At some point, you neglect infinitesimals. The notion of equality changes from something exact to something that's infinitely close.

Then 1 − 1 = 1 + (−1) = 000...001 + 111...111 = 000...000.000...001, which is the smallest representable number. Now let n be arbitrarily large, and this smallest number looks like an infinitesimal. If you want to see something neat, sum 2^n terms of Grandi's series like this.

Imagine an n.n-bit fixed-point number representation where addition is defined as usual for two's complement values, but any carryout on the left of the whole part wraps around to the far right of the fractional part.

Yes, there are algebras in which a − a is essentially an infinitesimal and not exactly zero. My favorite is this:

Here, ^★ and _★ are the right and left Hodge duals.

The expression b ∨ (a^★) is the definition of the right contraction, sometimes written b ⌊ a. Likewise, (a_★) ∨ b is the left contraction a ⌋ b.

Thanks!!

I wrote a very thorough answer to that question here:
www.amazon.com/dp/B0CXY8C72...

I feel very honored, and you can imagine the deep sense of satisfaction and justification this provides for me. The images above are from a recent NASA paper and a new textbook in aerospace engineering.

It's got lots of great stuff with two big chapters covering mathematics (including Grassmann / geometric algebra) and another chapter covering things like radiometry, reflectance models, scattering, and ray tracing. There's substantial overlap between this field and computer graphics.

I was recently considering this same question for the purpose of fitting horizontal strokes in a glyph to the pixel grid. I didn't find a satisfying solution.

• There is a lot of interest in editing equations with RP and then exporting the LaTeX code.
• The Greek letter in the RP logo is not obvious, but people love it when they see it.

You can download a free trial on the website:
radicalpie.com

Don’t forget the green one!