Repeating decimal as infinite geometric series | Precalculus | Khan Academy

Khan Academy
8 Nov 201308:02
EducationalLearning
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TLDRThe video script explores the representation of a repeating decimal, 0.4008..., as an infinite geometric series. It demonstrates how to express this repeating pattern as a series with a common ratio of 10^-4. The script then applies the formula for the sum of an infinite geometric series to convert the repeating decimal into a fraction, 4008/3333, showcasing a method to simplify complex decimals into more manageable forms.

Takeaways
  • ๐Ÿ”ข A repeating decimal 0.4008 can be represented as an infinite series.
  • ๐Ÿงฎ The repeating pattern 4008 can be broken down into smaller decimal parts (0.4008, 0.00004008, etc.).
  • ๐Ÿ”„ The infinite series is formed by continuously adding these smaller decimal parts.
  • ๐Ÿ“ This series can be identified as a geometric series.
  • ๐Ÿ” To confirm a geometric series, each term should be multiplied by the same common ratio.
  • ๐Ÿ”ข The common ratio here is 10^-4 or 0.0001.
  • โœ๏ธ The series can be expressed as 0.4008 * (10^-4)^k from k=0 to infinity.
  • ๐Ÿ“Š The sum of an infinite geometric series is given by a / (1 - r).
  • ๐Ÿ”ข Substituting the values, the sum is 0.4008 / (1 - 0.0001).
  • ๐Ÿ’ก Simplifying the fraction, we get 4008/9999.
  • ๐Ÿ” Further simplification leads to 1336/3333.
Q & A

  • What is a repeating decimal and how is it represented in the script?

    -A repeating decimal is a decimal number that has digits that repeat infinitely. In the script, it is represented as 0.4008,, where '4008' is the repeating pattern.

  • How does the script represent the repeating decimal as an infinite sum?

    -The script represents the repeating decimal as an infinite sum by breaking it down into a series of terms, each representing the repeating pattern '4008' with increasing numbers of leading zeros.

  • What is the common ratio in the infinite series of the repeating decimal?

    -The common ratio in the infinite series is 10 to the negative fourth power, which is equivalent to multiplying by 0.0001 each time to move the decimal point four places to the left.

  • Why is the infinite series considered a geometric series?

    -The series is considered geometric because each term is obtained by multiplying the previous term by a constant ratio, which in this case is 10 to the negative fourth power.

  • How is the infinite geometric series expressed in sigma notation in the script?

    -The series is expressed in sigma notation as the sum from k equals 0 to infinity of 0.4008 times 10 to the negative fourth power to the k-th power.

  • What is the formula for the sum of an infinite geometric series?

    -The formula for the sum of an infinite geometric series is a over 1 minus r, where 'a' is the first term and 'r' is the common ratio.

  • How does the script express the repeating decimal as a fraction?

    -The script expresses the repeating decimal as a fraction by using the formula for the sum of an infinite geometric series, resulting in 4008/9999 after simplification.

  • What simplification is performed on the fraction 4008/9999?

    -The simplification involves dividing both the numerator and the denominator by 3, resulting in the fraction 1336/3333.

  • Why is the fraction 4008/9999 considered to be in its simplest form?

    -The fraction is considered simplest because the numerator and denominator have no common factors other than 1, and the sum of the digits in the numerator and denominator is not divisible by 3.

  • What insight does the script provide about the relationship between repeating decimals and fractions?

    -The script demonstrates that repeating decimals can be represented as fractions, specifically by using the sum formula for an infinite geometric series.

  • How can the process described in the script be applied to other repeating decimals?

    -The process can be applied to any repeating decimal by identifying the repeating pattern, setting up an infinite geometric series with the pattern as the first term and the appropriate common ratio, and then using the sum formula to express it as a fraction.

Outlines
00:00
๐Ÿ”ข Representing a Repeating Decimal as an Infinite Geometric Series

The paragraph introduces the concept of representing a repeating decimal, specifically 0.4008, as an infinite sum or series. It challenges the viewer to consider the nature of this series and whether it can be classified as a geometric series. The explanation unfolds by breaking down the repeating pattern into an infinite series, where each term represents a shift of the decimal point by four places, effectively multiplying by 10 to the negative fourth power. This establishes a common ratio, confirming the series as geometric. The paragraph concludes with a transition to express this series using sigma notation and to find its sum as a fraction.

05:07
๐Ÿ“ Calculating the Sum of an Infinite Geometric Series as a Fraction

This paragraph delves into the process of calculating the sum of the previously introduced infinite geometric series. It begins by applying the formula for the sum of an infinite geometric series, which is 'a' over '1 - r', where 'a' is the first term and 'r' is the common ratio. The paragraph demonstrates the calculation by substituting the values for 'a' and 'r', simplifying the expression to find the sum as a fraction. It then proceeds to simplify the fraction further by identifying common factors and reducing the fraction to its simplest form. The final result is a simplified fraction that represents the original repeating decimal, showcasing the conversion of a repeating decimal into a fraction through the properties of geometric series.

Mindmap
Keywords
๐Ÿ’กRepeating Decimal
A repeating decimal is a decimal number that has a digit or a sequence of digits after the decimal point that repeat indefinitely. In the video, the repeating decimal 0.4008,, where '4008' is the sequence that repeats, is used to illustrate the concept of infinite series and geometric progressions. The script uses this repeating pattern to demonstrate how it can be represented as an infinite sum, which is a foundational concept in the video's explanation.
๐Ÿ’กInfinite Sum
An infinite sum refers to the sum of an infinite sequence of numbers. In the context of the video, the infinite sum is represented by the repeating decimal 0.4008,, which is written out as an endless series of the pattern '4008' with increasing numbers of leading zeros. This concept is central to the video's exploration of representing repeating decimals as infinite series.
๐Ÿ’กInfinite Series
An infinite series is the sum of the terms of an infinite sequence. The video script discusses representing the repeating decimal as an infinite series, where each term corresponds to the repeating pattern '4008' with additional leading zeros. This concept is crucial for understanding how the repeating decimal can be mathematically manipulated and eventually expressed as a fraction.
๐Ÿ’กGeometric Series
A geometric series is a series where each term after the first is found by multiplying the previous term by a constant called the common ratio. The video explains that the infinite sum of the repeating decimal can be considered a geometric series because each term is obtained by multiplying the previous term by 0.0001, which is 10 to the negative fourth power.
๐Ÿ’กCommon Ratio
The common ratio in a geometric series is the constant multiplier used to obtain each term from the previous one. In the script, the common ratio is identified as 10 to the negative fourth power, which is the factor by which each term of the series is multiplied to get the next term, moving the decimal point four places to the left.
๐Ÿ’กSigma Notation
Sigma notation is a way of expressing the sum of a series using the Greek letter 'ฮฃ'. In the video, sigma notation is used to represent the infinite geometric series of the repeating decimal, where the series is written as the sum from k equals 0 to infinity of the first term multiplied by the common ratio raised to the k-th power.
๐Ÿ’กSum of an Infinite Geometric Series
The sum of an infinite geometric series is a specific value that the series approaches as the number of terms goes to infinity. The video script explains that the sum can be found using the formula a / (1 - r), where 'a' is the first term and 'r' is the common ratio. This formula is applied to the repeating decimal series to express it as a fraction.
๐Ÿ’กFraction
A fraction represents a part of a whole, expressed as the ratio of two integers. In the context of the video, the repeating decimal is expressed as a fraction by using the sum formula for an infinite geometric series. The script demonstrates that the repeating decimal 0.4008,, can be simplified to the fraction 4008/9999, which is a key takeaway of the video.
๐Ÿ’กSimplification
Simplification in mathematics refers to reducing a complex expression to a simpler form, often by dividing both the numerator and the denominator by their greatest common divisor. The video script includes a step where the fraction 4008/9999 is simplified by dividing both the numerator and the denominator by 3, resulting in the simplified fraction 1336/3333.
๐Ÿ’กDecimal Place
A decimal place is the position of a digit in a number to the right of the decimal point. The video script discusses moving the decimal point to the left by four places to multiply by the common ratio of 0.0001, which is essential for understanding how the repeating pattern '4008' is represented in the infinite geometric series.
Highlights

Introduction of a repeating decimal 0.4008,, where the digits 4008 keep repeating.

Representation of the repeating decimal as an infinite sum or series.

Explanation of how each term of the infinite series represents a repeating pattern of 4008.

Illustration of the repeating pattern through the example of 0.4008 and its subsequent terms.

Identification of the pattern of adding four zeroes before the decimal in each term.

Introduction of the concept of a geometric series and its criteria.

Analysis of whether the infinite series is a geometric series by examining the common ratio.

Calculation of the common ratio as 10 to the negative fourth power.

Rewriting the series with the common ratio to clarify its geometric nature.

Introduction of sigma notation to express the infinite geometric series.

Derivation of the sum of an infinite geometric series using the formula a / (1 - r).

Application of the formula to find the sum of the given repeating decimal series.

Expression of the repeating decimal as a fraction using the derived formula.

Simplification of the fraction by multiplying the numerator and denominator by 10,000.

Further simplification of the fraction by dividing both the numerator and denominator by 3.

Final expression of the repeating decimal as a simplified fraction 1336/3333.

Conclusion on the representation of a repeating decimal as an infinite geometric series and its conversion to a fraction.

Transcripts

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