Imaginary Numbers

• Intuitive
• Concrete
• Abstract
• Why is it interesting?
• History

Intuitive

In the beginning mathematicians only used natural numbers: $1,2,3, \ldots$. Then, negative numbers were invented to represent things like debt. For example, $+5$ means a profit of $5$ units, while $-5$ means $5$ units of debt.

The invention of negative numbers then lead to a surprising observation:

• Multiplying a positive number with a negative number yields another negative number: $-3 \times 5 = -15$.
• Multiplying a negative number with another negative number yields a positive number: $-3 \times -5 = 15$.
• However, there is no number that yields a negative number when multiplied by itself. Therefore, with positive and negative numbers alone, simple equations like $x^2 = -1$ can't be solved.

For this reason, the imaginary number $i$ was invented. This new number is defined by the property: $i \times i=-1$. In this sense, $i$ was introduced to fill the gap that we discovered above since now we have a number that yields something negative when multiplied by itself.

In the beginning, $i$ was merely a convenient tool to help simplify calculations and its introduction was criticised by many mathematicians. This early criticism is also where the name "imaginary numbers" comes from, which was meant as a derogatory term.

Nowadays, imaginary numbers are an essential tool. Combinations of real and imaginary numbers like, for example, $4+3i$, are known as complex numbers. Complex numbers are the standard number system that physicists use.

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Concrete

• The best introduction is A Visual, Intuitive Guide to Imaginary Numbers by Kalid Azad
• See also https://math.stackexchange.com/questions/199676/what-are-imaginary-numbers

Abstract

The complex numbers can be constructed from the reals in several equivalent ways.

• As a quotient of the polynomial ring: $\mathbb{C} = \mathbb{R}[x]/(x^2 + 1)$. Since $x^2 + 1$ is irreducible over $\mathbb{R}$, the quotient is a field, and $i$ is the class of $x$.
• As $\mathbb{R}^2$ with the product $(a, b)(c, d) = (ac - bd,\, ad + bc)$.
• As the real $2 \times 2$ matrices of the form

$$a + bi \;\longleftrightarrow\; \begin{pmatrix} a & -b \\ b & a \end{pmatrix},$$

where $i$ corresponds to the rotation by $\pi/2$ and complex multiplication becomes matrix multiplication.

$\mathbb{C}$ is a field extension of $\mathbb{R}$ of degree 2. Its Galois group $\mathrm{Gal}(\mathbb{C}/\mathbb{R}) \cong \mathbb{Z}_2$ consists of the identity and complex conjugation $z \mapsto \bar z$, so $i$ and $-i$ cannot be told apart by any algebraic property over $\mathbb{R}$. The fundamental theorem of algebra states that $\mathbb{C}$ is algebraically closed: every non-constant polynomial with complex coefficients has a root in $\mathbb{C}$. This is the reason every linear operator on a finite-dimensional complex vector space has an eigenvalue, which real vector spaces do not guarantee (a rotation of $\mathbb{R}^2$ has none).

The multiplicative group splits as $\mathbb{C}^\times \cong \mathbb{R}_{>0} \times U(1)$ through the polar form $z = r e^{i\theta}$ (see Euler's formula). The unit circle $U(1)$ is the gauge group of electromagnetism, and multiplying a wave function by $e^{i\theta}$ is the corresponding phase transformation.

On a real vector space $V$, a linear map $J$ with $J^2 = -1$ is called a complex structure. It makes $V$ a complex vector space with $(a + bi)v = av + bJv$, which for finite-dimensional $V$ requires $\dim_{\mathbb{R}} V$ to be even. Complex structures appear in quantization, where one chooses a $J$ on the classical phase space compatible with the symplectic form. In quantum mechanics complex conjugation reappears as part of the antiunitary time-reversal operator.

Continuing the doubling construction $\mathbb{R} \to \mathbb{C}$ (the Cayley-Dickson construction) gives the quaternions, which are no longer commutative, and then the octonions, which are no longer associative.

Why is it interesting?

Imaginary numbers are essential in modern theories of physics like quantum mechanics and quantum field theory. In these theories, we describe a physical system using complex functions, which means functions that contain combinations of imaginary numbers as arguments.

Moreover, imaginary numbers can often be used to make calculations simpler.

For some further motivation, see the nice list here.

The shortest path between two truths in the real domain passes through the complex domain. J. Hadamard

Important Related Concepts:

• complex_analysis

History

The name "imaginary numbers" was introduced by Descartes as a derogatory term.

Leads to: Euler's Formula, Quaternions, Schrödinger Equation.