Question

Show that a quaternion is pure iff its square is a nonpositive real number.

Short answer

It has been shown that a quaternion is pure if and only if its square is a nonpositive real number. This was done by defining a pure quaternion, calculating the square of a pure quaternion to show its result as a nonpositive real number, and then showing that given that a quaternion square is a nonpositive real number, it must be pure.

Step-by-step solution

Define a pure quaternion

A quaternion is a number of the form \( q = a + b\mathbf{i} + c\mathbf{j} + d\mathbf{k} \) where \(a, b, c, d\) are real numbers and \(\mathbf{i}, \mathbf{j}, \mathbf{k}\) are the fundamental quaternion units. A quaternion is pure if \( a = 0 \). Thus a pure quaternion has the form \( q = b\mathbf{i} + c\mathbf{j} + d\mathbf{k} \).

Compute the square of a pure quaternion

The square of a quaternion is given by \( q^2 = (b\mathbf{i} + c\mathbf{j} + d\mathbf{k})^2 = b^2\mathbf{i}^2 + c^2\mathbf{j}^2 + d^2\mathbf{k}^2 + 2bci\mathbf{i}\mathbf{j} + 2bcd\mathbf{i}\mathbf{k} + 2cd\mathbf{j}\mathbf{k} \). Using the properties of quaternion units (\(\mathbf{i}^2 = \mathbf{j}^2 = \mathbf{k}^2 = -1\) and \(\mathbf{i}\mathbf{j}\mathbf{k} = -1\)), we get that \( q^2 = -(b^2 + c^2 + d^2) \), which is a nonpositive real number.

Show that if \(q^2\) is nonpositive real, \(q\) is pure

Suppose that \( q^2 \leq 0 \). Then \( a^2 - (b^2 + c^2 + d^2) \leq 0 \), which implies that \( b^2 + c^2 + d^2 \geq a^2 \). The right-hand side is nonnegative, which means that \(a = 0\) for \( q^2 \leq 0 \). So, \(q\) must be pure.

Key concepts

Pure Quaternion

A quaternion is a special type of number used primarily in three-dimensional calculations and rotations. It is generally written in the form \( q = a + b\mathbf{i} + c\mathbf{j} + d\mathbf{k} \), where \( a, b, c, \) and \( d \) are real numbers, and \( \mathbf{i}, \mathbf{j}, \mathbf{k} \) represent the fundamental quaternion units, which extend complex numbers into four dimensions.

A quaternion is called "pure" if its real part is zero. That means in the notation \( a + b\mathbf{i} + c\mathbf{j} + d\mathbf{k} \), the coefficient \( a = 0 \). Therefore, a pure quaternion looks like this:

• \( q = b\mathbf{i} + c\mathbf{j} + d\mathbf{k} \)

This form shows that a pure quaternion can be visualized as a three-dimensional vector since it entirely represents spatial direction without any scalar component. Understanding pure quaternions lays the groundwork for their application in describing rotations and orientations in 3D space.

Square of a Quaternion

The square of a quaternion involves multiplying it by itself. If we have a pure quaternion \( q = b\mathbf{i} + c\mathbf{j} + d\mathbf{k} \), calculating its square involves applying the standard rules of algebra, with the inclusion of quaternion multiplication rules. These rules are given by the properties \( \mathbf{i}^2 = \mathbf{j}^2 = \mathbf{k}^2 = -1 \) and \( \mathbf{i}\mathbf{j}\mathbf{k} = -1 \).

When squaring the quaternion, you perform the multiplication:

• \( q^2 = (b\mathbf{i} + c\mathbf{j} + d\mathbf{k})^2 = b^2\mathbf{i}^2 + c^2\mathbf{j}^2 + d^2\mathbf{k}^2 + 2bc\mathbf{i}\mathbf{j} + 2bd\mathbf{i}\mathbf{k} + 2cd\mathbf{j}\mathbf{k} \)

Using the aforementioned properties of quaternion units, the expression simplifies to \( q^2 = -(b^2 + c^2 + d^2) \). What you'll notice is that the square of a pure quaternion results in a nonpositive real number, specifically \(-1\) times the sum of the squares of its components. This gives it an interesting role in quaternion operations, especially considering its negativity and real nature.

Quaternion Units

Quaternion units are the building blocks of quaternions, similar to how imaginary number \(i\) is used alongside real numbers. In the quaternion system, there are three fundamental units:

• \( \mathbf{i} \)
• \( \mathbf{j} \)
• \( \mathbf{k} \)

These units allow quaternions to encode complex rotations and orientations in three-dimensional space.

The interesting part is how these units multiply with each other:

• \( \mathbf{i}^2 = \mathbf{j}^2 = \mathbf{k}^2 = -1 \)
• \( \mathbf{i}\mathbf{j} = \mathbf{k} \)
• \( \mathbf{j}\mathbf{k} = \mathbf{i} \)
• \( \mathbf{k}\mathbf{i} = \mathbf{j} \)

Moreover, the product \( \mathbf{i}\mathbf{j}\mathbf{k} = -1 \) indicates non-commutativity, meaning the order of multiplication affects the outcome. These unique properties enable quaternions to perform roles in computing three-dimensional rotations and handling the physics of spinning objects.

Nonpositive Real Number

A nonpositive real number is a concept rooted in the set of real numbers where the number is zero or negative. These numbers are important when working with quaternions, especially in our given exercise about pure quaternions' squares.

In the context of the square of a quaternion, if \( q^2 \) results in a nonpositive real number, as shown with pure quaternions where \[ q^2 = -(b^2 + c^2 + d^2) \], it implies that \( q^2 \) is either zero or less than zero due to this negative value. This outcome is critical because it confirms that the original quaternion was pure.

Recognizing and understanding nonpositive real numbers are key in capturing why square results of pure quaternions have the specific properties they do, helping illustrate how part of their geometric representation gets encoded in negative real terms.