Question

Finding which number supports the idea that the rational numbers are dense in the real numbers? an integer between $-11$ and $-10$,a whole number between $1$ and $2$,a terminating decimal between$\text{–3.14 and –3.15}$

Answer 1

Step-1:Explain the concept of density of rational number:

The density of the rational numbers in the real numbers, meaning that there is a rational number strictly between any pair of distinct real numbers (rational or irrational), however close together those real numbers may be.

Step-2: Find rational number between $-11$ and $-10$:

$-11$ and $-10$ can be written as $\frac{-11}{1}\mathrm{and}\frac{-10}{1}$.

Multiply numerator and denominator by $7$.

$$\begin{gathered} \frac{-11\times 7}{7},\frac{-10\times 7}{7} \\ \frac{-77}{7},\frac{-70}{7} \end{gathered}$$

So, the rational numbers between $\frac{-77}{7}\mathrm{and}\frac{-70}{7}$are:

$\frac{-76}{7},\frac{-75}{7},\frac{-74}{7},\frac{-73}{7},\frac{-72}{7},\frac{-71}{7}$

Step-3: Find rational number between $1\mathrm{and}2$:

$1\mathrm{and}2$.can be written as $\frac{1}{1},\frac{2}{1}$.

Multiply numerator and denominator by $6$.

$$\begin{gathered} \frac{1\times 6}{1\times 6},\frac{2\times 6}{1\times 6} \\ \frac{6}{6},\frac{12}{6} \end{gathered}$$

.So the rational numbers between $\frac{1}{1},\frac{2}{1}$are:

$\frac{7}{6},\frac{8}{6},\frac{9}{6}.\frac{10}{6},\frac{11}{6}$.

Step-4: Find rational number between a terminating decimal between $3.14\mathrm{and}3.15$.

$-3.14\mathrm{and}3.15$. can be written as$\frac{-314}{100},\frac{315}{100}$.

Let $\mathrm{p}=\frac{-314}{100},\mathrm{q}=\frac{315}{100}$.

Apply $\frac{\mathrm{p}+\mathrm{q}}{2}$.

$$\begin{gathered} \frac{1}{2}.\left(\frac{-314+315}{100}\right)=\frac{1}{2}.\left(\frac{1}{100}\right) \\ =\frac{1}{200} \\ =0.005 \end{gathered}$$

This is the rational number between $-3.14\mathrm{and}3.15$ .

Therefore, all the three number supports the idea that the rational numbers are dense in the real numbers.