Rational Numbers

Definition

Rational numbers are numbers that can be expressed as a ratio of two integers, where the denominator is not equal to zero. They include both whole numbers and fractions, and are a subset of the real number system.

5 Must Know Facts For Your Next Test

1. Rational numbers can be expressed in the form $\frac{a}{b}$, where $a$ and $b$ are integers and $b \neq 0$.
2. All integers are rational numbers, as they can be expressed as a fraction with a denominator of 1.
3. Fractions, including both positive and negative fractions, are examples of rational numbers.
4. Decimal representations of rational numbers either terminate or repeat infinitely.
5. The set of rational numbers is closed under the four basic arithmetic operations: addition, subtraction, multiplication, and division (by a non-zero rational number).

Review Questions

• Explain how rational numbers are related to integers and fractions.
  • Rational numbers include both integers and fractions. All integers can be expressed as a fraction with a denominator of 1, making them a subset of rational numbers. Fractions, which have a numerator and denominator, are also considered rational numbers as long as the denominator is not equal to zero. Therefore, the set of rational numbers encompasses both whole numbers (integers) and fractional quantities.
• Describe the properties of rational numbers in the context of simplifying expressions with roots.
  • When simplifying expressions with roots, rational numbers play a crucial role. The square root of a rational number is also a rational number, as long as the radicand (the number under the square root) is a perfect square. Additionally, the product or quotient of rational numbers is also a rational number. This property allows for the simplification of expressions involving roots of rational numbers, as the resulting expression will still be a rational number.
• Analyze how the properties of real numbers, specifically the closure property, apply to rational numbers.
  • The set of real numbers is closed under the four basic arithmetic operations: addition, subtraction, multiplication, and division (by a non-zero real number). This property also applies to the subset of rational numbers. The set of rational numbers is closed under these same operations, meaning that the result of performing any of these operations on two rational numbers will always yield another rational number. This closure property is essential in understanding and working with rational numbers, as it allows for the manipulation and simplification of expressions involving rational numbers.

Related terms

Integer: An integer is a whole number, positive, negative, or zero, that has no fractional part.

Fraction:

A fraction is a numerical quantity that is not a whole number, expressed as the quotient of two integers with the second one not equal to zero.

Real Number: A real number is a number that can be represented on the number line, including rational numbers and irrational numbers.