THE COMPLEX NUMBER LIST - INSPIRED BY SBIIS SAIBIAN'S CGNL - IN PROGRESS
+ 0.000 + 0.000i | Magnitude = 0 | Angle = undefined
gooiolseptuplex / imagi-octuplex
approx. 10^(-10^10^3222.347) | Magnitude = approx. 10^(-10^10^3222.347) | Angle = ?.???
So crazy small. GOOGOLOGICALLY small. It was actually a fairly recent (2024) computation with 3222 (and a few more) digits of precision that found out that imagi-septuplex was in Quadrant III, making imagi-octuplex a really really small number, not a really really large number.
above this to know if THIS number is in which quadrant we need a computation involving 10^3222 digits of precision. It's hard to compute much about numbers after this here.
See gooiol (iplex), imagi-septuplex, and imagi-nonuplex
decexpion
approx. 10^-17,743,925 | Magnitude = approx. 10^-17,743,925 | Angle = 3.781
+ 0.065988035845 + 0.000i | Magnitude = 0.066 | Angle = 0
z^^∞ -> 1/e ~ 0.368
Smallest positive real number that converges in infinite tetration. See e^e^-1.
Zeta of minus one
- 0.083333333333 + 0.000i | Magnitude = 0.083 | Angle = π
quintexpion
+ 0.08260953 + 0.02833136i | Magnitude = 0.087 | Angle = 0.33038496
fzi
+ 0.207879576350761 + 0.00000000i | Magnitude = 0.208 | Angle = 0
Hmm... i, to the i-th power... outputs a real number.
gooiolplex / iduplex
- 0.030442565536859 + 0.2125140413392i | Magnitude = 0.215 | Angle = 1.713078033
approx. + 0.438 + 0.360i | Magnitude = 0.568 | Angle = 0.688
gooiolduplex / itriplex
+ 0.82289529895 + 0.43821641086i | Magnitude = 0.932 | Angle = 0.48933166
@Googology101 did not continue after gooiolplex, leaving the rest as an exercise for the reader.
undecexpion
1 -10^-53,231,778 -10^-35,487,852i | Magnitude = 1 - 4.7x10^-17,743,926 | Angle = 2π - 3.5x10^-17,743,926
the number is GOOGOLOGICALLY CLOSE to 1, due to the tiny value of decexpion.
+ 1.000 + 0.000i | Magnitude = 1 | Angle = 0
+ 0.947158998 + 0.320764450i | Magnitude = 1 | Angle = 0.327
+ 0.7071067812 + 0.7071067812i | Magnitude = 1 | Angle = π/4
One of the TWO square roots of i, the other one being -sqrt(i).
expion
+ 0.5403023059 + 0.8414709848i | Magnitude = 1 | Angle = 1
exactly cos(1 radian) + isin(1 radian). that also means that angle is exactly 1.
i-illion (long scale)
+ 0.316138425964 + 0.948713073394i | Magnitude = 1 | Angle = 1.249
The "long" version of i-illion... which turns out to be much smaller :/
+ 0.000 + 1.000i | Magnitude = 1 | Angle = π/2
Why not include the imaginary unit? It's the basic idea of complex numbers.
Did you know that the name "imaginary" came from a mathematician named Descartes calling complex numbers "imaginary" as an insult?
gooiol / iplex
- 0.66820151019031 + 0.74398033695749i | Magnitude = 1 | Angle = 2.30258509299 = ln(10)
The number itself can be computed easily as cos(ln(10)) + isin(ln(10)).
minus one / gari
- 1.000 + 0.000i | Magnitude = 1 | Angle = π
iminex
- 0.66820151019031 - 0.74398033695749i | Magnitude = 1 | Angle = 2.30258509299 = ln(10)
If we can have an iplex, why can't we have an iminex?
- 0.7071067812 - 0.7071067812i | Magnitude = 1 | Angle = 3π/4
this is the other square root of i. note that +sqrt(i) and -sqrt(i) are NOT conjugates. they are on the opposite sides.
+ 0.000 - 1.000i | Magnitude = 1 | Angle = 3π/2
the oft-neglected other square root of -1.
+ 1.0856817634 + 0.0307670673i | Magnitude = 1.086 | Angle = 0.028
this number hovers a little above and further than 1.
fugai
+ 1.38615880891 + 0.00000000000i | Magnitude = 1.386 | Angle = 0.000
not surprising anymore that there are more real i exponentiation results.
+ 1.000 + 1.000i | Magnitude = 1.414 | Angle = π/4
this number is actually the same as +sqrt(i) * (sqrt(2)/2).
+ 1.4446678610 + 0.0000000000i | Magnitude = 1.445 | Angle = 0
z^^∞ -> e ~ 2.718
This is the BIGGEST REAL NUMBER that converges when doing infinite tetration.
This value was known and proven to exist by Leonard Euler. For numbers in [e^(-e),e^e^(-1)] the infinite power tower will converge to a single real value. Outside of this however all bets are off. It should be noted that for reals just slightly larger than this value, even though its divergent, it divergence will be incredibly slow. It will take a very long time to pass e. Once it does however, it will quickly accelerate and grow "tetrationally" from there. Interestingly, you can still get converging power towers with larger real components than this, if there is a certain amount of imaginary component. In fact it goes out slightly past Re(z) = 2.
duexpion
+ 1.144 + 1.280i | Magnitude = 1.717 | Angle = 0.841
quadrexpion
- 2.438030 + 0.330385i | Magnitude = 2.460 | Angle = 3.007
septexpion
+ 2.9600565784 + 0.0911010075i | Magnitude = 2.961 | Angle = 0.031
threxpion
+ 0.900289 + 3.006900i | Magnitude = 3.139 | Angle = 1.280
gooioltriplex / imagiquadruplex
+ 3.5429321925 + 5.6289549426i | Magnitude = 6.651 | Angle = 1.009
"ity"
+ 0.000 + 10.000i | Magnitude = 10 | Angle = π/4
"iteen"
+ 10.000 + 1.000i | Magnitude = 10.050 | Angle = 0.100
octexpion
+ 19.2190337462 + 1.7557331999i | Magnitude = 19.326 | Angle = 0.091
i-illion (short)
+ 811.21465284 + 584.748481843i | Magnitude = 1000 | Angle = 0.625
gooiolquadruplex / imagi-quintuplex
+ 3222.3493201896 + 1342.5926281541i | Magnitude = 3490.8580759607 | Angle = 0.395
nonexpion
approx. - 40,856,898 + 218,399,070i | Magnitude = 222,187,848 | Angle = 1.756
Finally something big. though it is not in Quadrant I, we are headed BACK TO COMPLEX ZERO. See decexpion.
gooiolquintuplex / imagi-sextuplex
approx. + (2.223 * 10^3222) + (2.378 * 10^3221)*i | Magnitude ~ 2.235 * 10^3222 | Angle = 0.107
More precisely the magnitude of this number is 2.2352195620*10^3222, the angle is 0.106600418937.
Finally something "LARGE". The TI-89 calls this inf+inf*i. this is tho due to overflow. See imagi-septuplex.
gooiolsextuplex / imagi-septuplex
approx. (- 0.408 - 0.913i)*10^10^3222.347 | Magnitude ~ 10^10^3222.347 | Angle = 4.292
See imagi-octuplex, and imagi-sextuplex.
gooioloctuplex / imagi-nonuplex
approx. + 1 ± 10^(-10^10^3222.347)i | Magnitude ~ 1± | Angle ~ 0±
gooiolnonuplex / imagi-decuplex
approx. + 10 ± 10^(-10^10^3222.347)i | Magnitude ~ 10± | Angle ~ 0±
gooioldecuplex / imagi-undecuplex
approx. + 10,000,000,000 ± 10^(-10^10^3222.347)i | Magnitude ~ 10,000,000,000± | Angle ~ 0±
gooiolundecuplex / imagi-duodecuplex
approx. + (10^10^10) ± 10^(-10^10^3222.347)i | Magnitude ~ (10^10^10)± | Angle ~ 0±
gooiolduodecuplex / imagi-tredecuplex
approx. + (10^10^10^10) ± 10^(-10^10^3222.347)i | Magnitude ~ (10^10^10^10)± | Angle ~ 0±
gooioltredecuplex / imagi-quattuordecuplex
approx. + (10^10^10^10^10) ± 10^(10^10^10^10-10^10^3222.347)i | Magnitude ~ 10^10^10^10^10± | Angle = ???
gooiolquattuordecuplex / imagi-quindecuplex
approx. 10^^6 * (cos(???)+isin(???)) | Magnitude ~ 10^^6± | Angle = ???
gooiolquindecuplex / imagi-sexdecuplex
approx. 10^(10^^6±) * (cos(???)+isin(???)) | Magnitude ~ 10^(±10^^6)± | Angle = ???
This is way beyond defined... so now onto a plethora of more (real!) googolisms... [LOCKED]