Perform the long division for $\dfrac{1}{13}$. Identify the repeating block of digits. Does it show cyclic properties if you evaluate $\dfrac{2}{13}$? Now compute $\dfrac{3}{13}$, $\dfrac{4}{13}$, etc. What do you notice?

Classify the following numbers as rational or irrational:
(i) $\sqrt{81}$
(ii) $\sqrt{12}$
(iii) $0.33333\ldots$
(iv) $0.123451234512345\ldots$
(v) $1.01001000100001\ldots$ (Notice the pattern: Is it repeating a single block?)
(vi) $23.560185612239874790120$

Find the explicit fractions in case they are rational.

We have seen that the repeating block of $\dfrac{1}{7}$ is a cyclic number. Try to find more numbers (n) whose reciprocals $\left(\dfrac{1}{n}\right)$ produce decimals with repeating blocks that are cyclic.

Consider this puzzle: What is the square root of –1? We know that $1 \times 1 = 1$. We also know that $(-1) \times (-1) = 1$. There is no Real Number that, when multiplied by itself, results in a negative number. Thus, $\sqrt{-1}$ cannot exist on number line.