Problem 47 - Intermediate Algebra

Question

When expressed as a decimal, is \(\frac{7}{9}\) a terminating or repeating decimal?

Step-by-Step Solution

Answer

\( \frac{7}{9} \) is a repeating decimal with the digit 7 repeating.

Step 1: Understanding Decimals

To determine if a fraction like \( \frac{7}{9} \) is a terminating or repeating decimal, we first need to understand the decimal representations. A terminating decimal has a finite number of digits after the decimal point, while a repeating decimal has one or more repeating digits after the decimal point.

Step 2: Calculate the Decimal

Convert \( \frac{7}{9} \) to a decimal by performing long division of 7 divided by 9. Start dividing 7 by 9, which gives 0. The remainder becomes 70 when you bring down 0 to the tenths place. Divide 70 by 9, which equals 7 with a remainder of again 7, forming a repeating sequence.

Step 3: Identifying the Pattern

Observe the outcome of the division. After performing the division, you notice that the decimal part is 0.777... The digit 7 repeats indefinitely.

Step 4: Classifying the Decimal

Since the decimal form of \( \frac{7}{9} \) is 0.777..., and the digit 7 repeats indefinitely, \( \frac{7}{9} \) is classified as a repeating decimal.

Key Concepts

Fraction to Decimal Conversion

Converting a fraction into a decimal involves dividing the numerator by the denominator. This transformation helps to express values in decimal form, which is often more intuitive for calculations. When we convert fractions like \( \frac{7}{9} \), this process involves quite simple arithmetic. The key is performing the division of 7 by 9. Here, since 7 is smaller than 9, we know its decimal equivalent will be less than 1. Using long division or a calculator, we find the decimal result. Often, this step is fundamental to understanding if your fraction's decimal form ends abruptly (making it a terminating decimal) or keeps going in a repeat, indicative of a repeating decimal.

Long Division

Long division is a method used to divide larger numbers that gives you a clearer understanding of the decimal conversion process. It involves a series of smaller, simplified step-by-step calculations that handle one digit of the dividend at a time. Let's see this in action with \( \frac{7}{9} \):

• Start by writing down the division symbol with 7 inside and 9 outside.
• Recognize that 9 does not fit into 7, so begin with a few decimal places.
• Bring down a 0, making it 70. Divide 70 by 9, which equals 7, with 7 as the remainder.
• Repeat this process. You keep bringing down zeros and dividing, noticing that the remainder cycles back, repeating the process indefinitely.

Once you see the repeating sequence, you can stop as it's evident the decimal won’t terminate.

Terminating Decimals

Terminating decimals come to an end after a certain number of digits. They are neat and concise because they eventually stop, unlike repeating decimals that go on forever. This happens when the fraction's denominator can be divided cleanly (after canceling any common factors) into parts of 10 (i.e., 2 or 5 as the prime factors), like \( \frac{1}{2} = 0.5 \).In comparison, a fraction like \( \frac{7}{9} \) has a denominator with a prime factor of 3, which cannot be expressed as a finite decimal. As a result, this fraction converts to an endless decimal string, 0.777... repeatedly, categorizing it as a repeating decimal. Terminating decimals are simpler to work within arithmetic calculations, thanks to their compact nature.