Repeating decimals

Definition

Repeating decimals are numbers that have a decimal representation which eventually repeats a specific sequence of digits indefinitely. This concept is crucial in understanding the relationship between fractions and their decimal equivalents, particularly in the realm of real numbers, where repeating decimals illustrate the idea that certain fractions cannot be expressed as terminating decimals.

5 Must Know Facts For Your Next Test

1. Any rational number can be expressed as either a terminating or a repeating decimal.
2. Repeating decimals can be represented using bar notation, where a line is drawn over the repeating digits to indicate they continue indefinitely.
3. An example of a repeating decimal is 0.333..., which is equivalent to the fraction \(\frac{1}{3}\).
4. Not all decimals are repeating; some decimals, like \(\pi\) or the square root of 2, are non-repeating and non-terminating, classifying them as irrational numbers.
5. The length of the repeating block in a decimal can vary, with some numbers having short cycles like 0.666... (which repeats every one digit) and others like 0.142857142857... (which repeats every six digits).

Review Questions

• How can you determine if a decimal representation is repeating or terminating?
  • To determine if a decimal is repeating or terminating, analyze its fractional form. If the denominator of the fraction can be expressed as a product of only the prime factors 2 and/or 5, then the decimal will terminate. If there are other prime factors present in the denominator, the decimal representation will be repeating. For example, \(\frac{1}{4}\) results in 0.25 (terminating), while \(\frac{1}{3}\) results in 0.333... (repeating).
• Describe how to convert a repeating decimal into a fraction.
  • To convert a repeating decimal into a fraction, let 'x' represent the repeating decimal. For instance, if x = 0.666..., multiply both sides by 10 to shift the decimal: 10x = 6.666.... Next, subtract the original equation from this new equation: 10x - x = 6.666... - 0.666..., resulting in 9x = 6. Therefore, x = \(\frac{6}{9}\), which simplifies to \(\frac{2}{3}\). This method can be applied to any repeating decimal.
• Evaluate how understanding repeating decimals contributes to a deeper grasp of real numbers and their properties.
  • Understanding repeating decimals helps clarify the structure of real numbers by showing how they encompass both rational and irrational numbers. Repeating decimals highlight that certain fractions yield infinite but predictable patterns, bridging an important connection between arithmetic operations and their decimal representations. This insight into number properties not only enhances computational skills but also reveals the intricate relationship between numbers, supporting broader mathematical concepts like limits and convergence.

Related terms

terminating decimal: A terminating decimal is a decimal number that has a finite number of digits after the decimal point, meaning it does not continue indefinitely.

rational number: A rational number is any number that can be expressed as a fraction where both the numerator and denominator are integers, with the denominator not equal to zero.

non-repeating decimal: A non-repeating decimal is a decimal that does not have a repeating pattern of digits and continues infinitely without repeating.