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Prove that √5 is an irrational number.

Convert the following rational numbers in the form of a terminating decimal or non-terminating and repeating decimal, whichever the case may be, by the process of long division:
\(\frac{2}{9}\)

Convert the following rational numbers in the form of a terminating decimal or non-terminating and repeating decimal, whichever the case may be, by the process of long division:
\(\frac{3}{50}\)

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.

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.

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.

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?

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.

Can you think of other way(s) to find a rational number between any two rational numbers?

Find three rational numbers between 3.1415 and 3.1416.