This page describes various numbers to make calculations easier. In digital systems, division and exponentiatials are difficult and time-consuming. Therefore, simple combinations of numbers are often easier to work with. The idea behind this is to scale the local calculation with a shortend number or even a binary number. Possibly it makes sense to scale a greater part of the whole calculation and rescale it afterwards. This can siginificantly increase speed.

 

Tuning word

A most simple relation between two adjecent notes in a music keyboard had been found trying around with VC20 in the 80-tees:

f ( "Cis") / f (" C ") =  196 / 185

This works for a whole octave and raises only some parts per million of deviation.

PI and 1/ PI

Pi is an important mathematical constant. It not only describes the ratio of a circle’s circumference to its diameter – it is also used to calculate the volume of spheres and cylinders. In physics, pi features in waves, oscillations and probability theory. PI ist often required to fit sine waves. The table shows binary combinations as well as standard INTs for both PI and 1/PI. Many people go for 22/7. The best number I found was 355/113. It has only 0,1 ppm deviation.

 

mul	div	div bits	PI*	rel / ppm
3217	1024	10	3,141601563	2,83579
205887	65536	16	3,141586304	-2,02123
411775	131072	17	3,141593933	0,40728
1647099	524288	19	3,141592026	-0,19985
3294199	1048576	20	3,141592979	0,10372
355	113		3,141592920	0,08491
63878	20333		3,141592485	-0,05362
6588397	2097152	21	3,141592503	-0,04806
13176795	4194304	22	3,141592741	0,02783
26353589	8388608	23	3,141592622	-0,01012
94053	29938		3,141592625	-0,00918
52707179	16777216	24	3,141592681	0,00885
105414357	33554432	25	3,141592652	-0,00063
1686629713	536870912	29	3,141592653	-0,00004
mul	div	div bits	1 / PI*	rel / ppm
20861	65536	16	0,318313599	11,66300
83443	262144	18	0,318309784	-1,00915
113	355		0,318309859	-0,26676
2670177	8388608	23	0,318309903	0,16740
20333	63878	16	0,318309903	0,16847
10680707	33554432	25	0,318309873	-0,12674
29938	94053		0,318309889	0,02883
21361415	67108864		0,318309888	0,02033
85445659	268435456	28	0,318309885	-0,01644
170891319	536870912	29	0,318309886	0,00195

Eulers number

 Euler’s number "e" is one of the most important constants in mathematics. It describes the principle of continuous, self identical growth and decay and is the basis for the natural exponential function. It describes a number of decay processes and is required for oscillator calculations. Binary representations work pretty good.

 

mul	div	div bits	e	rel / ppm
178145	65536	16	2,7182769775	-1,784550
356291	131072	17	2,7182846069	1,022147
712581	262144	18	2,7182807922	-0,381202
1425163	524288	19	2,7182826996	0,320473
2850325	1048576	20	2,7182817459	-0,030365
22802601	8388608	23	2,7182818651	0,013490
45605201	16777216	24	2,7182818055	-0,008437
91210403	33554432	25	2,7182818353	0,002526
182420806	67108864	26	2,7182818353	0,002526
364841611	134217728	27	2,7182818279	-0,000214
2918732889	1073741824	30	2,7182818288	0,000128
5837465777	2147483648	31	2,7182818283	-0,000043
11674931555	4294967296	32	2,7182818286	0,000042
23349863109	8589934592	33	2,7182818284	0,000000

sqrt (2)
 

 The number √2 describes, geometrically, the diagonals of a square. In electrical engineering, it is the ratio of the peak value of an alternating voltage to its root mean square value.

mul	div	div bits	sqrt(2)	rel / ppm
11585	8192	13	1,414184570	-20,5004826
46341	32768	15	1,414215088	1,0787038
741455	524288	19	1,414213181	-0,2699954
2965821	2097152	21	1,414213657	0,0671794
11863283	8388608	23	1,414213538	-0,0171143
47453133	33554432	25	1,414213568	0,0039592
189812531	134217728	27	1,414213561	-0,0013092
759250125	536870912	29	1,414213562	0,0000079
1451	1026		1,414230019	11,6369412
1492	1055		1,414218009	3,1445785
1632	1154		1,414211438	-1,5018228
1731	1224		1,414215686	1,5018251
1970	1393		1,414213927	0,2576723

sqrt (3)

The cube root comes into play when calculating the height of an equilateral triangle or the distance between opposite sides of a hexagon. In electrical engineering, it is required in three-phase alternating current networks, where it acts as the conversion factor between line-to-line voltage and phase-to-phase voltage.
 

mul	div	div bits	sqrt(3)	rel / ppm
14189	8192	13	1,7320556641	2,803898
227023	131072	17	1,7320480347	-1,600935
454047	262144	18	1,7320518494	0,601481
908093	524288	19	1,7320499420	-0,499727
1816187	1048576	20	1,7320508957	0,050877
14529495	8388608	23	1,7320507765	-0,017948
29058991	16777216	24	1,7320508361	0,016465
58117981	33554432	25	1,7320508063	-0,000742
929887697	536870912	29	1,7320508081	0,000334
1859775393	1073741824	30	1,7320508072	-0,000204
3719550787	2147483648	31	1,7320508077	0,000065
mul	div	div bits	sqrt(3)	rel / ppm
1978	1142		1,7320490368	-1,022367
2340	1351		1,7320503331	-0,273942
3691	2131		1,7320506804	-0,073403
5042	2911		1,7320508416	0,019668

more to come ...