Chapter 6.6, Problem 32WE

(a)

To determine

To Find:

If x and y are rational numbers , what can you conclude about their arithmetic mean.

Answer to Problem 32WE

The arithmetic mean of two rational numbers will also be rational number.

Explanation of Solution

Given:

The arithmetic mean of two numbers x and y is the number $\frac{x+y}{2}$ .

x and y are rational numbers.

Calculation:

x and y are rational numbers.

Let

  $\begin{array}{l}x=\frac{p}{q};\text{ p and q are integers and q}\ne \text{0}\\ y=\frac{r}{s};\text{r and s are integers and s}\ne \text{0}\\ \text{Then, arithmetic mean :}\\ \frac{x+y}{2}=\frac{\frac{p}{q}+\frac{r}{s}}{2}=\frac{\left(\frac{ps+rq}{qs}\right)}{2}=\frac{ps+rq}{2qs}\\ \text{Since p,q,r,s are integers and integers are closed under addition and}\\ \text{multiplication }\text{. So, }ps+rq\text{ and }2qs\text{ are integers }\text{.}\\ \text{q}\ne \text{0 and s}\ne \text{0}\Rightarrow 2qs\ne 0.\\ So,\frac{ps+rq}{2qs}\text{ is a rational number}\text{.}\end{array}$

Hence, the arithmetic mean of two rational numbers is also a rational number.

(b)

To determine

To Find:

If x and y are positive rational numbers , what can you conclude about their geometric mean.

Answer to Problem 32WE

Depending on the positive rational numbers , the geometric mean can be rational or irrational.

Explanation of Solution

Given:

The geometric mean of two positive rational numbers x and y is the number $\sqrt{xy}$ .

x and y are positive rational numbers.

Calculation:

x and y are positive rational numbers.

Let

  $\begin{array}{l}x=2\\ y=3\\ \text{Both x and y are positive rational numbers}\text{.}\\ \text{Geometric mean = }\sqrt{2\cdot 3}=\sqrt{6}\text{ , which is irrational number}\text{.}\end{array}$

Hence, the geometric mean of two positive rational numbers can be rational or irrational.