key term - Proper Fraction

Definition

A proper fraction is a fraction where the numerator is less than the denominator, resulting in a value that is less than 1. Proper fractions are an important concept in the context of partial fractions, as they represent the fractional components of a rational expression that can be broken down into simpler terms.

5 Must Know Facts For Your Next Test

1. Proper fractions are essential in the decomposition of rational expressions using the method of partial fractions.
2. The numerator of a proper fraction must be less than the denominator, ensuring that the resulting value is less than 1.
3. Proper fractions can be added, subtracted, multiplied, and divided like any other fractions, following the standard rules of fraction operations.
4. Proper fractions can be converted to decimal form, where the decimal value will be less than 1.
5. Proper fractions play a crucial role in the integration of rational functions, as the partial fraction decomposition allows for easier integration of the individual terms.

Review Questions

• Explain the relationship between proper fractions and rational expressions in the context of partial fractions.
  • Proper fractions are a key component of the partial fraction decomposition of rational expressions. When a rational expression is broken down into a sum of simpler terms using the method of partial fractions, the resulting terms are often proper fractions. These proper fractions represent the fractional components of the original rational expression, allowing for easier manipulation and integration of the expression.
• Describe the properties of proper fractions and how they differ from improper fractions.
  • The defining characteristic of a proper fraction is that the numerator is less than the denominator, resulting in a value that is less than 1. This is in contrast to an improper fraction, where the numerator is greater than or equal to the denominator, leading to a value that is greater than or equal to 1. Proper fractions can be easily converted to decimal form, where the decimal value will be less than 1, while improper fractions may result in a decimal value greater than 1. Proper fractions follow the standard rules of fraction operations, such as addition, subtraction, multiplication, and division.
• Analyze the role of proper fractions in the integration of rational functions using the method of partial fractions.
  • The decomposition of a rational function into a sum of proper fractions through the method of partial fractions is a crucial step in the integration of rational functions. By breaking down the original rational expression into a sum of simpler terms, each of which is a proper fraction, the integration process becomes much more straightforward. The individual proper fraction terms can be integrated using standard integration techniques, and the results can then be combined to obtain the final antiderivative of the original rational function. This approach leverages the properties of proper fractions to simplify the integration of complex rational expressions.

Related terms

Improper Fraction:

An improper fraction is a fraction where the numerator is greater than or equal to the denominator, resulting in a value that is greater than or equal to 1.

Mixed Number: A mixed number is a combination of a whole number and a proper fraction, such as 3 1/4 or 7 2/3.

Rational Expression:

A rational expression is a quotient of two polynomials, which can be broken down into a sum of proper fractions using the method of partial fractions.