Question

Prove that the set \(\mathbb{C}\) of complex numbers is uncountable.

Short answer

The set of complex numbers \(\mathbb{C}\) is uncountable, as shown by constructing a complex number that doesn't correspond to any natural number, contradicting the assumption that \(\mathbb{C}\) is countable.

Step-by-step solution

Understanding Countability

Recall the definition of countability. A set is countable if its elements can be placed in one-to-one correspondence with the set of natural numbers \(\mathbb{N}\). In other words, a set is countable if we can enumerate its elements as \(a_1, a_2, a_3, \ldots\). If there is no possible way to do this enumeration, then the set is uncountable.

Argue by Contradiction

Assume for the sake of contradiction that \(\mathbb{C}\) is countable. This would mean there exists a surjective (onto) function \(f:\mathbb{N}\rightarrow \mathbb{C}\), associating each natural number with a unique complex number.

Define a Subset of Complex Numbers

Consider the set of all complex numbers with rational real and imaginary parts, denote it as \(Q[i]\). This set is countable, because both the set of rational numbers \(\mathbb{Q}\) and the set \(\mathbb{Q}[i]=\{a+bi:a,b\in \mathbb{Q}\}\) are countable. This is because we can establish a bijection between them and the set of pairs of natural numbers, which is countable.

Construct a Complex Number Not Mapped by the Function

Now, construct a complex number \(z\) with non-rational coordinates, such that \(z\not \in \{f(n):n\ge1\}\). Given our function \(f\), we know that there should be no natural number \(n\) such that \(f(n)=z\). If \(\mathbb{C}\) had countability, such a number could not exist, because every complex number should be pairable with a natural number.

Conclusion

Since we have found a complex number \(z\) which is not part of the range of \(f\), we have a contradiction assuming \(\mathbb{C}\) was a countable set. Therefore, we can conclude that the set of all complex numbers \(\mathbb{C}\) is uncountable.

Key concepts

Countability in Mathematics

Countability is a fundamental concept in set theory, a branch of mathematical logic that deals with the size and comparability of sets. A set is considered countable if we can list its elements in a sequence that pairs each element with exactly one natural number. Imagine you were assigning each member of a team a unique jersey number; if you can do this without any jerseys left over or any players without jerseys, the team is countably infinite.

For example, the set of natural numbers \( \mathbb{N} \) is countable because we can list them as \(1, 2, 3, \ldots \). Similarly, the set of rational numbers is also countable, even though it seems 'bigger' since between any two natural numbers there are infinitely many rational numbers. However, through clever arrangements, like pairing each positive and negative rational number and arranging them in a certain sequence, we can also enumerate them using the natural numbers.

Once a set is demonstrated to be uncountable, it implies that its cardinality, or 'size', is larger than that of the set of natural numbers. This revelation has profound implications in various fields of mathematics, including analysis, probability, and even computer science.

Complex Numbers

Complex numbers are an extension of the real numbers and are fundamental in advanced mathematics and engineering. A complex number is composed of a real part and an imaginary part and is commonly written in the form \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit satisfying \( i^2 = -1 \).

Visualizing Complex Numbers

The set of all complex numbers, denoted by \( \mathbb{C} \), can be visualized as a two-dimensional plane. This is known as the complex plane, where the x-axis represents the real part and the y-axis represents the imaginary part of the complex numbers. Each point on this plane corresponds to a unique complex number.

Applications of Complex Numbers

Complex numbers are incredibly useful in various fields. In physics, they simplify the mathematics of quantum mechanics and electrical engineering. In control systems, they help design stable system responses. As you can see, the leap to understand and use complex numbers unlocks a higher realm of both mathematical sophistication and practical application.

Proof by Contradiction

Proof by contradiction is a powerful and often used technique in mathematics to establish the truth of a statement. The essence of this method lies in assuming the opposite of what you want to prove, and then showing that this assumption leads to an illogical result, a contradiction. When you hit that wall of impossibility, that very wall confirms the original statement must be true, as the opposite scenario is untenable.

A classic example is proving the irrationality of the square root of 2 by assuming it can be expressed as a fraction of two integers and then showing this leads to the conclusion that the same number is both odd and even, which is absurd. The trick here is not only to recognize the contradiction but to make sure that the steps leading to it are logically sound and that no other possibilities exist that could avoid the contradiction.

Rational Numbers

Rational numbers, denoted as \( \mathbb{Q} \), are numbers that can be expressed as a fraction where both the numerator and the denominator are integers. The denominator cannot be zero because division by zero is undefined. The set includes positive fractions, negative fractions, and whole numbers, which can be regarded as fractions with a denominator of one.

Properties of Rational Numbers

Rational numbers are dense, meaning between any two rational numbers there’s another rational number. They can also be repeating or terminating decimals. These properties make them useful for measuring and providing precise values in everyday life and in various fields of science and engineering. However, despite their dense nature, rational numbers are countable, which seems counterintuitive yet is a fascinating result proved through the construction of a bijection with the natural numbers.