Problem 25 - Introductory and Intermediate Algebra for College Students 4th

Question

Express each rational number as a decimal. $$\frac{7}{8}$$

Step-by-Step Solution

Answer

The rational number \(\frac{7}{8}\) as a decimal is 0.875.

Step 1: Identify the numerator and the denominator

From the given fraction \(\frac{7}{8}\), we can identify that 7 is the numerator and 8 is the denominator.

Step 2: Perform the division

Divide the numerator by the denominator, i.e., calculate \( \frac{7}{8} \).

Step 3: Write the result as a decimal

After performing the division, express the result in decimal notation.

Key Concepts

Numerator and Denominator

Understanding the roles of the numerator and denominator is crucial when working with fractions and converting them to decimals. In a fraction, the numerator, which is located above the fraction line, represents the number of equal parts that are being considered, while the denominator, situated below the line, signifies how many of those equal parts make up a whole.

For instance, in the fraction \( \frac{7}{8} \), 7 is the numerator, indicating 7 parts are taken out of the 8 parts that make up a whole. When attempting to convert a fraction to a decimal, it is the numerator that we effectively 'divide' by the denominator to determine what portion of a single unit we have. With the understanding of these roles, the conversion process becomes a straightforward task of division.

Division of Fractions

Division is the process used to convert a fraction into decimal format. This involves dividing the numerator by the denominator. The approach mirrors the operation we conduct with natural numbers: the numerator is treated as the dividend and the denominator as the divisor.

In essence, to transform \( \frac{7}{8} \) into a decimal, calculate 7 \( \div \) 8. If the division doesn’t result in a whole number, you will end up with a decimal point followed by a string of numbers, representing the fractional part. This decimal outcome might be exact or repeating, depending on whether the fraction simplifies to a finite or repeating decimal. For fractions that do not simplify neatly, long division can be applied to achieve an approximation to the desired degree of accuracy.

Decimal Notation

Decimal notation is a way of writing numbers that combines whole numbers and fractions of ten. We use a decimal point to separate the whole number part from the fractional part. Each place to the right of the decimal point represents a successively smaller unit: tenths, hundredths, thousandths, and so on.

When converting a fraction like \( \frac{7}{8} \) to decimal notation, the goal is to express the value as a combination of whole numbers and decimal fractions. After division, you would write out the whole numbers as is, and then add the decimal fraction part, for example, 0.875. Decimals are a very efficient way to portray exact values or close approximations of fractions, facilitating easier calculations and comparisons in various fields of mathematics and real-life applications.