A Peek Beyond | Class 7 Mathematics Notes

Rational Numbers and Their Decimal Expansions

This section revisits rational numbers and focuses on their decimal expansions. Rational numbers are defined as numbers that can be expressed in the form p/q where p and q are integers and q ≠ 0. The decimal expansion of a rational number either terminates after a finite number of digits or repeats a pattern of digits infinitely. For example, 1/2 = 0.5 (terminating decimal), and 1/3 = 0.333... (repeating decimal). The section explains why this happens by discussing division and remainders. When dividing p by q, if the remainder becomes zero at some point, the decimal expansion terminates. Otherwise, the remainders repeat in a cycle, causing the decimal to repeat. The section also explains how to identify the repeating part in a decimal expansion and how to write rational numbers as decimals and vice versa. It emphasizes that every rational number has a decimal expansion that is either terminating or repeating, and this property is used to distinguish rational numbers from irrational numbers. The section includes examples and exercises to convert fractions to decimals and decimals to fractions, reinforcing the understanding of rational numbers and their decimal forms.

📊 Diagram: Diagrams show long division steps for fractions like 1/2 and 1/3, highlighting when the remainder becomes zero (terminating) or repeats (repeating decimal).

🧪 Activity: Activity to perform long division for given fractions and identify whether the decimal expansion terminates or repeats.

🔗 Connection: This section prepares students to understand the concept of irrational numbers by contrasting their decimal expansions with those of rational numbers.