Decimal Expansions and Fraction Conversions

This page delves into the intricacies of decimal expansions and the process of converting repeating decimals to fractions. The content is particularly relevant for students studying Liczby Klasa 7 pdf and those interested in Rozwinięcia dziesiętne liczb wymiernych.

The document begins by introducing the concept of approximation error, which is defined as the difference between the actual number and its approximation. This is crucial for understanding the limitations of decimal representations of certain numbers.

Definition: Approximation error = actual number - approximation

The main focus of the page is on the method of przedstaw liczbę w postaci ułamka zwykłego, which involves converting repeating decimals to fractions. This process is demonstrated through several examples, showcasing the algebraic approach to solving such problems.

Example: To convert 0.$49$ to a fraction:

1. Let x = 0.494949...
2. 100x = 49.4949...
3. Subtracting x from 100x: 99x = 49
4. Solving for x: x = 49/99

The document also touches on the distinction between finite and infinite decimal expansions.

Vocabulary:

• SKOŃCZONE $Finite$: Decimal expansions that terminate, e.g., 0.4, 0.420, 0.72
• NIESKOŃCZONE $okresowe$ $Infinite periodic$: Decimal expansions that repeat indefinitely, e.g., 0.$8$, 0.$24$, 0.$690$

This classification is essential for understanding which numbers can be represented as rational fractions and which cannot.

Highlight: The ability to convert between decimal and fractional representations is a fundamental skill in mathematics, particularly useful in Rozwinięcia dziesiętne liczb wymiernych klasa 7 zadania.

The page provides a solid foundation for students to tackle more complex problems involving Rozwinięcia dziesiętne liczb wymiernych zadania and to understand the relationship between different number representations.