R3 – Quaternion Comparison

Comparison of Wave Number and Quaternion Multiplication Tables:

An earlier post makes a detailed comparison of the Wave Number Multiplication table and the Quaternion Multiplication table.

The only difference occurs when Quaternions are multiplied together, as in i*i, = -1, but in R3, 1^{^}*1^{^} = 1^{^}.

In R3, multiplying 1^{^} and 1^{^} = 1^{^} and multiplying 1^{^} and 1^v = 1^v, leaving the operand unchanged.

Rotation Comparison

Another post explains the calculation of the rotation of 60^{o^} ↺^{(4v+ 5iv + 6j^)} (1^v + 2i^{^} + 3j^v) = (0.419^{^} + 0.003i^{^} + 3.718j^v) and compares it to the equivalent Quaternion rotation. The result of Quaternion rotation is not in 3d, so a final step is required to convert the Quaternion result into an R3 vector. This is done by multiplying the result with the conjugate of the Quaternion of Rotation to get the R3 vector. The Quaternion and Wave Number results match.

Wave Number and Quaternion Efficiency

A third post gives a comparison between the efficiency of rotation of Wave Numbers and Quaternions. This comparison makes two conclusions. The first is that the efficiency of Wave Numbers is greater. The second is that the Wave Number representation is clearer than Quaternions as Wave Numbers are always in 3d space.

Squares Comparison with Quaternions

A further comparison looks at squares in R3 Wave Numbers and Quaternions.

Quaternions are expressed in the form $(a + bi + cj +dk)$ . A quaternion with only a value for $a$ is a scalar, whereas a quaternion with $a = 0$ is a vector or pure quaternion. They represent vectors in real $\mathbb{R}3$ space. For example, $(2i + 3j + 4k)$ is a pure quaternion, and -29 is a scalar.

In quaternion algebra, squaring a pure quaternion results in a scalar. For example, $(2i - 3j + 4k)(2i -3j +4k)= -29$ . This result directly follows from the rules of quaternion multiplication.

In Wave Numbers the equivalent squaring of (2^{^} + 3i^v + 4j^{^})*(2^{^} + 3i^v + 4j^{^}) = (4^{^} + 9iv + 16j^{^}).

This demonstrates that squaring an expression results in the squares of the individual Opposite Values. Consequently, the square root of an expression is the square root of the individual Opposite Values. This formula is surprisingly easy to use.

Intuitively, the Wave Number result seems more realistic, as it provides a point of greater magnitude in 3D space. In contrast, quaternions produce a scalar, which does not exist in 3D space.

Conjugates

Roots

The need for conjugates in classical mathematics can be traced back to square roots. In $\mathbb{R}2$ more than one root is available. When dealing with roots, especially in fractions, multiplying by the conjugate above and below the divisor line helps remove the root from the divisor, simplifying the expression. For example:

$\frac{1}{(a + \sqrt{b})} * \frac {(a - \sqrt{b})}{(a - \sqrt{b})} = \frac {(a - \sqrt{b})}{(a^2 - b)}$

There is no need for a conjugate in Wave Numbers R3 as multiplying above and below the divisor line with the divisor, in cases like the above example, eliminates the root from the divisor.

$\frac{(1\hat{ })} {(a + \sqrt{b})} * \frac{(a + \sqrt{b})} {(a + \sqrt{b})} = \frac {1\hat{ }(a + \sqrt{b})}{(a^2 + b)}$

Quaternion Conjugate

As demonstrated in the section on rotation, quaternions require the quaternion conjugate to complete the quaternion sandwich, which transforms the quaternion back into 3D space. In contrast, R3 Wave Numbers are always in 3D space and do not require this mechanism.

Eliminating Complex Numbers

Conjugates are used in a general fashion to simplify complex numbers. For example:

$\frac {(a + bi)} {(c + di)} * \frac {(c - di)} {(c - di)}= \frac {(ac + bd + (bc - ad)i} {(c^2 + d^2)}$

In contrast, R3 Wave Numbers are real numbers and do not require this mechanism.

Conclusion

It is now evident how the Wave Numbers system works in R3. No such system is available in classical maths. Wave Numbers reside naturally in R3 without the 4d complexities that Quaternions bring.

Try out the online calculator to check out any of these calculations.

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