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Quaternion conjugate versus complex conjugate

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The coupling of a composite number

Is the composite number

z ^ * = a - b

Taking the close becomes a composite number, taking its reflection in the true axis.

The multiplication stretches and rotates composite numbers, and a compound translates composite numbers. You can not make the complex plane by any series of extensions, turns and translations.

The situation is different for Quaternions. Quaternion Coupling

q = a + bi + cj + dk

he

q ^ * = a - bi - cj - dk

are you tin Transform the four-dimensional space by a series of extensions, rotations and translations. That is

q ^ * = -  frac 1 2 (q + iqi + jqj + kqk)

To prove this equation, let’s first see what happens when you multiply That On both sides by I am:

i (a + bi + cj + dk) i = -a - bi + cj + dk

That is, the effect of multiplication on both sides b I am Is to become the sign of the real component and I am component.

Doubling in both sizes b י or K Works analogically: it becomes the mark of the real component and its component, leaving the other two alone.

It follows

 begin align * q + iqi + jqj + kqk & =  phantom - a + bi + cj + dk \ & ,  phantom = - a - bi + cj + dk \ & ,  phantom = - a + bi - cj + dk \ & ,  phantom = - a + bi + cj - dk \ & = -2a + 2bi + 2cj + 2dk \ & = -2q ^ *  end align *

Therefore the result is due to a division in 2.

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