Rational Number

Definition

A rational number is a number that can be expressed as a ratio or fraction of two integers, where the denominator is not zero. Rational numbers are an important concept in both the topics of Rational and Irrational Numbers, as well as Integer Exponents and Scientific Notation.

5 Must Know Facts For Your Next Test

1. Rational numbers can be positive, negative, or zero, and include whole numbers, fractions, and terminating or repeating decimal representations.
2. All integers are rational numbers, as they can be expressed as a fraction with a denominator of 1.
3. The set of rational numbers is denoted by the symbol $\mathbb{Q}$, and is a subset of the real number system.
4. Rational numbers are important in scientific notation, as the exponents used are always integers, which are a type of rational number.
5. Rational numbers can be added, subtracted, multiplied, and divided, and the result will always be a rational number.

Review Questions

• Explain how rational numbers are defined and how they relate to the set of integers.
  • Rational numbers are numbers that can be expressed as a ratio or fraction of two integers, where the denominator is not zero. All integers, which are whole numbers, are considered rational numbers because they can be expressed as a fraction with a denominator of 1. This means that the set of integers is a subset of the set of rational numbers, and all integers are also rational numbers.
• Describe the importance of rational numbers in the context of scientific notation and integer exponents.
  • Rational numbers are crucial in the topic of scientific notation and integer exponents because the exponents used in scientific notation are always integers, which are a type of rational number. This allows for the representation of very large or very small numbers in a compact and standardized format. Additionally, the laws of exponents, such as multiplication and division, rely on the properties of rational numbers to ensure that the resulting values are also rational numbers.
• Analyze how the properties of rational numbers, such as the ability to be added, subtracted, multiplied, and divided, contribute to their importance in mathematics and problem-solving.
  • The key properties of rational numbers, including the ability to perform arithmetic operations such as addition, subtraction, multiplication, and division, make them essential in a wide range of mathematical applications and problem-solving contexts. These operations allow for the manipulation and transformation of rational numbers, enabling the simplification of expressions, the solving of equations, and the modeling of real-world scenarios involving quantities that can be represented as ratios or fractions. The versatility of rational numbers and the predictability of their behavior under these operations are fundamental to the development and application of mathematical concepts and techniques across various fields, from algebra and calculus to statistics and physics.

Related terms

Integer:

An integer is a whole number, including positive, negative, and zero numbers, that can be expressed without a fractional component.

Fraction:

A fraction is a way of representing a part of a whole, expressed as a ratio of two integers, the numerator and the denominator.

Irrational Number:

An irrational number is a number that cannot be expressed as a ratio of two integers, and has a non-repeating, non-terminating decimal representation.