Question

Use a calculator to find the decimal form of the rational number. If the number is a nonterminating decimal, write the repeating pattern.\(\frac{14}{111}\)

Short answer

The decimal form of \(\frac{14}{111}\) is 0.126 with '126' as the repeating pattern.

Step-by-step solution

Set Up the Division Operation

Begin by setting up the division operation. Here, the numerator 14 is the dividend and the denominator 111 is the divisor, meaning you'll divide 14 by 111.

Perform Long Division

Perform the long division operation. Since 111 does not divide evenly into 14, add zeros to the right of 14 (as many as needed until you observe a repeating pattern or until it terminates). Remember that adding zeros to the right does not change the value of the number.

Identify the Decimal Form and Any Repeating Pattern

After you have completed the division operation, write down the decimal and if it is a non-terminating decimal, identify and denote the repeating pattern.

Key concepts

Long Division

Long division is a traditional method of dividing two numbers to find the quotient in decimal form. It is an essential skill for handling divisions that result in decimal numbers, especially when the divisor is larger than the dividend. Long division is best understood through step-by-step execution:

1. **Divide:** Begin by determining how many times the divisor can fit into the dividend or the selected portion of it. 2. **Multiply:** Multiply the divisor by the quotient obtained from the previous step and write this product below the dividend or the selected portion of it. 3. **Subtract:** Subtract the product from the dividend or selected portion, and bring down the next digit, if applicable. 4. **Repeat:** Repeat these steps until reaching a remainder of zero or a limited number of decimal places.

• When the dividend is smaller than the divisor, bring down zeros after the decimal point to continue the process.
• This method provides clarity on whether a decimal terminates or repeats.

For instance, when dividing 14 by 111, you'll note that 111 can't go into 14 even once, requiring you to adjust by considering it as 140. This illustrates how long division allows for the systematic breaking down of numbers into manageable parts.

Repeating Decimals

When performing long division, some numbers might not result in a clean termination but rather into a repeating decimal pattern. Repeating decimals are decimals that have a portion of digits that repeat endlessly.

Understanding repeating decimals is important because they tell us about the nature of certain rational numbers:

• Repeating decimals are indicated using a bar over the digits that repeat, such as \(0.126\ar{126}\).
• They arise when the division results in a remainder that repeats during the long division process.
• These decimals might look different, but they often represent a recurring cycle of numbers.

In the division of 14 by 111, you realize a repeating pattern emerges. The quotient eventually stabilizes into a repeated group of numbers that reflect a cyclic pattern in its division process.

Rational Numbers

Rational numbers are numbers that can be expressed as the fraction of two integers, where the denominator is not zero. Every rational number can be represented as either a terminating or repeating decimal, which unveils a clear character of the number:

- **Terminating decimals** are decimals that have a finite number of digits after the decimal point.- **Repeating decimals** have one or more repeating sequences of digits.
Rational numbers are central in mathematics since they can be precisely represented as either form of decimals. It is key to recognize

• that operations like division can convert these fractions into decimals.
• The division can expose underlying repeating features as seen in fractions like \(\frac{14}{111}\).

Understanding that every fraction is a rational number helps us discern its behavior when converted, as through the use of long division.