Why do we need quaternions?

Quaternions are vital for the control systems that guide aircraft and rockets. Let us think of an aircraft in flight. Changes in its orientation can be given by three rotations known as pitch, roll and yaw, represented by three arrays of numbers called matrices.

Is there an extension of complex numbers?

The quaternions and octonions are number systems that extend the complex numbers. Together with the complex numbers and the real numbers themselves, these form the only normed division algebras over the real numbers.

Why do you think we need complex numbers?

Complex numbers are used in electronics and electromagnetism. A single complex number puts together two real quantities, making the numbers easier to work with. For example, in electronics, the state of a circuit element is defined by the voltage (V) and the current (I).

Why are quaternions used in rotations?

Quaternions are very efficient for analyzing situations where rotations in R3 are involved. A quaternion is a 4-tuple, which is a more concise representation than a rotation matrix. Its geo- metric meaning is also more obvious as the rotation axis and angle can be trivially recovered.

Why are quaternions used for rotation?

A rotation of Euler angles is represented as a matrix of trigonometric functions of the angles. While quaternions are much less intuitive than angles, rotations defined by quaternions can be computed more efficiently and with more stability, and therefore are widely used.

What is above complex numbers?

We have gradually expanded the set of numbers we use: first from finger counting to the whole set of positive integers, then to positive rationals, ir- rational reals, negatives and finally to complex numbers.

Are quaternions complex?

The quaternions are structured like the complex numbers, but with additional square roots of –1, which Hamilton called j and k. Every quaternion has the form a + bi + cj +dk, where a, b, c and d are real numbers, and i^2=j^2=k^2=-1.

What quaternion means?

A quaternion represents two things. It has an x, y, and z component, which represents the axis about which a rotation will occur. It also has a w component, which represents the amount of rotation which will occur about this axis. In short, a vector, and a float.

What are quaternion vectors?

Quaternions are basically just four-dimensional vectors, who’s orthonormal basis lies in some weird four-dimensional existence. That sounds like a mouthful, and to be honest, it kind of is. Let’s take a step back and look at complex numbers.

Do complex numbers have an orthonormal basis?

Essentially each component of the basis represents a different dimension. There are many other orthonormal bases, for example: 2D Cartesian coordinates, polar coordinates (2D), cylindrical coordinates (3D), and spherical coordinates (3D) to name a few. As it turns out, complex numbers also form an orthonormal basis.

How do you know if a quaternion is pure?

If a quaternion’s imaginary components are all equal to zero, then the quaternion is said to be “real”: Alternatively, if a quaternion’s real component is equal to zero, then the quaternion is said to be “pure”:

What is a complex number?

Complex numbers are just two-dimensional vectors which are composed of both real and imaginary dimensions. In the 2D Cartesian coordinate system, vectors are composed of the y y dimensions.