Problem 28 - Introductory and Intermediate Algebra for College Students 4th

Question

Express each rational number as a decimal. $$\frac{3}{11}$$

Step-by-Step Solution

Answer

The decimal form of \(\frac{3}{11}\) is \(0.\overline{27}\)

Step 1: Conversion from Fraction to Decimal

Conversion from a fraction to a decimal is done by dividing the numerator by the denominator. Hence, we divide 3 by 11: \(3 ÷ 11 = 0.272727...\). Here, '27' is repeating continuously after the decimal.

Step 2: Representing Repeating Decimal

When the same number repeats after the decimal, we represent it as 'bar' over the repeating part. So, 0.272727.. can be written as \(0.\overline{27}\)

Key Concepts

Repeating Decimals

When we talk about repeating decimals, we might encounter numbers that don't just stop but continue indefinitely. These decimals have a specific pattern that continuously repeats. In the case of the fraction \(\frac{3}{11}\), the division results in the decimal 0.272727..., where '27' keeps repeating.
You can recognize a repeating decimal because the digits follow a predictable pattern indefinitely. Instead of writing out the entire sequence each time, mathematicians use a 'bar' notation over the repeating digits as an abbreviated form. Thus, 0.272727... becomes \(0.\overline{27}\). This not only makes it easier to write but also instantly communicates the repeating nature of the sequence. Understanding this concept helps in distinguishing terminating decimals from non-terminating repeating ones.

Fraction to Decimal Conversion

Converting a fraction to a decimal involves a straightforward division process where you divide the numerator (the top number) by the denominator (the bottom number). For example, with \(\frac{3}{11}\), you would divide 3 by 11 to get a decimal result.
The resulting decimal might either terminate–meaning it comes to a complete stop, or as in this case, repeat indefinitely. To perform these calculations, you can use long division. Long division is a method where you divide the numerator by the denominator, writing it out as you would simple division, but after the decimal point, keeping a close eye on repeating numbers.

• Begin with the division of the first digit of the numerator.
• Extend the decimal by adding zeros to the numerator as needed, continuing the division.
• Note the repeating cycle when it occurs.

This helps to break down and accurately convert fractions to their equivalent decimal forms.

Numerator and Denominator Division

The division between the numerator and the denominator is the key step in expressing a rational number as a decimal. The numerator is the number on top, which you are dividing, and the denominator is below, representing how many times the numerator is divided.
Using the example \(\frac{3}{11}\), you would start with 3 as the dividend and 11 as the divisor. The goal is to see how many times 11 can fit into the digits of 3 when expressed as a decimal number.
This calculation process involves:

• Placing the denominator outside and the numerator inside the division bracket.
• Performing the division similarly as with whole numbers, starting with bringing down zeros after a decimal point in the dividend.
• Repeating the division steps until a pattern emerges or it terminates.

Each division step gives a clearer picture of the repeating or terminating nature of the decimal. This concept is foundational in converting fractions to decimals, providing insight into the relationship between the numbers involved.