Write a repeating decimal as a fraction

Learning Objectives Identify the non-repeating and repeating parts (repetend) of a decimal. Express a repeating decimal as the sum of a terminating decimal and an infinite geometric series. Determine the first term (a) and the common ratio (r) for the geometric series that represents a repeating decimal. Apply the formula for the sum of an infinite geometric series to find the value of the repeating part. Convert any repeating decimal into a simplified fraction (a rational number in the form p/q). Justify why a repeating decimal must be a rational number using the concept of a convergent geometric series. Is 0.999... really equal to 1, or just incredibly close? 🤔 This lesson will give you the mathematical tools to prove the surprising answer! In this tutorial, you will dis...

TermDefinitionExample Repeating DecimalA decimal number in which a digit or a block of digits repeats infinitely after the decimal point. The repeating part is often indicated with a bar over the digits (vinculum).0.333... is written as 0.$\overline{3}$. 0.1272727... is written as 0.1$\overline{27}$. RepetendThe specific digit or block of digits that repeats in a repeating decimal.In 0.$\overline{5}$, the repetend is 5. In 0.1$\overline{234}$, the repetend is 234. Infinite Geometric SeriesThe sum of the terms of a geometric sequence that continues without end.4 + 2 + 1 + 0.5 + 0.25 + ... First Term (a)The initial term in a sequence or series.In the series representing 0.444..., which is 0.4 + 0.04 + 0.004 + ..., the first term 'a' is 0.4. Common Ratio (r)The constant factor by w...

Decomposition of a Repeating Decimal Repeating Decimal = (Non-repeating part) + (Infinite geometric series of the repeating part) This is the foundational principle. We split the decimal into its stable, non-repeating component and its infinite, repeating component, which can be modeled as a series. Sum of an Infinite Geometric Series S_{\infty} = \frac{a}{1-r}, \text{ where } |r| < 1 This is the key formula used to calculate the sum of the infinite series that represents the repeating part of the decimal. 'a' is the first term of the series, and 'r' is the common ratio.