Problem 105 - Introductory Algebra for College Students

Question

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

Step-by-Step Solution

Answer

Rational numbers can be expressed as fractions and their decimals either terminate or repeat. On the other hand, irrational numbers cannot be expressed as fractions and their decimals neither terminate nor repeat.

Step 1: Definition of Rational Numbers

Rational numbers can be defined as a number which can be expressed in the form of \(p/q\) where \(p\) and \(q\) are integers and \(q\) is not equal to zero. For example, numbers such as 1/2, 2, -5, etc. are rational.

Step 2: Properties of Rational Numbers

Rational numbers have two key properties: firstly, they can be expressed as a fraction, and secondly, when rational numbers are expressed as a decimal it either terminates after a finite number of digits or begins to repeat the same finite set of digits.

Step 3: Definition of Irrational Numbers

Irrational numbers are the numbers which cannot be expressed in the form of \(p/q\), where \(p\) and \(q\) are integers and \(q\) is not equal to zero. For example, numbers such as \(\sqrt{2}\), \(\pi\) etc., are irrational.

Step 4: Properties of Irrational Numbers

Irrational numbers also have some unique properties - they cannot be expressed as a fraction and when expressed as a decimal, they neither terminate nor repeat.