Let a and b be two non-zero rational numbers such that $a + \dfrac{1}{b} = 0$. Without assigning any numerical values, determine whether ab is positive or negative. Justify your answer.

A rational number has a terminating decimal expansion whose last non-zero digit occurs in the 4th decimal place. Show that such a number can be written in the form $\dfrac{p}{10^4}$, where p is an integer not divisible by 10. Is it necessary that the denominator of this rational number, when written in the lowest form, is divisible by 2⁴ or 5⁴? Give reasons.

A rational number in its lowest form has denominator 2³ × 5. How many decimal places will its decimal expansion have? Explain your answer.

Let $a = \dfrac{7}{12}$ and $b = \dfrac{5}{6}$. Express both a and b in the form $\dfrac{k$1}{m} and $\dfrac{k2}{m}$ where $k$1, $k2$ and m are integers and $k$2 - k1 > 6. Using the same denominator m, write exactly five distinct rational numbers lying between a and b keeping an integer numerator. Explain why the condition $k$2 - k1 > n + 1 is necessary to find n such rational numbers between the two rational numbers a and b using this method.