Rational and Irrational Numbers

Rational Numbers

Rational numbers are numbers that can be expressed as the quotient or fraction p/q of two integers, where q is not equal to zero. In other words, a rational number is any number that can be written in the form p/q, where p and q are integers and q is not equal to zero.

Examples of rational numbers include: 1/2, 3, -5/7, 0, 2.25, and -4. Rational numbers can be positive, negative, or zero. They can also be terminating decimals or repeating decimals.

Irrational Numbers

Irrational numbers are numbers that cannot be expressed as a simple fraction p/q where p and q are integers. In other words, irrational numbers cannot be represented as terminating or repeating decimals.

Examples of irrational numbers include: √2, π, and √7. These numbers cannot be expressed as fractions, and their decimal representations go on forever without repeating.

Study Guide

Here are some key points to remember about rational and irrational numbers:

1. Rational numbers can be expressed as fractions, terminating decimals, or repeating decimals.
2. Irrational numbers cannot be expressed as fractions and have non-repeating, non-terminating decimal representations.
3. All integers are rational numbers (since they can be expressed as a fraction with a denominator of 1).
4. The set of rational numbers is denoted by the symbol ℚ.
5. The set of irrational numbers is denoted by the symbol 𝕀.

Understanding the distinction between rational and irrational numbers is important in various mathematical applications, including algebra, geometry, and calculus.

Practice identifying and working with rational and irrational numbers through exercises and real-world examples to reinforce your understanding of this fundamental concept.

Remember, rational and irrational numbers together form the set of real numbers, which encompasses all numbers on the number line.