Problem 30 - Introductory Algebra for College Students

Question

Express each rational number as a decimal. $$-\frac{1}{4}$$

Step-by-Step Solution

Answer

The rational number \(-\frac{1}{4}\) expressed as a decimal is -0.25.

Step 1: Identify the Numerator and Denominator

The fraction \(-\frac{1}{4}\) has -1 as the numerator and 4 as the denominator.

Step 2: Perform the Division Operation

To express the given rational number as a decimal, divide the absolute value of the numerator by the denominator. In this case, divide 1 by 4, which gives 0.25.

Step 3: Determine the Sign

Since the numerator of the given rational number was negative, the decimal will also be negative, so the resulting decimal is -0.25.

Key Concepts

Understanding Rational Numbers

Rational numbers are an essential part of mathematics, bridging the gap between integers and real numbers. A rational number is any number that can be expressed as the quotient or fraction of two integers, where the numerator is the integer on top and the denominator is the integer on the bottom. The denominator cannot be zero, as division by zero is undefined. Examples of rational numbers include fractions like \( \frac{1}{2} \), integers like 3 (as \( \frac{3}{1} \)), and even decimals that terminate or repeat, such as 0.25 or 0.333... .

When working with rational numbers, it is crucial to understand that they represent ratios or parts of a whole. This allows us to convert them to decimals for easier comparison or computation in many situations.

Numerators and Denominators: The Building Blocks of Fractions

When dealing with fractions, understanding the components is crucial. The numerator and denominator serve as the building blocks of these expressions. In a fraction \( \frac{a}{b} \), the numerator "a" represents how many parts we have, while the denominator "b" indicates how many parts make a whole.

• Numerator: Positioned above the fraction bar, it counts the number of parts.
• Denominator: Positioned below the fraction bar, it indicates the total number of equal parts the whole is divided into.

For example, in the fraction \( -\frac{1}{4} \), the numerator is -1, showing that we have one part out of four, but in a negative sense, and the denominator is 4, meaning the whole is divided into four equal parts.

The Division Operation: Converting Fractions to Decimals

To convert a fraction into a decimal, the division operation is used. This process involves dividing the numerator by the denominator. This simple mathematical operation helps translate a fraction into a decimal form, which often makes numbers easier to understand or visualize.

Consider the fraction \( -\frac{1}{4} \):

• Ignore the negative sign temporarily and divide 1 by 4.
• The division yields 0.25.

After performing the division, don't forget about the original sign of the fraction. If the fraction is negative, the resulting decimal will also be negative, converting \( \frac{1}{4} \) to \'-0.25\' as a decimal.

Handling Negative Numbers in Decimal Conversion

Negative numbers may seem confusing, but they are simply the opposite of positive numbers, lying to the left of zero on a number line. When converting rational numbers to decimals, it's important to keep track of negative signs.

• If a fraction starts with a negative numerator or denominator, the resulting decimal will also be negative.
• A negative sign in the numerator flips the direction of the value from positive to negative.

In the given example, \( -\frac{1}{4} \), the numerator is negative, which results in a negative decimal when converted. Thus, when performing the conversion, maintain the negative sign throughout the process, resulting in \(-0.25\) as the final value.