Question

Show how to modify Cantor's second diagonal argument so that the real number produced is always irrational. (Hint: There is more than one way to do this.)

Short answer

Alter digits in Cantor's diagonal method to prevent terminating or repeating sequences.

Step-by-step solution

Understand Cantor's Diagonal Argument

Cantor's diagonal argument demonstrates that the set of all real numbers between 0 and 1 is uncountable. This argument assumes you have a list of real numbers and constructs a new real number by changing the nth digit of the nth number to ensure it differs from all numbers in the list.

Identify the Issue with Rational Numbers

The real number constructed by the usual diagonal method might terminate or repeat, making it rational. For example, if you change the nth digit to a '9' and the other digits are zeros after it, the number could be rational.

Modify the Construction to Avoid Rationality

One simple modification is to ensure that every digit is altered such that it doesn't create a terminating or repeating sequence. For example, change the digit to '1' if the original digit is anything but '1', and change it to '2' if it is '1'.

Implement the Modified Construction

Create the new number by taking each digit in the nth decimal place of the nth number. If it's a '1', change it to '2'. If it's not '1', change it to '1'. This ensures that no digit repeats consistently enough to result in a pattern that could make the number rational.

Addressing Edge Cases

Consider cases where sequences may appear rational due to repetitive conversions. The choice to convert '1' to '2' and all others to '1' will maintain the irrationality because it avoids patterns like recurring threes or terminal zeros.

Key concepts

Irrational Numbers

Irrational numbers are fascinating creatures in the world of mathematics. They cannot be expressed as a simple fraction. Unlike rational numbers, which can be written as a ratio of two integers, irrational numbers have non-repeating and non-terminating decimal expansions. Some well-known examples of irrational numbers include

• \( \pi \), which is the ratio of a circle's circumference to its diameter.
• \( \sqrt{2} \), which is the length of the diagonal of a square with side length 1.

Understanding irrational numbers helps in grasping how vast and complex the set of real numbers is since irrational numbers make up a significant part of this set. When applying Cantor’s Diagonal Argument, one of the goals is to ensure that the created number remains in this category by having a non-repeating, non-terminating decimal form.

Real Numbers

Real numbers encompass both the rational and irrational numbers, stretching across an infinite line of values. Imagine a continuous line without any breaks or gaps—that's what the set of real numbers looks like. They include:

• Whole numbers like 1, 2, and 3
• Fractions like \(\frac{1}{2}\) or \(\frac{3}{4}\)
• Irrational numbers such as \(\pi\) and \(\sqrt{2}\)

Because they include every possible number that can occur on the number line, they are used to measure continuous quantities. In the context of Cantor's work, understanding real numbers is crucial, as his diagonal argument focuses on proving that even within the seemingly dense set of numbers between, say, 0 and 1, there are still more numbers than can be counted.

Uncountability

In mathematics, uncountability is a concept that shows just how large certain sets can be. A set is countable if you can list its elements in a sequence (like counting sheep one by one). The set of real numbers, however, is uncountable. This is due to the fact that there are infinitely more real numbers than natural numbers. Cantor's diagonal argument is a famous method that proves the uncountability of the set of real numbers. It shows that no matter how you try to list all the real numbers, you will always find some number that is not included in your list. This discovery was groundbreaking, as it revealed that some infinities are bigger than others! By modifying Cantor's argument slightly, it ensures that the newly formed real numbers are irrational, therefore reinforcing the uncountability of the set without including any rational, repeatable structures.

Diagonalization Method

The diagonalization method is a powerful tool introduced by Georg Cantor to demonstrate the uncountability of the real numbers. The basic idea is to assume that all numbers can fit into a list and then show a contradiction by constructing a number that cannot appear on that list. In practice, you start with a hypothetical list of real numbers, each having an infinite decimal expansion. For each number in the list, look at its nth decimal place. Then, create a new number that differs in at least that one digit for each corresponding number in the list. By systematically changing each digit, you ensure that the new number does not match any number on the list. To produce a number that is surely irrational, adjust each digit to avoid creating a repeating or terminating sequence. For example, change all '1's to '2's and the rest to '1's. This careful alteration prevents rationality, ensuring the number generated by the diagonal method is irrational. Hence, Cantor's diagonalization method ingeniously demonstrates how some infinities are larger because you can always find a number not listed in any supposed complete list of real numbers.