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. This 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 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 wanted to find a way to do math with points in three-dimensional space. He worked on this for years. His big discovery 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 studied them for much of his life. After he passed away, others kept working 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 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, and i, j, and k are special symbols.

These symbols have special rules when you multiply them. For example, i × i = -1, and i × j = k. Quaternions can be added and multiplied, but the order matters when you multiply them.

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

All quaternions form a special mathematical space with four parts, made from real numbers. When you multiply quaternions, the order matters, so they are not the same forward and backward. This makes them useful for solving certain 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 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.

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. This makes quaternions valuable in many areas where spatial vectors are used.

Matrix representations

Just like complex numbers, quaternions can 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. Both ways keep the rules of adding and multiplying quaternions the same.

One way uses 2 × 2 complex matrices. This method is 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 ways to set up these 4 × 4 matrices, and they all follow certain patterns.

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 can be written as the sum of four squares of whole numbers. This idea is useful in many parts of math. The proof uses special quaternions called Hurwitz quaternions.

Quaternions as pairs of complex numbers

Main article: Cayley–Dickson construction

Quaternions can be seen as two complex numbers together. This idea uses a method called the Cayley–Dickson construction, which creates new number systems from older ones.

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

Square roots

Square roots of −1

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

Each pair of opposite square roots of −1 creates a different copy of the complex numbers inside the quaternions. These copies are linked to the quaternions through special math rules. Every non-real quaternion makes a smaller part of the quaternions that acts like the complex numbers.

Square roots of arbitrary quaternions

Any quaternion has at least one square root. For a quaternion with a real part and a vector part, the square roots can be found using special formulas. 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.

Functions of a quaternion variable

Functions that work with quaternions, like those with complex numbers, help us understand physics. For example, quaternions were used to describe electric and magnetic fields. We can also use quaternions to create shapes like the Mandelbrot and Julia sets in four dimensions.

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. This rule connects to angles on a four-dimensional sphere. We can find this distance using the dot product of the quaternions.

Three-dimensional and four-dimensional rotation groups

Quaternions help us understand turns and spins in three-dimensional space. A special kind of quaternion, called a unit quaternion, can show a rotation by a certain angle around a specific direction. This method has benefits over other ways to describe rotations. It avoids problems with other systems, is faster and smaller, and always gives clear results.

Unit quaternions form a group that matches 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 special math rules for multiplying numbers. These algebras depend on certain choices, 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. They were first described by an Irish mathematician named William Rowan Hamilton in 1843. These numbers help us understand how things move in three-dimensional space, like turning or spinning objects.

Quaternions are linked to a bigger idea in math called Clifford algebra. This system has special rules that help describe reflections and rotations. For example, reflecting a direction twice can create a turn. These ideas are closely related to how quaternions work, showing their connection to geometry and movement.

Brauer group

Further information: Brauer group

Quaternions are special because they are a main example of a math structure called a central simple algebra over the real numbers. This means any similar structure over the real numbers connects to either the real numbers or quaternions. These structures extend fields but have extra rules that limit them. Quaternions are unique because they are the only complex example of this kind over the real numbers, similar to how complex numbers extend the real numbers.