Problem 9 - Calculus Single Variable

Question

Use long division to convert the rational fraction to a (possibly nonterminating) decimal with a repeating block. Identify the repeating block.\(5 / 3\)

Step-by-Step Solution

Answer

The decimal form of \( \frac{5}{3} \) is 1.666..., where '6' is the repeating block.

Step 1: Set Up Long Division

Write the division in long division format where 5 is the dividend and 3 is the divisor. It should look like this: 3 | 5.000. Add a decimal point and zeros to the dividend (5) to make it 5.000 for easier division.

Step 2: Divide the Whole Number

Divide 5 by 3. 3 goes into 5 one time. Write 1 above the division bar, over the first digit of 5. Multiply 1 by 3 and subtract from 5. This leaves a remainder of 2.

Step 3: Bring Down the Next Digit

Bring down a 0 next to the remainder, making it 20. Divide 20 by 3. 3 goes into 20 six times (since 3 * 6 = 18). Write 6 next to 1 above the division bar.

Step 4: Continue the Division Process

Subtract 18 from 20, leaving a remainder of 2 again. Bring down another 0 to make it 20. Divide 20 by 3 to get 6 again. Add another 6 above the division bar.

Step 5: Identify the Repeating Decimal

Recognize that the division process will continue to generate the remainder 2, followed by another 0, and the recurring quotient digit 6. Therefore, the repeating block in the decimal representation is '6'.

Key Concepts

Rational Fractions

A rational fraction is a mathematical way to represent a division between two integers. In simple terms, it is a fraction where the numerator and denominator are both whole numbers. Rational fractions can always be expressed in the form \( \frac{a}{b} \), where \( a \) and \( b \) are integers, and \( b eq 0 \).

• The fraction \( \frac{5}{3} \) is a rational fraction because both 5 and 3 are integers, and the denominator, 3, is not zero.
• Rational fractions can often be expressed as either terminating or non-terminating repeating decimals.

Whenever you encounter a rational fraction, converting it into a decimal can offer a different perspective and might make some calculations easier. This process often involves long division, especially when dealing with non-terminating decimals.

Repeating Decimals

Repeating decimals are decimal numbers that have one or more digits that repeat infinitely. They arise from the division of two integers when there is a remainder that cycles through the same sequence.

• For example, with the division of 5 by 3 using long division, we arrive at 1.666...; this is an example of a repeating decimal.
• The repeating decimal can be indicated by placing a bar over the repeating digit(s), such as \(1.\overline{6}\).

Understanding repeating decimals is crucial because they represent a precise mathematical value despite their extended form. Recognizing the repeating block allows us to write and work with these numbers effectively.

Decimal Conversion

Decimal conversion involves transforming a fraction into its decimal form, which can either terminate or repeat. This conversion helps in analyzing and using numbers more effectively in different mathematical situations.

• To convert \( \frac{5}{3} \) to its decimal form, long division can be employed to reveal a repeating sequence of 6s in the decimal.
• Some fractions will not convert to a simple terminating decimal but will instead extend indefinitely with a repeating pattern.

Understanding decimal conversion is essential for grasping how fractions relate to decimals, thereby enhancing numerical literacy and problem-solving skills.

Mathematical Division Process

The mathematical division process, particularly long division, is a methodical way to divide numbers, offering a step-by-step approach to reaching an answer. It is especially useful for converting fractions to decimals.

• Long division sets the stage by placing the dividend (5) and divisor (3) in the correct format, preparing for division.
• By bringing down digits sequentially and finding where the divisor fits within the current number, the method reveals both the quotient and the repeating pattern.
• Repetition of remainders leads to the identification of repeating decimals.

This process is particularly effective in handling divisions that produce non-terminating decimals, making it a foundational skill in mathematics.