Using a number line, what whole number and rational number to the nearest half is 58−−√ between?

Answer 1

To determine the whole number and rational number to the nearest half that is closest to $\sqrt{58}$, we can compare it to the square root of the surrounding perfect squares. Since $7^2=49$ and $8^2=64$, we have $7<\sqrt{58}<8$. Writing the square root as a fraction, we get $\frac{\sqrt{58}}{1}$. To simplify this fraction, we multiply by $\frac{1}{\sqrt{58}}$: $$\frac{\sqrt{58}}{1}\cdot\frac{\sqrt{58}}{\sqrt{58}}=\frac{58}{\sqrt{58}}.$$ Rationalizing the denominator, we obtain $\frac{58}{\sqrt{58}}=\frac{58}{\sqrt{58}}\cdot\frac{\sqrt{58}}{\sqrt{58}}=\frac{58\sqrt{58}}{58}=\sqrt{58}$. Notice that $\frac{7}{\sqrt{58}}<1$ since $\sqrt{58}<7$. Similarly, $\frac{8}{\sqrt{58}}>1$ since $\sqrt{58}<8$. Thus, $\boxed{\frac{7}{2}}$ and $\boxed{\frac{8}{2}=4}$ are the whole number and rational number to the nearest half, respectively.

Answer 2

Using a number line, what whole number and rational number to the nearest half is −−√ 58 between?