# Math

## Irrational Numbers

• I read that Kronecker opposed the use of irrational numbers because they can't be constructed from integers by a finite number of steps. But you can construct pi as the ratio not only as an infinite series, but as the ratio of circumference to diameter. What's wrong with that?

## Line Integrals

I should set up a latex file in a rasmapedia directory to explain line integrals.

They are really "curve integrals".

1. You want to integrate x^2 over all points between 2 and 6.

2. You want to integrate x^2 + y^2 over all points between y=0, x in [2,6].

3. You want to integrate x^2 + y^2 over all points on the straight line between (2,2) and (6,6) (not intervals now--- vectors).

4. You want to integrate x^2 + y^2 over all points on the curve y = x^3 between (2,8) and 3,27). This is the real stuff.

5. You want to integrate x^2 + y^2 over all points in the area bounded by (0,0) and (0,2) and (2,0) and (2,2).

6. You want to integrate x^2 + y^2 over all points in the area bounded by y = x^3 and y = log x (or some two curves that cross).

## Slide Rules

December 6, 2021: In late 2021, Russian video appeared that showed a heavy bomber in flight with one of the crew using s slide rule, apparently to calculate course and/or fuel consumption rate. Most pilot training, especially for the crews of long-range aircraft, includes instruction on how to use special slide rules for such calculations in the event of problems with the electronic navigation and flight management instruments that do this automatically. For generations, ever since long-range flight became possible, manual tools were used for these calculations, along with a special bubble sextant to obtain the location of an aircraft. On the surface the original sextant is used for this but in the air, there is no fixed horizon to base these calculations. The bubble-sextant creates an artificial horizon that enables aerial navigation that shows position within ten kilometers or less. Surface sextant navigation is even more accurate and electronic navigation, especially using GPS, is accurate enough for landing aircraft.

Media tends to refer to EMP (Electromagnetic Pulse) created by a distant nuclear weapon detonation as the reason for these manual backups.