Question

Yes or No? If No, give a reason. Assume that$a$ and$b$are non-zero real numbers.

(a) Is the distance between any two different real numbers always positive?

(b) Is the distance between$a$ and$b$the same as the distance between role="math" localid="1649490736430" $b$ and$a$?

Short answer

(a)Yes,the distance between any two different real numbers is always positive.

(b) Yes, the distance between$a$ and$b$ is the same as the distance between$b$ and $a$.

Step-by-step solution

Part a. Step 1. State the concept.

The distance between two real numbers$a$ and$b$ on the real line is given as $d\left(a,b\right)=\left|b-a\right|$.

Part a. Step 2. List the objective.

The objective is to determine if the distance between any two different real numbers is always positive.

Part a. Step 3. Determine the given objective.

Since the distance is an absolute value, which by definition is always non-negative, and since for any two different real numbers, the distance is the absolute value of a non-zero number, it follows that the distance between any two different real numbers is always positive.

Part b. Step 1. State the concept.

The distance between two real numbers $a$and $b$on the real line is given as $d\left(a,b\right)=\left|b-a\right|$.

The absolute value of a real number $x$ has the property $\left|-x\right|=\left|x\right|$.

Part b. Step 2. List the objective.

The objective is to determine if the distance between$a$ and$b$ is the same as the distance between$b$ and $a$.

Part b. Step 3. Determine the given objective.

According to the formula, the distance between $a$and $b$is $d\left(a,b\right)=\left|b-a\right|$.

Then, the distance between $b$and $a$is,

$$\begin{gathered} d\left(b,a\right)=\left|a-b\right| \\ =\left|-\left(b-a\right)\right| \end{gathered}$$

So,

$d\left(b,a\right)=\left|-\left(b-a\right)\right|$

Put$x=b-a$ in$\left|-x\right|=\left|x\right|$ to get,

$\left|-\left(b-a\right)\right|=\left|b-a\right|$

Put $d\left(a,b\right)=\left|b-a\right|$ and $d\left(b,a\right)=\left|-\left(b-a\right)\right|$ in $\left|-\left(b-a\right)\right|=\left|b-a\right|$ to get,

$d\left(b,a\right)=d\left(a,b\right)$

This implies that the distance between $a$ and $b$ is the same as the distance between $b$ and $a$.