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Chapter 4: Problem 30

Find decimal notation. \(\frac{8}{9}\)

Short Answer

Expert verified
0.\bar{8}

Step by step solution

01

Understand the Problem

The goal is to express the fraction \( \frac{8}{9} \) in decimal notation. This requires dividing the numerator (8) by the denominator (9).
02

Perform the Division

Carry out the division of 8 by 9. Since 8 is less than 9, the quotient will be less than 1. Begin with 0 and add a decimal point followed by adding zeroes to the numerator.
03

Long Division

Perform the long division: 8.000 divided by 9. The process is as follows:1. 9 goes into 80 (the initial 8 with an added zero) 8 times.2. 8 times 9 is 72. Subtract 72 from 80, which leaves a remainder of 8.3. Bring down another 0, making it 80 and repeat the process.This will continue indefinitely with the remainder always being 8, creating a repeating decimal.
04

Conclude the Result

Since the remainder continues to be 8 and the division continues endlessly, the solution is a repeating decimal. The final decimal notation for \( \frac{8}{9} \) is 0.888... or simply \( 0.\bar{8} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Fraction to Decimal
Converting a fraction to a decimal is a common task in math. This means rewriting a fraction like \(\frac{8}{9}\) as a decimal number. To do this, we divide the numerator (8) by the denominator (9). It's important to remember that fractions like this, where the denominator is not a power of 10, often result in repeating decimals. Whenever you see a fraction, think about the division sign (\(/\)) and prepare to divide the top number by the bottom one.
Long Division
Long division is the method used to divide bigger numbers or when the division doesn't come out perfectly. Let's apply it to \(\frac{8}{9}\).
  • First, notice that 8 is less than 9.
  • Start with 0 and add a decimal point: 0. and then add a zero to 8 making it 80. Now we divide 9 into 80.
  • 9 goes into 80 exactly 8 times because 9 x 8 is 72.
  • We subtract 72 from 80 which leaves us with a remainder of 8 again.
  • Bring down another zero making it 80 again.
  • Notice the pattern?
This process will keep repeating, meaning the decimal repeats indefinitely.
Repeating Decimal
A repeating decimal is a decimal number that goes on forever in a repeating pattern. For \(\frac{8}{9}\), after performing long division, we observe that the remainder always becomes 8 and restarts the cycle. This results in 0.888... which is written as \(0.\bar{8}\) using a bar notation. The bar above the 8 indicates that 8 repeats indefinitely. Understanding repeating decimals is crucial because it helps us recognize patterns and makes complex divisions simpler.

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