I had this thought on my mind considering the "meaning" of complex numbers in terms of them being graphed on a plane.

It is obvious that complex numbers "should" be graphed with its real part on the x-axis and its imaginary part graphed on the y-axis, but how did someone arrive at the conclusion that it is the "best" way to graph them. This reasoning makes statements such as Euler's Formula and the concepts such as "modulus" and "argument" understood since they display clear geometric meaning.

The question is: How did someone reason that complex numbers are graphed the way they are on a plane? Why did they decide so? What is so special about the imaginary unit that gives complex numbers geometric meaning on the plane?

  • 2
    $\begingroup$ Look at the times: Descartes, Argand, and Wessel. The first 'invented' (made popular) the coordinate method. Just a matter of throwing-the-method-to-everything-you-see, is all it takes to make complex numbers fall in the picture. $\endgroup$ – orole Jan 18 '18 at 22:47
  • $\begingroup$ Perhaps one explanation is that the numbers $1$ and $i $ are independent. You cannot obtain one from the other. The natural way to geometrically represent this is by perpendicular axes, i.e. the real and imaginary axes. $\endgroup$ – Pixel Jan 18 '18 at 23:18

What is so special about the imaginary unit that gives complex numbers geometric meaning on the plane?

The imaginary unit $i$ has the property that for real numbers $x$ and $y$, the complex number $x+iy$ has the absolute value $|x+iy|=\sqrt{x^2+y^2}$. This absolute value function ties together two structures:

  • It is compatible with complex multiplication: $|z||w|=|zw|$ for all complex numbers $z$ and $w$.
  • It is equal to the distance between $(0,0)$ and $(x,y)$ in the Euclidean plane.

This connection is elaborated in another answer to a similar question: https://math.stackexchange.com/a/1326984/87023

By contrast, consider the primitive cube root of unity $\omega=\frac{-1+i\sqrt3}{2}$. It obeys $\omega^3=1$, instead of $i^4=1$, so it's a pretty nice number, and in particular $|\omega|=1$. We can also express any complex number as $a+b\omega$. However, you can't tell the same story with $|a+b\omega|$ -- go ahead and try to make it work! -- so you wouldn't want to identify the number $a+b\omega$ with the point $(a,b)$.


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