Quaternion

Quaternions are a special kind of number system used in mathematics. They were first created by an Irish mathematician named William Rowan Hamilton in 1843. Quaternions help us work with numbers in three-dimensional space, which means they can describe things like spinning and turning in the world around us.

Unlike regular numbers, quaternions have four parts: one real number and three imaginary parts called i, j, and k. This makes them very useful for solving problems that involve turning and rotating objects. They are used in many areas, such as computer graphics, robotics, and medical imaging.

Quaternions are important because they can describe rotations in a way that is easier to work with than other methods. They are used in pure math and also in practical technology, helping computers and machines understand how things move and change direction.

History

Main article: History of quaternions

Quaternions were created by the mathematician William Rowan Hamilton in 1843. Before this, other mathematicians like Euler and Olinde Rodrigues had worked with similar ideas, but they did not connect them into a single system.

Hamilton was trying to find a way to do math with points in three-dimensional space, like how we do with points on a flat surface. He struggled for years to figure out how to multiply these points together. His big breakthrough came on October 16, 1843, in Dublin. While walking to a meeting, he suddenly understood how quaternions worked. He carved the main rule into a bridge: i² = j² = k² = ijk = −1.

Hamilton called these four-number groups "quaternions" and spent much of his life studying them. After he passed away, others continued to work with quaternions. For a time, they were very important in physics and geometry. Later, a simpler system called vector analysis became more popular, but quaternions found new uses in the late 20th century, especially in computer graphics and controlling spacecraft.

Definition

A quaternion is a special kind of number, first created by a mathematician named William Rowan Hamilton in 1843. It looks like this: a + b i + c j + d k, where a, b, c, and d are regular numbers (called real numbers), and i, j, and k are special symbols.

These symbols have special rules when you multiply them together. For example, i × i = -1, and i × j = k. Quaternions can be added and multiplied, but their multiplication is not the same order both ways, which makes them interesting for studying three-dimensional spaces.

Quaternions help us understand movements in three dimensions, like turning and rotating objects.

Conjugation, the norm, and reciprocal

Conjugation of quaternions works like conjugation of complex numbers. For a quaternion written as ( q = a + b\mathbf{i} + c\mathbf{j} + d\mathbf{k} ), its conjugate is ( q^* = a - b\mathbf{i} - c\mathbf{j} - d\mathbf{k} ). Conjugating twice brings you back to the original quaternion.

The norm of a quaternion is found by multiplying it with its conjugate and then taking the square root. For ( q ), the norm is ( |q| = \sqrt{a^2 + b^2 + c^2 + d^2} ). This norm is always a positive real number.

Multiplying a quaternion by a real number changes its norm by the absolute value of that number. The norm also has a special property: the norm of a product of two quaternions is the product of their norms. This helps define the reciprocal of a nonzero quaternion, which is ( q^{-1} = \frac{q^*}{|q|^2} ).

Algebraic properties

The set of all quaternions is a special kind of mathematical space with four parts, built using real numbers. When you multiply quaternions, the order matters, so they are not the same forward and backward. This makes them interesting for solving certain types of problems.

Quaternions can also be divided, which makes them a division algebra. They are one of only three such structures that work over real numbers. Because of this, they are also used in special kinds of mathematical spaces.

Quaternions and three-dimensional geometry

The vector part of a quaternion can be thought of as a coordinate vector in three-dimensional space. This makes quaternions useful for understanding spatial vectors and their operations. They can help in computer graphics by smoothly changing the angles between key positions.

Quaternions include special units i, j, and k that act like basis vectors. By changing their signs, we can find the opposite direction of a vector. Quaternions can also help calculate the dot product and cross product of vectors, which are important in understanding how vectors relate to each other in space. This makes quaternions valuable in many areas where spatial vectors are used.

Matrix representations

Just like complex numbers, quaternions can also be shown using special arrangements of numbers called matrices. There are two main ways to do this: using smaller 2 × 2 complex matrices or larger 4 × 4 real matrices. Each way keeps the rules of adding and multiplying quaternions the same as adding and multiplying these matrices.

One way uses 2 × 2 complex matrices to represent a quaternion. This method keeps things simple and connects quaternions to other math ideas.

Another way uses 4 × 4 real matrices. This method offers more options and can show quaternions in different forms. There are many different ways to set up these 4 × 4 matrices, and they all follow certain patterns to work correctly.

Lagrange's four-square theorem

Main article: Lagrange's four-square theorem

Quaternions help prove a special math rule called Lagrange's four-square theorem. This rule says that any whole number that is zero or more can be written as the sum of four squares of whole numbers. Besides being interesting on its own, this rule is useful in many parts of math, like designing patterns. The proof using quaternions involves special quaternions called Hurwitz quaternions, which follow rules similar to a method used to find the greatest common factor.

Quaternions as pairs of complex numbers

Main article: Cayley–Dickson construction

Quaternions can be thought of as two complex numbers put together. This idea comes from a method called the Cayley–Dickson construction, which builds new number systems from older ones.

Imagine a special space with two building blocks, labeled 1 and j. We can create new numbers by mixing these blocks with complex numbers. By setting some rules for how these blocks multiply, we can link these mixed numbers to quaternions, which have four parts: a, b, c, and d. This shows how quaternions relate to pairs of complex numbers.

Square roots

Square roots of −1

In the complex numbers, there are exactly two numbers, i and −i, that give −1 when squared. In quaternions, there are infinitely many square roots of minus one. The quaternion solution for the square root of −1 is the unit sphere in three-dimensional space. To satisfy the equations for a quaternion squaring to −1, the real part must be zero, and the vector part must have a norm of one. This means that any quaternion that squares to −1 is a vector quaternion with norm 1, forming the unit sphere.

Each pair of opposite square roots of −1 creates a distinct copy of the complex numbers inside the quaternions. These copies are linked to the quaternions through special mathematical functions. Every non-real quaternion generates a smaller part of the quaternions that acts like the complex numbers, forming a flat space inside the larger quaternion structure.

Square roots of arbitrary quaternions

Any quaternion has at least one square root. For a quaternion represented by a real part and a vector part, the square roots can be found using specific formulas involving the norm of the quaternion and its parts. Nonzero, non-real quaternions have exactly two square roots. The number zero has exactly one square root (zero itself). Negative real quaternions have infinitely many square roots, forming a special set of vector quaternions.

Functions of a quaternion variable

Functions that work with quaternions, like those with complex numbers, can help us understand physics. For example, the way electric and magnetic fields were first described used quaternions. We can also extend shapes like the Mandelbrot and Julia sets into four dimensions using quaternions.

Exponential, logarithm, and power functions

We can create special math rules for quaternions using series of numbers. For a quaternion made from parts a, b, c, and d, we can find its exponential, logarithm, and power using trigonometric functions like cosine and sine.

Geodesic norm

We can measure the distance between two unit quaternions using a special math rule that connects to angles on a four-dimensional sphere. This distance can be found using the dot product of the quaternions.

Three-dimensional and four-dimensional rotation groups

Quaternions can help us understand turns and spins in three-dimensional space. When we use a special kind of quaternion called a unit quaternion, it can show a rotation by a certain angle around a specific direction. This method has several benefits over other ways to describe rotations, such as avoiding problems that can happen with other systems, being faster and smaller in size, and always giving clear results.

Unit quaternions form a group that matches closely with the rotations in three-dimensional space. They also help us understand rotations in four-dimensional space. These quaternions can be linked to special shapes and symmetry groups in geometry.

Quaternion algebras

Quaternion algebras are a way to extend the idea of quaternions. They use a special kind of math where you can create new rules for multiplying numbers. These algebras depend on certain choices made, and they can look like special tables of numbers or have unique properties based on those choices.

Quaternions as the even part of Cl_3,0(ℝ)

Main article: Spinor § Three dimensions

Quaternions are special numbers used in math and science, first described by Irish mathematician William Rowan Hamilton in 1843. They help us understand movements in three-dimensional space, like turning or spinning objects.

Quaternions are connected to a bigger mathematical idea called Clifford algebra. In this system, special rules help describe reflections and rotations. For example, reflecting a direction twice can create a specific turn. These ideas match up closely with how quaternions work, showing their deep connection to geometry and movement in space.

Brauer group

Further information: Brauer group

Quaternions are special because they are the main example of a type of math structure called a central simple algebra over the real numbers. This means that any similar structure over the real numbers is connected to either the real numbers themselves or to quaternions. These structures are like extensions of fields but have extra rules that make them more limited. The quaternions stand out because they are the only complex example of this kind over the real numbers, much like complex numbers are the main extension of the real numbers.