A fraction is a mathematical representation of a part of a whole. It is used to express the relationship between a numerator, which represents the number of parts, and a denominator, which represents the total number of equal parts that make up the whole.
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Fractions can be used to represent parts of a whole, measurements, and ratios.
Fractions can be classified as proper, improper, or mixed, depending on the relationship between the numerator and denominator.
Operations with fractions, such as addition, subtraction, multiplication, and division, follow specific rules and procedures.
Simplifying fractions involves reducing the numerator and denominator to their lowest common terms.
Rational expressions are fractions where the numerator and denominator are polynomials, and they can be manipulated using the same rules as regular fractions.
Review Questions
Explain how the concepts of numerator and denominator are used to represent a fraction.
The numerator of a fraction represents the number of parts being considered, while the denominator represents the total number of equal parts that make up the whole. For example, in the fraction $\frac{3}{5}$, the numerator 3 indicates that we are considering 3 parts, and the denominator 5 indicates that the whole is divided into 5 equal parts. The fraction $\frac{3}{5}$ represents the relationship between the number of parts being considered (3) and the total number of equal parts that make up the whole (5).
Describe the role of fractions in the context of rational expressions.
Rational expressions are fractions where the numerator and denominator are polynomials. In the context of rational expressions, fractions are used to represent the relationship between the numerator and denominator, which can be manipulated using the same rules and procedures as regular fractions. For example, in the rational expression $\frac{2x^2 + 3x - 1}{x^2 - 4}$, the numerator and denominator are both polynomials, and the fraction represents the relationship between these two polynomial expressions.
Analyze how the properties of fractions, such as simplification and operations, can be applied to rational expressions.
The properties of fractions, such as simplification and operations, can be directly applied to rational expressions. To simplify a rational expression, you can reduce the numerator and denominator to their lowest common terms, just as you would with a regular fraction. Similarly, the same rules for addition, subtraction, multiplication, and division of fractions can be used when working with rational expressions. For instance, to add two rational expressions, you would find a common denominator and then add the numerators. These fundamental properties of fractions are crucial in manipulating and simplifying rational expressions, which are essential in the context of 1.6 Rational Expressions.