I was reading about how for the imaginary numbers now called complex numbers Gauss found a geometrical interpretation and were "legalized" in math.
So basically $i$ is a rotation through $90^\circ$ and the complex number $a + bi$ and its operations "move" a point across the cartesian plane.
Assuming my description is correct, it seems to me that essentially the complex numbers are an aggregate of all the pairs of a cartesian plane but what I am not sure if this goes together with the complex number interpretation or not. I think we could also have considered all pair of reals as defining a plane but why was the rotation interpretation required?
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$\begingroup$ Does this answer your question? Geometric interpretation of the multiplication of complex numbers? $\endgroup$– Lee MosherApr 18, 2022 at 17:30
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$\begingroup$ @LeeMosher: No because I am not asking how the operations are defined. From what I understand $i$ was introduced as a rotation and the rest was built on that $\endgroup$– JimApr 18, 2022 at 17:34
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$\begingroup$ Perhaps look at that link in more detail. It explains the geometric nature of rotations arising from all complex numbers, not just from $i$. It's not accurate to say that just $i$ alone was introduced as a rotation. $\endgroup$– Lee MosherApr 18, 2022 at 17:37
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$\begingroup$ @LeeMosher: Even if there is some similarity in the basis of the question of the other post, the answers provided there are not helpful for what I am trying to understand $\endgroup$– JimApr 18, 2022 at 18:34
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"imaginary numbers now called complex numbers" Strictly speaking the imaginary numbers, numbers of the form bi for some real number a, are a subset of the complex numbers, numbers of the form a+ bi for some real numbers a and b.
Yes, the Cartesian coordinate system associates every point in the plane with a pair of numbers, (x, y). The difference is that while the Cartesian coordinate system is purely "geometric", the "complex plane" also has an "algebraic" structure- we can add and multiply complex numbers which we cannot do with points.
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$\begingroup$ " the "complex plane" also has an "algebraic" structure" but how does the rotation come into play here? $\endgroup$– JimApr 18, 2022 at 16:59