Problem 10 - Algebra 2 and Trigonometry

Question

In \(8-12,\) write each rational number as a repeating decimal. $$ \frac{5}{7} $$

Step-by-Step Solution

Answer

\( \frac{5}{7} = 0.\overline{714285} \)

Step 1: Begin Division

To write the rational number \( \frac{5}{7} \) as a repeating decimal, start by performing long division with 5 as the dividend and 7 as the divisor. Since 5 is less than 7, we begin by adding a decimal point and a zero to extend the process.

Step 2: Divide and Find the First Digit

Perform the division: 7 goes into 50 (the first extended part of 5.0) a total of 7 times (since 7 x 7 = 49). Place 7 in the tenths place of the quotient and subtract 49 from 50, which leaves a remainder of 1.

Step 3: Extend the Division

Bring down another zero to make the remainder 10. Divide 10 by 7, which goes 1 time (since 7 x 1 = 7). Place 1 in the hundredths place of the quotient. Subtract 7 from 10 to get a remainder of 3.

Step 4: Continue the Process

Bring down another zero to make the remainder 30. Divide 30 by 7, which goes 4 times (since 7 x 4 = 28). Add 4 to the quotient. Subtract 28 from 30 to leave a remainder of 2.

Step 5: Identify the Repeating Pattern

Continue this process by bringing down zeros and dividing until you see the pattern repeat. You will find that the sequence will repeat every 6 digits as 714285. This is the repeating part in the decimal.

Step 6: Write the Repeating Decimal

Express the fraction as a repeating decimal: \( 5 \div 7 = 0.714285714285 \ldots \). Therefore, \( \frac{5}{7} = 0.\overline{714285} \), where 714285 is the repeating sequence.

Key Concepts

Long Division

Long division is a method used to divide numbers, particularly when dividing two numbers results in a decimal. This technique is crucial for converting fractions to decimal form, especially for repeating decimals like \( \frac{5}{7} \). Long division involves the following steps:

• Division: Look at the first number of the dividend (the number you are dividing). If it's smaller than the divisor, add a decimal point and zeros.
• Multiply and Subtract: Multiply the divisor by a number that brings you close to the dividend without exceeding it. Write this number above the division bar. Subtract the result from the dividend, then bring down the next digit of the dividend.
• Repeat the Process: Continue this cycle of division, multiplication, and subtraction, each time bringing down the next digit, until you identify a repeating pattern or reach a remainder of zero.

This process transforms the fraction \( \frac{5}{7} \) into a decimal by repeatedly performing these steps until the pattern \( 714285 \) is observed to repeat, resulting in the repeating decimal \( 0.\overline{714285} \). Long division is fundamental for understanding the precise value of rational numbers expressed as decimals.

Rational Numbers

Rational numbers are quantities that can be expressed as the quotient or fraction \( \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b eq 0 \). This is why they are often referred to as fractions. Any fraction or rational number can be converted to a decimal.
Rational numbers include:

• Positive and negative integers: Examples include \( -5, 2, 0 \)
• Proper and improper fractions: Such as \( \frac{2}{3} \) or \( \frac{7}{4} \).
• Mixed numbers: Like \( 1 \frac{1}{2} \) which can also be expressed as an improper fraction.

The fraction \( \frac{5}{7} \) is a rational number. When converted into a decimal using long division, we discover its repeating decimal form \( 0.\overline{714285} \). Every rational number has a decimal representation, which can be either terminating or repeating. Understanding this concept helps distinguish rational numbers from irrational numbers, which have non-repeating, non-terminating decimals.

Decimal Representation

The decimal representation of numbers is a way of expressing numbers in base 10, using digits 0 through 9. Decimals can be categorized into two groups:

• Terminating decimals: These stop after a finite number of digits. For example, \( \frac{1}{4} = 0.25 \).
• Repeating decimals: These have a sequence of digits that repeat infinitely. A good example is \( \frac{5}{7} = 0.\overline{714285} \).

Repeating decimals are typically represented by a bar over the repeating sequence to denote the pattern continues forever, like \( 0.\overline{714285} \). Understanding decimal representation is important as it allows us to better grasp the size and value of a number. Decimals make it easier to compare these values and use them in calculations. When working on converting fractions to decimals, recognizing whether a decimal will terminate or repeat helps us better interpret and analyze rational numbers.