Three Number Systems (thanks to John Baez for reminder)

John Baez deserves a huge credit for educating us tirelessly over the years. There is a treasure of knowledge in his website and blogs.

In the October 15, 2019 diary there was this reminder.

We call numbers z = x + i y with x, y real

The conjugate of z = x + i y is defined as \bar{z} = x - i y

The definition of “exponential” is

John Baez also reminds us that:

  • In physics, the complex numbers describe points in 2d space. The split complex numbers describe points in 2d Minkowski spacetime. The dual numbers describe points in 2d Galilean spacetime – the version of spacetime in classical mechanics before special relativity.
  • In the complex numbers, multiplying by exp(ix) describes a rotation. In the split complex numbers, it describes a Lorentz transformation in 2d spacetime. In the dual numbers, a “Galilei boost” – a transformation to a moving frame of reference in Galilean spacetime.
  • We can take the real numbers and throw in a number i that squares to any real number q. By changing q, we can “morph” the complex numbers to the dual numbers and then split complex numbers. The circle flattens out to ellipse, then parallel lines, then a hyperbola!
gif image credit: Tomasz Stachowiak (refresh the page to repeat the animation)
  • In terms of physics, this number q is 1/c², where c is the speed of light. When q hits zero, the speed of light becomes infinite and special relativity reduces to the physics of Newton and Galileo. When q goes negative, time turns into another dimension of space!

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