Problem 105 - Introductory and Intermediate Algebra for College Students 4th

Question

Describe the difference between a rational number and an irrational number.

Step-by-Step Solution

Answer

Rational numbers can be expressed as a ratio or fraction of two integers, whereas irrational numbers cannot be expressed as a ratio of two numbers and aren't exact fractions. Rational numbers either terminate or repeat in decimal form, while irrational numbers do not; their decimal expansions go on forever without repeating.

Step 1: Describe Rational Numbers

Rational numbers are numbers that can be expressed as the quotient or fraction \( p/q \) of two integers, with the denominator \( q \) not equal to zero. Since integers are also rational, we might say that a rational number is a number which can be represented in a ratio form.

Step 2: Describe Irrational Numbers

Irrational numbers cannot be expressed as a ratio between two numbers and it is not an exact fraction. In other words, they cannot be written as \( p/q \) with \( p \) and \( q \) as integers and \( q \) not equal to zero. They are usually the result of non-repeating, non-terminating decimals.

Step 3: Differentiate Between the Two

The fundamental difference between rational and irrational numbers is that rational numbers can be expressed as a fraction with integers in both the numerator and the denominator while irrational numbers cannot be expressed in such a way. Sometimes another main difference is that while rational numbers either terminate or repeat in decimal form, irrational numbers do neither; their decimal expansions continue without repeating indefinitely.