $b$ belong to the interval, we will say that it is a closed interval and will denote it as $[a,b]$.

If $x$ is a real number that belongs to $[a,b]$, the point that represents $x$ on the line is on the right of $a$ and on the left of $b$; this means that $a < x < b$, and since $a$ and $b$ belong to the interval as well, it is possible that $x=a$ or $x=b$, so a real number $x$ belongs to the closed interval $[a,b]$ if $a \leq x \leq b$. We will write this algebraic definition in the following way: $$[a,b]=\{x \in \mathbb{R} | a \leq x \leq b\}$$

If the endpoints do not belong to the interval, we call it an open interval and we will denote it as $(a,b)$. If $x$ is a real number that belongs to $(a,b)$, it is necessary that $a < x < b$, and we will write it in algebraic language as: $$(a,b)=\{x \in \mathbb{R} | a < x < b\}$$

If only one of the endpoints belongs to the interval we say that it is a semiopen interval and we will denote it as $(a,b]$ or $[a,b)$, depending on which endpoint belongs to the interval:

$$(a,b]=\{x \in \mathbb{R} | a < x \leq b\}$$ $$[a,b)=\{x \in \mathbb{R} | a \leq x < b\}$$

In any kind of interval, $a$ is the lower endpoint , and $b$ the upper endpoint. And $|b-a|$ is the length of the interval.

The point $C$ at the same distance from $a$ to $b$, we will call center of the interval. We will call the distance between the center of the interval and the endpoints the radius .

The center of an interval of endpoints $a$ and $b$ is the point $\dfrac{a+b}{2}$; in fact:

$$d\Big(a,\dfrac{a+b}{2}\Big)=\Big|\dfrac{a+b}{2}-a\Big|=\Big|\dfrac{a+b-2a}{2}\Big|=\dfrac{b-a}{2}$$

$$d\Big(\dfrac{a+b}{2},b\Big)=\Big|b-\dfrac{a+b}{2}\Big|=\Big|\dfrac{2b-a-b}{2}\Big|=\dfrac{b-a}{2}$$

On the other hand, the points of an interval of endpoints $a$ and $b$ can be defined in terms of the distance to the center of the interval.

If $x\in [a,b]$, the distance of $x$ to the center is less than or equal to the radius of the interval, and as $d(x,C)=|C-x|$, we have: $$[a,b]=\{x \in \mathbb{R} \ | \ |C-x|\leq r \}$$ where $r$ represents the radius of the interval $(r=d(a,b))$, and, similarly, for open intervals: $$(a,b)=\{x \in \mathbb{R} \ | \ |C-x| < r \}$$

To determine the endpoints of an interval given the center and the radius, we apply the properties of the absolute value:

$$|C-x| < r \Rightarrow |x-C| < r \Rightarrow$$ $$-r < x-C < r \Rightarrow -r+C < x < r+C$$

Therefore the endpoints of an interval of center $C$ and radius $r$ are $C-r$ and $C+r$.

The length of an interval is equal to the distance between its two endpoints: $$long([a,b])=d(a,b)$$ And as it depends on the endpoints, the length is the same whether the interval is open or closed: $$long((a,b))=long([a,b])=long((a,b])=long([a,b))$$

Let's observe that the length of an interval depends on the distance used when calculating it, so, continuing with the previous notation, if we use the p-adic distance to calculate the length of an interval, we will denote it as: $$long_p((a,b))=d_p(a,b)$$

The interval $\Big[\dfrac{1}{3},\dfrac{2}{5}\Big]$ is a closed interval bounded by lower endpoint $\dfrac{1}{3}$ and upper endpoint $\dfrac{2}{5}$.

The center of the interval is a point $C$: $$C=\dfrac{a+b}{2}=\dfrac{\dfrac{1}{3}+\dfrac{2}{5}}{2}=\dfrac{5+6}{15\cdot 2}=\dfrac{11}{30}.$$

And the radius is: $$d(a,C)=\Big|\dfrac{11}{30}-\dfrac{1}{3}\Big|=\Big|\dfrac{11}{30}-\dfrac{10}{30}\Big|=\dfrac{1}{30}.$$

The length of this interval is: $$long\Big(\Big[\dfrac{1}{3},\dfrac{2}{5}\Big]\Big)=d\Big(\dfrac{1}{3},\dfrac{2}{5}\Big)=\Big|\dfrac{1}{3}-\dfrac{2}{5}\Big|=\Big|\dfrac{5-6}{15}\Big|=\dfrac{1}{15}$$

Unbounded intervals

If we consider an interval that does not have lower endpoint or upper endpoint, we obtain a set of the kind: $$\{x\in \mathbb{R} \ | \ x \leq b\}, \ \mbox{or} \ \{x\in \mathbb{R} \ | \ a \leq x\} $$

Graphically, these sets are represented as all those that are on the left of $b$, or on the right of $a$, respectively.

We call these sets unbounded intervals and to denote them we use the infinity symbol $\infty$ as an endpoint. Although $\infty$ is not a number, we will use $-\infty$ to denote that it is less than any number and $+\infty$ to denote that it is greater than any number, in such a way that a lower unbounded interval is denoted by:

$$(-\infty,a)=\{x\in\mathbb{R} \ | \ x < a\}$$

if it is opened, whereas if it is closed:

$$(-\infty,a]=\{x\in\mathbb{R} \ | \ x \leq a\}$$

If the interval does not have upper endpoint, we call it an unbounded upper, and we write it as:

$$(a,+\infty)=\{x\in\mathbb{R} \ | \ a < x\}$$

if it is opened, and

$$[a,+\infty)=\{x\in\mathbb{R} \ | \ a \leq x\}$$

if it is closed.

$$[5, +\infty)=\{x\in\mathbb{R} \ | \ 5 \leq x\}$$

$$\Big(-\infty,\dfrac{\sqrt{2}}{3}\Big)=\Big\{ x \in \mathbb{R} \ \Big| \ \dfrac{\sqrt{2}}{3} < a \Big\}$$