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Chapter 9: Problem 79

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

Short Answer

Expert verified
Rational numbers can be written as fractions with integers. Irrational numbers cannot be written as fractions.

Step by step solution

01

Define Rational Numbers

A rational number is a number that can be expressed as the quotient or fraction \(\frac{p}{q}\), where \(p\) and \(q\) are integers and \(q eq 0\). Examples of rational numbers include 1/2, 3, and -5.
02

Define Irrational Numbers

An irrational number is a number that cannot be expressed as a fraction \(\frac{p}{q}\) where \(p\) and \(q\) are integers and \(q eq 0\). Examples include \(\sqrt{2}\), \(\frac{\pi}{e}\), and \(e\).
03

Comparison

The primary difference between rational and irrational numbers is that rational numbers can be written as fractions with integer numerators and non-zero integer denominators, while irrational numbers cannot be accurately represented as simple fractions.
04

Summarize with Examples

For clarity, summarize with examples: \(\frac{3}{4}\) is rational because it is the fraction of two integers, and \(\sqrt{3}\) is irrational because it cannot be precisely written as a fraction.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rational Numbers Definition
Rational numbers are those that can be written as a fraction, where both the numerator (top number) and the denominator (bottom number) are integers, and the denominator is not zero. This means that any number you can write as \(\frac{p}{q}\) where \p\ and \q\ are whole numbers (with \q\ not being zero) is a rational number.

For example:
  • \(\frac{3}{4}\) - a fraction with whole numbers
  • \5\ - which can be written as \(\frac{5}{1}\)
  • \-7\ - can be written as \(\frac{-7}{1}\)
Rational numbers can also show up as repeating or terminating decimals. For instance, \0.75\ can be written as \(\(\frac{3}{4}\)\) and \0.666...\ can be written as \(\frac{2}{3}\).

This makes them easy to identify once you know they must fit this fractional form.
Irrational Numbers Definition
Irrational numbers, on the other hand, are numbers that cannot be written as a simple fraction of two integers. They have decimals that go on forever without repeating.

Some well-known examples include:
  • \(\text{\sqrt{2}}\)\ - the square root of 2
  • \(\pi\)\ - approximately 3.14159...
  • \(\text{e}\)\ - Euler's number, approximately 2.71828...
These numbers cannot be pinned down as exact fractions between two integers. They are continuous, with never-ending, non-repeating decimal parts. Because of this property, irrational numbers are tricky to work with in comparison to rational numbers.
Comparison of Rational and Irrational Numbers
Understanding the difference between rational and irrational numbers comes down to knowing their definitions:
  • Rational numbers can be expressed as \(\frac{p}{q}\) with integers
  • Irrational numbers cannot be written as \(\frac{p}{q}\) with integers
This difference means that rational numbers can be represented as simple, finite or repeating decimals, whereas irrational numbers are marked by non-terminating and non-repeating decimal expansions.

A few examples to clarify:
  • \(\frac{5}{2}\) is rational - it equals 2.5
  • \(\text{\pi}\) is irrational - it approximately equals 3.14159...
  • \(\frac{7}{3}\) is rational - it equals 2.333...
  • \(\text{\text{\sqrt{2}}\) is irrational - approximately 1.4142...
Recognizing these differences helps in categorizing and working with various types of numbers in mathematics more effectively.

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Most popular questions from this chapter

Find the amount of the exhibition and sales space devoted to exhibits. Round to the nearest hundred. Rolex opened a new. . .8,600 square-foot exhibition and sales space, the Rolex Experience, along the Bund, the riverfront avenue in Shanghai. About two-thirds of the surface area is devoted to exhibits, tracing the history of Oyster and other watches that have established Rolex's popularity. (Source: www.nytimes.com, Feb. 21, 2012)

\((2-3 i)-(6-9 i)\)

The voltage drops in an AC circuit are \(15-26.6491 i\) volts, \(-9.7294-6.8813 i\) volts, and \(9.7452-19.7729 i\) volts. Find the sum of these voltages.

\(\sqrt{8^{2}-4(4)(8)}\)

\(7 a^{2}=-252\)
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