Classify and operate on real numbers with confidence.
The real numbers are organized in a hierarchy: natural numbers â whole numbers â integers â rational numbers â real numbers. Rational numbers can be expressed as a ratio p/q (q â 0), while irrational numbers cannot â their decimal expansions never terminate or repeat.
Real Number Hierarchy
Each set is contained within the next larger set.
Irrational numbers are real but cannot be written as a simple fraction. Common examples include square roots of non-perfect squares and transcendental numbers like Ī.
The sum or product of a rational and an irrational number is always irrational. For example, 2 + â3 is irrational.
Classify each: â7 is an integer (and rational). 3/4 is rational but not an integer. â5 is irrational. 0.333âĻ = 1/3 is rational.
Remember This!
A decimal that repeats is always rational. Write 0.777âĻ = 7/9 by using the equation x = 0.777âĻ, then 10x = 7.777âĻ, subtract: 9x = 7, so x = 7/9.
Which of the following numbers is irrational?