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Chapter 1: Problem 4

Show that \(\sqrt{3}\) and \(\sqrt[3]{2}\) are not rational.

Short Answer

Expert verified
\(\sqrt{3}\) and \(\sqrt[3]{2}\) are irrational numbers.

Step by step solution

01

Assume for Contradiction

Let's first assume that \(\sqrt{3}\) is a rational number. By definition, a rational number can be expressed as a fraction \(\frac{a}{b}\), where \(a\) and \(b\) are integers, and \(b eq 0\). Additionally, \(\frac{a}{b}\) is in its simplest form, meaning that \(a\) and \(b\) have no common factors other than 1.
02

Squaring Both Sides

Given \(\sqrt{3} = \frac{a}{b}\), we square both sides to eliminate the square root: \((\sqrt{3})^2 = \left(\frac{a}{b}\right)^2\). This simplifies to \(3 = \frac{a^2}{b^2}\). Therefore, \(3b^2 = a^2\).
03

Resulting Contradiction for \(\sqrt{3}\)

Since \(3b^2 = a^2\), \(a^2\) is a multiple of 3, implying \(a\) itself must also be a multiple of 3 (if a number squared is divisible by 3, so is the number itself). Let \(a = 3k\) for some integer \(k\). Substitute \(a = 3k\) into the equation, resulting in \(3b^2 = (3k)^2 = 9k^2\). This simplifies to \(b^2 = 3k^2\), indicating \(b^2\) is also a multiple of 3. Thus, \(b\) is a multiple of 3. Since both \(a\) and \(b\) are multiples of 3, they have a common factor of 3, contradicting the initial assumption that \(\frac{a}{b}\) was in simplest form. Therefore, \(\sqrt{3}\) is not rational.
04

Assume for Contradiction (Cube Root)

Now, assume \(\sqrt[3]{2}\) is a rational number. It can be expressed as \(\frac{c}{d}\), where \(c\) and \(d\) are integers with no common factors other than 1. \(deq 0\).
05

Cubing Both Sides

Given \(\sqrt[3]{2} = \frac{c}{d}\), cube both sides: \((\sqrt[3]{2})^3 = \left(\frac{c}{d}\right)^3\). This simplifies to \(2 = \frac{c^3}{d^3}\), leading to \(2d^3 = c^3\).
06

Resulting Contradiction for \(\sqrt[3]{2}\)

The equation \(2d^3 = c^3\) indicates that \(c^3\) is even, so \(c\) must be even (since only even numbers can produce an even cube). Let \(c = 2m\) for some integer \(m\). Substitute \(c = 2m\) into the equation, yielding \(2d^3 = (2m)^3 = 8m^3\) and simplifying to \(d^3 = 4m^3\). Therefore, \(d^3\) is even, and \(d\) must also be even, which means \(c\) and \(d\) have a common factor of 2. This contradicts the assumption that \(\frac{c}{d}\) is in simplest form. Thus, \(\sqrt[3]{2}\) is not rational.

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Key Concepts

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

Rational Numbers
Rational numbers are numbers that can be expressed as a fraction \( \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b eq 0 \). This definition is key to understanding the difference between rational and irrational numbers.

  • The integers \( a \) and \( b \) must be whole numbers.
  • The fraction must be in its simplest form, meaning that \( a \) and \( b \) have no common factors other than 1.
  • Examples include fractions like \( \frac{1}{2} \), whole numbers like 3 (which is \( \frac{3}{1} \)), and repeating decimals like 0.333... (which is \( \frac{1}{3} \)).
Understanding rational numbers helps us grasp why certain numbers, such as the square root of 3 or the cube root of 2, cannot be rational.
These are called irrational numbers because they cannot be expressed as a fraction of two integers.
Contradiction Method
The contradiction method is a powerful tool in mathematics for proving a statement by demonstrating that assuming the opposite leads to a contradiction.

To use this method:
  • Assume the opposite of what you want to prove. For example, assume \( \sqrt{3} \) is rational.
  • Logically work through the mathematical consequences of this assumption.
  • Reach a contradiction that disproves your assumption, thus proving the original statement.
In the case of showing that \( \sqrt{3} \) is irrational, we started by assuming it was rational. Then, by analyzing the implications of this assumption and finding that it leads to a number not being in simplest form, we proved that our initial assumption was incorrect.
Thus, \( \sqrt{3} \) must be irrational.
Square Roots
Square roots are the mathematical operation of finding a number that, when multiplied by itself, equals the given number.

For example:
  • The square root of 4 is 2 because \( 2 \times 2 = 4 \).
  • Not all numbers have clean square roots. For example, \( \sqrt{3} \) cannot be simplified to a whole number.
When working with square roots in the context of rational and irrational numbers, identify if the root can be exactly expressed as a fraction. In cases where it cannot, like \( \sqrt{3} \), it is considered irrational.
Applying the contradiction method reveals why some square roots, such as \( \sqrt{3} \), cannot be written as a fraction of two integers.
Cube Roots
Cube roots work similarly to square roots but involve finding a number that, when multiplied by itself three times, equals the original number.

To understand cube roots:
  • The cube root of 8 is 2 because \( 2 \times 2 \times 2 = 8 \).
  • Not all numbers have a straightforward cube root. For instance, \( \sqrt[3]{2} \) cannot be simplified into a clean integer.
In evaluating cube roots regarding rationality, it's essential to test if the root can be expressed as a fraction. Using the contradiction method, if assuming \( \sqrt[3]{2} \) could be rational leads to contradictory consequences, then \( \sqrt[3]{2} \) cannot be a rational number.
This reasoning helps us understand that some cube roots, like \( \sqrt[3]{2} \), are inherently irrational.

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Most popular questions from this chapter

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