Expressions Containing pi and e That Have Numerical Values Very Close to Rational Numbers

Robert A. Warren

When I got my first electronic scientific calculator back in the early 1970s, I was so fascinated by its number crunching ability that I spent a lot of time playing around with it. While checking the numerical value of some expressions involving pi and e, I discovered the expressions below.

The appearance of the digital sequence 9997 in the numerical value of all but two of these randomly discovered expressions is interesting. Also, by first multiplying and then subtracting, combinations of these expressions can be produced that have long strings of zeros in their respective numerical values.

Perhaps some of these randomly discovered expressions or their combinations could be the key to an important discovery? But at least they are “neat" mathematical novelties, especially combination #1.  

Randomly Discovered Expressions

1. (pi)ln(pi) = 3.59627499972915819808600175164636….

2. (pi)ln[(pi)ln(pi)] = 4.02091999973548485670457841754488…

3. e – 1/pi = 2.39997194227525456382251994460763…..

4. (e)^(pi)(e) = 5113.9849997519324952288345376649….

5. Arctan (pi ) + Arctan (e ) = 142.145681559014491710007920388483..….

6. 9[e^pi/4 + sin (pi/4) + cos (pi/4)] = 32.467442517999994548253125451383..….  

Some Combinations

1. (pi)ln[(pi)ln(pi)] – (pi)ln(pi) = 0.424645000006326658618576665898….

2. (e)^(pi)(e) – (1000)(pi)ln(pi) = 1517.710000022774297142832786018….

3. 3 (e)^(pi)(e) + ( 7000)(pi)ln(pi) – 10000(pi)ln[(pi)ln(pi)] = 306.680000005056305242731699...

4. (100000)(pi)ln(pi) - (e – 1/pi) = 359625.100000973544554036352644691…..

Copyright 11 March 2006 by Robert A. Warren.