And yes, it is a complex mathematical process.
Which, I must say, you have explained so well, that even myself, who is useless in math, can understand.
I'm new to this forum, kinda stumbled in and got interested.
I'm no pilot except for MSFS, which I started using to overcome my fear of flying. It helps a lot, because I now better understand what happens and why tons of steel do fly.
I remember having been on an Aer Lingus flight from EDDL (Dusseldorf, Germany) to EIDW (Dublin, Ireland) on a very hot and humid summer day (temperatur was 36 Celsius, don't know how much that is in Fahrenheit but I expect it to be in the hundreds).
We used a good deal more of the runway than I usually witness there in the same type of aircraft, the initial climb out of the airport was unstable and rough - just what I don't like - until about 4.000 ft. (taken into consideration my knowledge of the terrain, because I live in the aerea and the SID routes of the airport).
The guy next to me told me it was because of the heat. I assume this must have been "High Density"?!
Thanks for all the interesting information and the entertaining chatter between pilots, soon-to-be-pilots and other non-pilots like myself.