Real Numbers as Union of Rational and Irrational
Real numbers as the union of rational and irrational numbers is a foundational concept in Grade 11 Algebra 1 enVision Chapter 1. Every real number is either rational (like 100 or 17/4) or irrational (like -sqrt(15) or pi), but never both. The set of real numbers represents every point on the number line — the complete union of these two distinct, non-overlapping sets. Rational numbers have terminating or repeating decimal expansions; irrational numbers have non-terminating, non-repeating decimals. All numbers students encounter in Algebra 1 are real numbers.
Key Concepts
Real numbers are numbers that are either rational or irrational. The set of real numbers is formed by combining the set of rational numbers and the set of irrational numbers.
Common Questions
What are real numbers?
Real numbers are all numbers that can be represented on the number line. Every real number is either rational or irrational.
Is 100 rational or irrational?
100 is rational because it can be written as 100/1, a ratio of two integers. It is also a real number.
Is -sqrt(15) rational or irrational?
Irrational. sqrt(15) cannot be expressed as a fraction of integers, so -sqrt(15) is irrational — but still a real number.
Can a number be both rational and irrational?
No. Rational and irrational are mutually exclusive categories. Every real number is one or the other, never both.
Why is pi irrational?
Pi has a non-terminating, non-repeating decimal expansion (3.14159...). This means it cannot be expressed as a fraction of two integers.
What numbers are NOT real numbers?
Imaginary numbers like sqrt(-1) = i are not real numbers. They exist outside the real number line in the complex number system.