The Dawn of Mathematics: The Human Need to Count | Class 9 Mathematics Notes

3.6 REAL NUMBERS: DECIMALS AND CYCLIC PATTERNS

Real numbers (ℝ) combine rational and irrational numbers to form a continuous number line. Rational numbers have decimal expansions that either terminate or repeat cyclically. For example, 3/8 = 0.375 (terminating) and 5/11 = 0.454545... (repeating). The type of decimal expansion depends on the prime factors of the denominator: if the denominator’s prime factors are only 2 and/or 5, the decimal terminates; otherwise, it repeats. Non-terminating repeating decimals can be converted back into fractions using algebraic methods by multiplying by powers of 10 and subtracting. General repeating decimals have non-repeating parts followed by repeating blocks, and can also be converted to fractions using systematic steps. Cyclic numbers, such as 142857 from 1/7, exhibit fascinating properties where multiplying by numbers 1 to 6 results in cyclic permutations of the digits. Irrational numbers have non-terminating, non-repeating decimals, exemplified by √2 and π. Rational numbers can have two decimal representations, such as 1.000... = 0.999..., highlighting the non-uniqueness of decimal expansions.

📊 Diagram: No specific diagram for decimal expansions; Fig. 3.12 shows the real number line combining rational and irrational numbers.

🧪 Activity: Convert various repeating decimals to fractions and explore cyclic properties of 1/7 and other reciprocals.

🔗 Connection: Prepares for the conclusion on the evolution of numbers and the introduction to imaginary numbers.