Exercise 3.5

Without performing long division, determine which of the following rational numbers will have terminating decimals and which will be repeating: \(\frac{7}{20}\), \(\frac{4}{15}\) and \(\frac{13}{250}\). Then check your answers by explicitly performing the long divisions and expressing these rational numbers as decimals.

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

Classify the following numbers as rational or irrational: i) \( \sqrt{81} \) ii) \( \sqrt{12} \) iii) 0.33333... iv) 0.123451234512345... v) 1.01001000100001... (Notice the pattern: Is it repeating a single block?) vi) 23.560185612239874790120 Find the explicit fractions in case they are rational.

The number \( 0.\overline{9} \) (which means 0.99999...) is a rational number. Using algebra (let x = \( 0.\overline{9} \) , multiply by 10, and subtract), explain why \( 0.\overline{9} \) is exactly equal to 1.

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