Repeating decimals

Learning Objectives Define a repeating decimal and correctly identify its repetend. Prove that any repeating decimal represents a rational number. Convert a simple repeating decimal (e.g., 0.444...) into its equivalent fractional form using an algebraic method. Convert a complex repeating decimal (e.g., 0.1232323...) into its equivalent fractional form. Explain the logical sequence of steps required for the algebraic conversion process. Distinguish between rational and irrational numbers based on their decimal expansions. How can an infinitely long, repeating decimal be perfectly captured by a simple, finite fraction? 🤔 Let's use logic to unravel this mathematical magic! In this tutorial, we will explore the logical connection between repeating decimals and rational n...

TermDefinitionExample Rational NumberAny number that can be expressed as a quotient or fraction p/q of two integers, where p is the numerator and q is the non-zero denominator.1/2, -5/3, 7 (which is 7/1), 0.25 (which is 1/4) Repeating DecimalA decimal representation of a number whose digits are periodic and the infinitely repeated portion is not zero.1/3 = 0.333... or 1/7 = 0.142857142857... RepetendThe sequence of digits that repeats infinitely in a repeating decimal.In 0.523523..., the repetend is '523'. VinculumA horizontal line drawn over the repetend to indicate which digits repeat.0.3̅ means 0.333... and 0.12̅4̅5̅ means 0.12454545... Terminating DecimalA decimal number that has a finite number of digits after the decimal point. All terminating decimals are rational numbers...

Conversion Rule for Simple Repeating Decimals 1. Let x = the repeating decimal. \n2. Multiply x by 10^k, where k is the number of digits in the repetend. \n3. Subtract the first equation from the second: (10^k)x - x. \n4. Solve for x. Use this logical procedure when the repeating digits begin immediately after the decimal point. The goal is to create two equations where the decimal parts are identical, so they cancel out upon subtraction. Conversion Rule for Complex Repeating Decimals 1. Let x = the decimal. \n2. Multiply x by 10^m to move the decimal past the non-repeating part. \n3. Multiply the new equation by 10^k, where k is the length of the repetend. \n4. Subtract the equation from step 2 from the equation in step 3. \n5. Solve for x. Use this procedure when there are...