CGNL

Welcome to my Complex Googology Number List, perhaps the first of it's kind! Here you will find a list of complex numbers relevant to googology, sorted by their magnitude (and further sorted by their angle). A Complex Numbers magnitude is it's distance from Complex Zero. If two numbers have the same magnitude they are further sorted by their complex angle, which is the angle measured from the positive real direction going counter-clockwise measured in radians. Entries highlighted in red are those within the Shell-Thron Region of the Complex Plane. This is the area where z^z^... converges to a singular unique complex value. Entries highlighted in yellow are those of Period-2 centered around complex zero. Entries Highlighted in cyan are in neither region but have a known magnitude, and entries highlighted in green are either undefined or of unknown magnitude.

The List is broken into four parts: Sub-Normal, Normal, Large, and Unsorted. Sub-Normal are complex numbers whose modulus is less than 1. Normal are those whose Modulus is exactly 1. Large are those of modulus greater than 1. Unsorted are those whose modulus is unknown.

0

Complex Zero

Magnitude = 0.000 | Angle = [undefined] 

= 0 + 0 i

z^z^... -> 0,1

Let's start the list off at "complex zero". Complex Zero can be written as 0+0i. It's the only complex number whose modulus/magnitude is equal to 0. It is also the Complex Multiplicative Annihilator. That means that any complex number times complex zero equals complex zero. e^z does not equal complex zero, for any complex number z. 0 is not part of the Shell-Thron Region, however it can be treated as having a period of 2, where 0^0 = 1 and 0^1 = 0. Complex Zero is the only complex number which does not have a defined angle as well. 0 is the center of the Orbit of Alternation. Numbers of magnitude less than e^(-e) will alternate between 2 complex values.

10^i#8

gooiolseptuplex / imagi-octuplex

Magnitude = 10^(-10^10^3222) | Angle = ?.???

~ 10^(-10^10^3222) * [ cos( ?.??? ) + sin( ?.??? )*i ]

What a crazy number this is! Initially it was not known whether this was an extremely large number (about 10^10^10^3222) or an extremely small number (about 10^-10^10^3222). This was due to not being able to calculate the modulus of the rather large imaginary component of imagi-septuplex. However recently (2024) this computation was carried out using over 3222 digits of precision. This resulted in the discovery that imagi-septuplex lies in Quadrant III and therefore imagi-octuplex must be an extremely small number. Unfortunately the imaginary component of imagi-septuplex is approximately -10^10^3222, thus it is far too large for any computer on earth to compute up to the decimal point. This means we can not determine the angle of imagi-octuplex. All we know is that it lies somewhere on an extremely small circle of radius 10^(-10^10^3222). However since we know it's magnitude we can still place it on the sorted list. See imagi-nonuplex.

e^i#10

decexion

Magnitude ~ 10^-17,743,925 | Angle = 3.78182

~ ( - 4.7227 - 3.5180 i )x10^-17,743,926

This is currently the smallest non-zero complex number on this list. This is the first member of the Expion Sequence to create an underflow. That is, it's so small that it's rounded to Complex Zero. The actual value can never be 0, because again, |e^z| > 0 for all complex numbers "z", but this one gets frightfully close. It's about 17 million orders of magnitude below unity in size! Although it would barely be noticeable, decexpion would fall in the third quadrant. This number is so close to Complex Zero that it is no where near the Shell-Thron region which doesn't begin until a real value of approximately 0.06.

e^(-e)

Magnitude = 0.065988 | Angle = 0.000

~ + 0.065988 + 0.000000 i

z^z^... -> 0.368

This is the lowest possible point on the interval of convergence of infinite power towers on the real numbr line. The exact convergence value is 1/e ~ 0.368. See e^e^(-1). This is one of the useful extremity points of the boundary of the Shell-Thron Region to know. Small positive real numbers smaller than e^(-e) will no longer converge. Instead they enter The Period-2 Region centered at Zero, the boundary of which I'll call the Orbit of Alternation. If the number is a positive real number in this region it will bounce back and forth between 0+ and 1-. If the value is 0 itself it will bounce back and forth between 0 and 1.

-1/12

Zeta of minus one

Magnitude = 0.083 | Angle = 3.142

~ -0.083333 + 0.000000 i

This number arises from the Riemann-Zeta Function, an analytic continuation of the sums of the reciprocals of powers generalized to complex arguments. This value corresponds to the infinite series 1+2+3+4+5+... which is clearly divergent, yet this analytic continuation gives us this strange negative value. This is also one of only 3 negative numbers to appear on Robert Munafo's famous Numbers List. The other two are -7/4 and -1.

e^i#5

quintexion

Magnitude = 0.087 | Angle = 0.33038496

~ 0.08260953 + 0.02833136 i

This is the closest the expion sequence has gotten to 0 yet. Because the exponential map can not map to 0 however, the expion sequence is gauranteed to never actually hit 0, but it can become arbitrarily close, just as it can't ever hit infinity but can become arbitrarily "close" (that is to say arbitarily large yet finite). This number, being close to complex zero, will result in a number near complex unity. This number barely falls within the Shell-Thron Region.

i^i

fzi

Magnitude = 0.208 | Angle = 0.000

~ + 0.207879576351 + 0.000000000000 i

i^i is a commonly cited "complex number" on account of the surprising fact that raising an imaginary number to an imaginary power yields ... a real number? (And a positive real number at that). You would expect to get something really weird, perhaps something that isn't even a complex number anymore. But in fact we get a positive real value around 0.207. How is that possible. Well in order to compute b^z we first convert the base to e using b^z = e^(z*ln(b)). So in this case we get e^(i * ln(i)). How do we compute ln(i)? We note that i is 1 unit away from complex zero and it's pi/2 radians from the positive real direction. Thus we get ln(i) = i*pi/2. Now if we multiply this by i we get: i*i*pi/2 = i^2*pi/2 = -pi/2. Lastly since we now have a real exponent e^(-pi/2) yields a small positive real value of approximately 0.207. This is yet another number that combines the mathematical constants of pi and e. Given how often this number comes up in number discussions (also occurring on Robert Munafo's and Cookie Fonster's Number list), this would be reason enough to include it. However it also can be related to googology since we can use the "fz" prefix to give this number a name. Since fz(x) = x^x, we can assume that "fzi" = fz(i) = i^i. Thus it's one of the basic complex googolisms we can form with i.

10^10^i

gooiolplex / iduplex

Magnitude = 0.215 | Angle = 1.713078033

~ - 0.030442565536859 + 0.2125140413392 i

In The blogpost "Numbers", googology101 also coined gooiolplex explicitly. It is defined as 10^10^i naturally. Again an explicit value is not computed. This time the computation is more ... ahem ... complex. Complex numbers behave chaotically under the plex suffix. Their distance from 0 does not strictly increase, as you might expect. This is due to the erratic behavior of the imaginary part. From gooiol = -0.66...+0.74... i we have:

gooiolplex = 10^10^i = 10^(-0.66..+0.74... i) = e^(-0.66...*ln(10)+0.74... i*ln(10))

= e^(-1.5385...+1.7130... i) = e^(-1.5385...)*[cos(1.7130...)+sin(1.7130...)i]

= 0.214683412407...*[-0.141802131779...+0.989895022426... i]

= -0.030442565537...+0.212514041339... i

e^(-1.5485...) = 0.214683412407... is known as the modulus. It is the straight line distance of the complex number from 0 in the complex plane. Note that the modulus of gooiol is larger than the modulus of gooiolplex. The plex actually made it "smaller" in some sense. How is that possible? It's because gooiol was in the second quadrant and had a negative real part. Due to the definition, the real part controls the modulus independently of the imaginary part. Since ln(10) is positive, this means if the real part is negative, applying the next plex will cause the next number in the "gooiol sequence" to be very small instead of very large, which is what happens here. The imaginary part then controls the angle. If the angle is in the first or fourth quadrant then the next plex will result in a large number, but if the angle is in the second or third quadrant then the next plex will result in a small number. It follows, that since gooiol is a complex number in the second quadrant, that the next number will have a modulus smaller than 1 rather than larger than 1. This number also barely falls within the Shell-Thron Region.

i^i^i^i^...

Infinite power tower of i

Magnitude = 0.568 | Angle = 0.688

~ + 0.438282936727 + 0.360592471872 i

If we iterated i exponentiated to the i, that is, if we looked at the limit of an infinite power tower of i, it would converge to this value. This convergence shows that i is in the Shell-Thron Region, however it is also the case that this value is within the Shell-Thron Region. 

i^^i (Paulsen)

megafugai

Magnitude = 0.6003 | Angle = 0.5824

~ + 0.50134608238017127458 + 0.33020066879235626266 i

megafuga(n) is defined as n^^n. Typically n here is assumed to be restricted to a positive integer. If we generalize this to complex-values however this requires that we can raise to complex tetrates. However, there does exist limited extensions of tetration to complex tetrates ... the catch? We are limited to specific complex bases. As it so happens however, a certain William Paulsen has generalized complex tetration to various bases, including i. This specialized tetration I will call "Paulsen Tetration". Since under Paulsen Tetration we can raise i to any complex tetraponent, we can therefore calculate i tetrated to the i. This we use to define megafugai. The exact nature of Paulsen tetration is highly technical, but in simple terms, if the imaginary component is large the values to will converge to a special fixed value depending on the base. These convergence values are typically rather small. It's also worth noting that if there is any imaginary component in the tetrate the value will rapidly shrink (tetrationally) towards this small value. It sounds too good to be true. The problem is that it's only available for 9 bases, thus it is not quite the full complex tetration, but it does allow us to compute a few interesting values such as this one. See i^^i^^i (Paulsen). Although incidental it's worth noting that paulsen's i^^i ends up being close to i^^inf., being within 0.07 units between each other. In terms of magnitude megafugai (as defined here) is slightly larger.

-3/4

Magnitude = 0.75 | Angle = 3.142

= -0.75 + 0.00 i

This number is the point on the complex plane that connects the main body of the mandelbrot set to the head. For the purposes of orienting oneself, this is a good point to remember to have an idea of the relative position of other features. While this number behaves bounded under the iteration z^2+c, it does not behave bounded under c^z (iterated exponentiation). Thus while it is in the Mandelbrot Set, it's not in the Shell-Thron Set.

- 0.119196 + 0.750586 i

Smallest 10^z=z Fixed Point

Magnitude = 0.760 | Angle = 1.728

~ - 0.119196 + 0.750586 i

The accuracy here may be limited. This is another possible version of "unfathomillion". That is, this is a solution to 10^z=z. What makes this one notable however is that its the smallest possible solution to the problem, at a size of a mere 0.760. It's also the only solution with a negative real component. There are actually countably infinite solutions, and for every solution it's conjugate is also a solution. So this is actually part of a pair of smallest possible solutions. CookieFonster's infamous unfathomillion is actually part of the second smallest pair of solutions. The rest of solutions can be ranked in this way. See Unfathomillion.

i^^i^^i^^... (Paulsen)

Infinite tetra-tower of i

Magnitude = 0.774 | Angle = 0.744

~ + 0.5691... + 0.5242... i

A more exact value of this number is not easily obtained. Actually it's not even completely guaranteed that this value exists. However it does appear that an infinite iterated tetration of i converges, or at very least, is bounded to a very small area in the vicinity of 0.569+0.524 i. Not much is known about this "value", however it emerges from Paulsen Tetration when we feed the outputs of i^^z back into themselves starting with either 0, 1, or i. This might be the first time anyone has attempted to look into infinite iterated complex tetration. This number falls snuggly within the Shell-Thron Region (ironically). However if i is convergent it means that it not only falls in the Shell-Thron Region, but in some kind of "super Shell-Thron Region", that is, the complex numbers for which iterated tetration converges to an orbit of 1. For now, very little is known of this region. More investigation is required.

i^^^3 = i^^i^^i (Paulsen)

i pentated to the third

Magnitude = 0.907 | Angle = 0.703

~ + 0.69215738071124144030 + 0.58720987759037306095 i

You read that correctly ... complex pentation! With Paulsen Tetration it becomes possible for the first time to iterate it to create pentation with complex (paulsen) bases. So we can actually compute a few of these unusual values. Simply compute i^^i then feed that back as the complex tetrate to compute i^^(i^^i). Keep in mind this only valid for a specific implementation of complex tetration. Paulsen's Tetration however appears to be the most complete and sophisticated attempt at complex tetration to date. This number is in the Shell-Thron Region and a close neighbor of a gooiolduplex, better known as an itriplex.

10^10^10^i

gooiolduplex / itriplex

Magnitude = 0.932 | Angle = 0.48933166364

~ + 0.82289529895195 + 0.43821641086408 i

Googology101 didn't continue after gooiolplex, but concludes after it, "you get it", suggesting the continuation is trivial and left as an exercise for the reader. Thus we have gooiolduplex or itriplex / imagitriplex. Because imagiduplex has real part -0.03 we must get a "small number" but because it happened to be close to 0, this means we get a value close to 1, since e^0 = 1. The modulus of this number is about 0.932. Because the real part of imagitriplex is positive it means the modulus of the next number in the gooiol sequence will be large. The small imaginary part will be small enough that the next number will still fall within quadrant I. See gooioltriplex/imagiquadruplex.

- 0.380979728648 + 0.899700195546 i

North-West Terminal

Magnitude = 0.977 | Angle = 1.971

~ - 0.380979728648 + 0.899700195546 i

This is one of two points that are furthest to the left within the Shell-Thron Region. The real part here is negative, thus it obviously extends well below e^(-e) = 0.065988. See North-East Terminal.

e^i#11

undecexion

Magnitude = 1 - 4.7x10^-17,743,926 | Angle = 2π - 3.5x10^-17,743,926

~ 1 - 10^-53,231,778 - 10^-35,487,852 i

This number is unimaginably close to Complex Unity. It is ever so slightly smaller, due to the very small negative real component of decexpion, and it lies ever so slightly below the real number line, due to the very small negative imaginary component of decexpion. 

1

Complex Unity

Magnitude = 1.000 | Angle = 0.000

= 1 + 0 i

Complex Unity plays the same role as 1 in the real numbers. Complex Unity is the Multiplicative Identity of the Complex number system. It has a "scaling factor" of 1, and a "rotation factor" of 0, meaning that it neither squishes nor stretches nor rotates the complex number it is being multiplied by. Numbers of the same magnitude as Complex Unity I call rotoroids as they have the effect of rotating complex numbers without distorting them. Complex Unity may also be called the normoid, as it defines what "the normal" is. That is, it's magnitude is deciding factor in whether a complex number is large or small. 1 is also in the Shell-Thron Region, which means the power tower 1^1^... converges (to 1 obviously).

i^i^i

i to the i to the i

Magnitude = 1.000 | Angle = 0.327

~ + 0.947158998072 + 0.320764449979 i

Part of the sequence forming the limit of i^^n as n approaches infinity. It's interesting to note that this value is so different than i^i^(i-1). Here we get a complex number on the unit circle with a very small angle . If we keep recursively computing f(z) = i^z, we actually converge onto a definite value.

sqrt(i)

square root of i

Magnitude = 1.000 | Angle = 0.785

( 1+ i )/sqrt(2) ~ 0.707 + 0.707 i

One of two possible square roots of i. This number is notable for being the only non-real, non-imaginary complex number to appear on Robert Munafo's famous Numbers List. This number also shows up on Cookie Fonster's Pointless Gigantic List of Numbers. This is one of those numbers that likely gets thought about by almost anyone introduced to complex numbers. if "i" is the square root of -1, what is the square root of i? If finding the square root of -1 required a new kind of number, does the sqrt(i) involve some new kind of unit. As it turns out, no. And we can prove this easily enough. Take ( 1 + i )/sqrt(2) and square it. This yields (1+ 2i + i^2 )/2. Note that i^2 = -1 so we can cancel it with 1 and obtain 2i / 2 = i. This number falls on the unit circle and also falls within the Shell-Thron Region, which means it has it's own convergence value, which is approximately 0.652+0.367 i.

e^i

exion

Magnitude = 1.000 | Angle = 1.000

~ 0.540 +0.840 i

This looks very similar to e^(pi*i). I've coined the name expion for this number, based on the expression exp(i), where exp is the exponential function and i is the imaginary unit. This number is exactly equal to cos(1)+sin(1)*i. Since 1 radian falls in the first quadrant so does expion. Since it's in the first quadrant it means e^expion will have a magnitude greater than 1. (see duexpion)

10^(6i)

i-illion (long)

Magnitude = 1.000 | Angle = 1.249

~ + 0.316138425964 + 0.948713073394 i

The "long" version of i-illion ... which turns out to be much smaller :)

The idea is simple (n)-illion = 10^(6n) in the long scale, so if we generalize this to complex arguments we can say i-illion = 10^(6i). However because of the weird way complex exponentiation works this ends up doing nothing more than cycling around the "euler circuit" (the circle of radius 1 on the complex plane). In this case it rotates around 2 full rotations before stopping at approximately 1.249 radians, placing it in the first quadrant. However since it is simply cycling around the unit circle, it only has a magnitude of 1. Thus the long scale i-illion turns out to be smaller than the short scale i-illion. This value falls within the Shell-Thron Region.

i

Imaginary Unit

Magnitude = 1.000 | Angle = 1.571

= 0 + i

How could we not include the imaginary unit? It's basically where complex numbers begin. The imaginary unit is endowed with the "supernatural" property that it's square is equal to -1. That is i^2 = -1. From this simple new rule, combined with all the existing laws of real arithmetic, we obtain the arithmetic of the complex numbers! The Imaginary unit lies on the "Euler Circle", the unit circle of the complex plane, along with Complex Unity, -1, and it's conjugate -i. i lies within the Shell-Thron Region. An infinite power tower of i's yields the value of 0.43828+0.36059 i, a number which also falls within the Shell-Thron Region ...

10^i

gooiol / iplex

Magnitude = 1.000 | Angle = 2.30258509299

~ - 0.66820151019031 + 0.74398033695749 i

Back in the early days of googology, in the months just following the formation of the Googology Wiki on December 5th 2008, there was a blog that began called "A googol is a tiny dot". It's first post is dated January 11th 2009, and it's last is dated March 18th 2009. The blog is believed to have been created by Googology Wiki Founder, Nathan Ho. It still exists to this day, and serves as a museum of sorts, of what early googology looked like. The blog has a decidedly "Joycian" flavor to it, more interested in coining googolism's for its own sake, rather than specifically tring to create notations for generating large numbers.

In the interest of preserving the early history of the googology community, which I believe is in danger of being forgotten by the current and future generations of so called "googologists", I will be adding several entries on the ULNL from "A googol is a tiny dot".

The current entry in question comes from a blogpost dated February 18th 2009 simply called "Numbers", we are introduced to the gooiol. It is simply "googol" (10^100), with the 100 replaced with i, the imaginary unit, thus gooiol = 10^i. Given that there is an "eplex" and "piplex" (seen later in this list), it seems fitting there should also be an iplex somewhere on this list, especially in light of infamous formula e^(i*pi) = -1. There is just one problem: 10^i is a strictly non-real value, so there is no standard place to place it; yet gooiol is part of the early history of googology so I'd be remiss not to include it. To resolve this I have decided to place all non-real complex values at the beginning of my list before the negatives.

This is a very rare example of the intersection of googology with complex numbers.

In order to compute gooiol or iplex, we need a formula for complex exponentiation. For our purposes the formula: e^(a+bi) = e^a*[cos(b)+sin(b)i], where b is interpretted as an angle in radians, will be most useful. The blogpost does not bother to compute an explicit complex value for gooiol, but using this formula it is trivial:

gooiol = 10^i = e^(i*ln(10)) = e^0*[cos(ln(10))+sin(ln(10))i]

= -0.66820151019...+0.743980336957... i

This value is not "large" by any stretch of the imagination. Both it's real and imaginary part are of magnitudes less than 1, and it exists exactly 1 unit from 0 in the complex plane, on what I call the Euler-Circle, a circle of radius 1 centered at 0 on the complex plane. This number just falls out of the Shell-Thron Region.

-1

minus one / gari

Magnitude = 1.000 | Angle = 3.142

= -1 + 0 i

-1 is actually kind of important for a complex number. It is the result of the famous equation e^(pi*i) = -1. It is the square of i. An infinite power tower of -1's happens not to converge. It's one of the four cardinal points along the Euler Circle. It also performs some useful functions. When multiplying it by any complex number, it perserves the magnitude but reverses the direction, causing it to travel the same distance in the opposite direction. It's also one of the square roots of 1, which means that multiplying by it twice restores a number back to its original position. It can be used to calculate the reciprocal of a complex number in the form of z^-1. Amusingly we can also call -1 "gari" based on the idea that it's equal to i^2, using the gar prefix.

10^-i

iminex

Magnitude = 1.000 | Angle = 3.9806

~ -0.668 +0.744 i

If we can have an iplex, we can certainly have an iminex. The "minex" prefix was coined by Jonh Conway to be analogous to "plex" but yielding small numbers instead of large ones. Here it yields the complex-reciprocal of iplex. This also gives us the conjugate in this case of iplex since iplex has a magnitude of 1 and thus so does iminex. Like iplex it lies outside the Shell-Thron Region.

-sqrt(i)

minus square root of i

Magnitude = 1.000 | Angle = 3.927

= -(1+i)/sqrt(2) ~ -0.707 -0.707 i

This is the other square root of i. This number appears on CookieFonster's PGLN. It's interesting to note, that because sqrt(i) and -sqrt(i) are not conjugates, they don't necessarily both lie inside or outside the Shell-Thron Region. In this case sqrt(i) lies inside and -sqrt(i) lies outside.

-i

minus i

Magnitude = 1.000 | Angle = 4.712

= 0 - i

Minus i is the oft neglected other square root of -1. Since i^2 = -1, it follows that (-i)^2 = (-i)(-i) = --i^2 = +i^2 = -1. Huh. In general, in the complex numbers, every complex number gets 2 square roots (except for 0). These square roots will always be additive inverses, which means these pairs will always add up to 0. -i as some other interesting properties. It is the conjugate of i, which means when multiplied together it yields it's and i's magnitude: (-i)(i) = -i^2 = +1. It is also the third power of i. i^3 = -i. It is also the reciprocal of i. 1/i = i/i^2 = i/-1 = -i. One last very strange property ... i and -i are completely interchangable. That is to say, if we labled "-i" "i" and "i" "-i" nothing would change about the math. This implies that the notion of a positive and negative direction is abitrary on the imaginary line. We simply designate one as the "imaginary unit" and the other as it's negative, but either could be designated as the "imaginary unit". This property hints at the limits of our platonic ideals as we can't really say there is a canonical direction of the imaginary axis to the real axis. Mathematicians don't trouble themselves with such things ... but philosophers might. One last point of note, conjugate pairs have the same "Shell-Thron Status". The conjugate of any number in the Shell-Thron Region is also in that region. Since i is in that region, so is -i.

e^i#6

sextexion

Magnitude = 1.0861176289 | Angle = 0.02833136

~ 1.0856817634 + 0.030767067267 i

z^z^... -> 1.0927 + 0.0372 i

A complex number, that is almost exactly equal to 1. It hovers just slightly ahead and slightly above from 1. The small imaginary component means that the next member of the expion sequence will be close to e, the next will be close to e^e, the next e^e^e, and so on. We will see however however a small difference is magnified over iterated complex exponentiation. See septexpion. This number falls within the Shell-Thron Region. Because this value is close to 1, and infinite power tower of this number also converges to a number close to 1.

i^i^(i-1)

fugai

Magnitude = 1.386 | Angle = 0.000

~ + 1.38615880891 + 0.00000000000 i

This entry is a little debatable, but I'm adding it anyway. Allow me to explain. The root "fuga" was intended for "weak tetration", sometimes symbolized as "vv", where fuga(n) = n vv n. Weak Tetration expands to iterated exponentiation but evaluated from left-to-right. That is:

n vv n = ((...((n^n)^n)^n  ... )^n)^n w/n n's

For ease of computation it is noted that this can be simplified to n^n^(n-1) for positive integers (the only values for which the above definition makes sense). Just like with tetration, there is no generally agreed on generalization of weak tetration to real and complex arguments. Thus while this formula z^z^(z-1) is valid for positive integers, it's validity for real values and complex values is questionable. We would first need some understanding of what "weak tetration" means when using real and complex arguments. However, if we make the argument, that we can generalize it by assuming this formula applies to any real or complex value, we do get some kind of operation that has all the canonical values of weak tetration as well as any complex value as well.

If we assume this formula generalizes then we can define fuga(z) = z^z^(z-1), and then say that fugai = fuga(i) = i^i^(i-1). If we accept all that ... what does this value even come out to? Interestingly it works out to a real value. To see why recall that i = e^(i*pi/2). This yields:

e^(i*pi/2*e^(i^2*pi/2-i*pi/2)) = e^(i*pi/2*e^(-pi/2-i*pi/2))

= e^(i*pi/2*e^(-pi/2)*-i) = e^(-i^2*pi/2*e^(-pi/2)) = e^(pi/2*e^(-pi/2))

= e^(1.571*e^(-1.571))

Note that the last computation does not involve complex numbers at all and thus we get a completely real value in this case. See i^i^i.

1 + i

Magnitude = 1.414 | Angle = 0.785

= 1.0 + 1.0 i

z^z^... -> 0.641 + 0.524 i

The simplest example of a complex number one could basically give. If we try to combine the real unit with an imaginary unit, we can't simplify it and we get a complex number instead. If we have some real number, x, the easiest way to create a complex number is to create x + x * i. So if we want a "large" complex number, the simplest way is to take a large positive real number, M, and just create M + M i. I might have one example of such later in this list, but there are more interesting ways we can obtain large complex numbers. This number forms the famous 1-1-sqrt(2) right triangle. For this reason when we square 1+i we get a number of magnitude 2. Secondly because this is at a 45 degree angle if we square it we get a right angle. This thus predicts that (1+i)^2 should be 2i. This checks out if we carry out the arithmetic: (1+i)(1+i) = 1 + 2i + i^2 = 1 + 2i - 1 = 2i.

e^e^(-1)

Magnitude = 1.444 | Angle = 0.000

~ 1.44466786101 + 0.00000000000 i

z^z^... -> 2.71828182846

This exists at the "pinch point" of the Shell-Thron Region. It's the largest real number greater than 1 whose infinite power tower still converges. 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. 

1.52598338517 + 0.0178411853321 i

Sheldon's Base

Magnitude = 1.526 | Angle = 0.012

~ + 1.526 + 0.018 i

z^z^... -> + 2.05668... + 1.14480... i

Sheldon's Base, named after Sheldon Levenstein, is a complex mathematical constant that is very close to the boundary of the Shell-Thron Region near the point at e^e^(-1). It was a tetrational base believed to be particularly difficult to compute values for Complex heights. It's one of the only 9 available base options in the Paulsen Tetration Calculator. It's unusual in that it's one of the very few non-real mathematical constants to get a special name. It lies just slightly above and to the right of e^e^(-1), which is the limit of converging real power towers. Because it has a small imaginary component though it still converges. It's convergence however is very strange. It slowly spirals in large arcs, wildly swinging about. At first it appears to grow without bound, reaching a real value of 4.68 before collapsing back down. It then jumps down to a real value as low as -2.55666. It takes so long to settle down in fact and is so erratic that one would be excused for believing it doesn't converge at all! We can however estimate the convergence value using the Lambert W function. This yields a value of + 2.05668 + 1.1448 i.  

e^e^i

duexion

Magnitude = 1.717 | Angle = 0.841

~ 1.144 + 1.280 i

Another complex googolism I'm coining. Duexpion is a large complex number equal to exp(exp(i)) = exp^2(i) = e^e^i. It's magnitude is 1.717, placing it just outside of the unit circle. It lies within the first quadrant, ensuring the real component is positive. Because the real component is positive, we will again get a large number when we calculate exp(duexpion). See triexpion.

-7/4

Magnitude = 1.75 | Angle = 3.142

= 1.750 + 0.000 i

This is a number that occurs on Robert Munafo's list in connection with the Mandelbrot Set. He describes it as the "Nucleus of the Period-3 Island". What does this mean? Along the Antenna of the Mandelbrot set, you can find mini mandelbrots. This Period-3 Island, is actually the largest of the "minibrots" along the antenna ... by a long shot. Being Period 3 means that the iterative process of the Mandelbrot process should converge unto a cycle of 3 complex numbers.  In the case of -7/4, it cycles between 3 values of (very approximately): -1.74, 1.3,-0.04. Complex Numbers in this vicinity will also have a period of 3. Despite being in the Mandelbrot set, this is a long way from the Shell-Thron Region. In fact any negative real is not in the Shell-Thron Region and therefore a power tower of negatives will not converge no matter what!

2.04779052746 + 0.842045503531 i

North-East Terminal

Magnitude = 2.214 | Angle = 0.390

~ + 2.04779052746 + 0.842045503531 i

This is an estimation of rightmost point of the Shell-Thron Region (along with it's conjugate). That is, this point yields the maximal value of Re(z) within the Shell-Thron Region. If we go any further right then we lie outside the region. Note that 2.048 - 0.842 i also has the maximal real component. Since the border of the Shell-Thron Region is a continuous curve, there is a unique value to this, even though, it would be difficult to obtain. I decided to call this the "Shell-Thron North-East Terminal" on the analogy that the border of the Shell-Thron region can be thought of as a train circuit. In this case, important points along the "track" can be thought of as terminals. See North-West Terminal.

e^e^e^e^i

quadrexion

Magnitude = 2.4603 | Angle = 3.00690008

~ -2.43803034 +0.33038496 i

This is the first member of the expion sequence to wind up in the second quadrant. It's still a large number, but it's the first time there has been a decrease in magnitude. Because of the negative real component, this will be a small number, but because of the small imaginary component, it will be in the first quadrant again leading to a large number again (See quintexpion).

e^i#7

septexion

Magnitude = 2.96145814441 | Angle = 0.030767067267

~ 2.96005657844 + 0.09110100746 i

This number is approximately equal to e. Since the imaginary component is still so small the next member of this sequence will be close to e^e. See octexpion.

e^e^e^i

threxion

Magnitude = 3.138785 | Angle = 1.279883

~ 0.90028908 + 3.00690008 i

So far the magnitude has been steadily increasing with the expion sequence. Not only that, but every member thus far has fallen squarely in the first quadrant, which ensures numbers greater than 1. However this time the real component has actually gotten smaller. This means the next term in the sequence will actually be smaller. Worse yet, because the imaginary component is greater than pi/2 but less than pi, this means it will wind up in the second quadrant leading to a small number next (See quadrexpion).  

{ z | z = 10^z , 0 < Im(z) , 0.5 < Re(z) < 0.75 }

unfathomillion

Magnitude = 3.385 | Angle = 1.414 

~ 0.529474 + 3.342716 i

The name unfathomillion is due to CookieFonster. This number came about in discussions of naive googology in which someone might come up with a number such as a unfathomillion, defined as 1 followed by an unfathomillion zeroes, the implication being, that such a value would have to be infinite. While it is true that no finite real number can satisfy this equation, it turns out there are a countably infinite number of solutions in the complex numbers. This is one of the solutions that was found and which CookieFonster dubbed the "unfathomillion". The solutions come in conjugate pairs, so this means that the conjugate of unfathomillion, ie. 0.529 - 3.343 i will also work as a solution. It turns out however that 0.529+-3.343 i are not the smallest solution pair. See -0.119196+0.750586 i. It should be noted that any number of magnitude greater than 2.5 lies well outside the boundary of the Shell-Thorn Region, so this number will not converge to a unique value. It does appear however to converge to 3 unique values. Ironic that a number that was suppose to be "infinite" turn out to only be about the size of 3, and also that an infinite power tower of it wouldn't diverge to infinity.

10^10^10^10^i

gooioltriplex / iquadruplex

Magnitude = 6.651 | Angle = 1.009

~ + 3.5429321925322 + 5.6289549426484 i

The modulus of this complex number is 6.65112789437. No matter what happens, because the real part is 3.542932, the next modulus in the gooiol sequence should be 10^3.542932 = 3490.85807594, just like the real part of 0.822895298952 for imagitriplex implied the modulus of imagiquadriplex would be 10^0.822895298952 = 6.65112789437. On the other hand the imaginary part of imagiquadriplex is just shy of 2*pi. Multiplying it by ln(10) however brings it once again back to the first quadrant after rotating two full revolutions.

10i

ity

Magnitude = 10.000 | Angle = 1.571

= + 0 + 10 i

Another simple linguistic construction we can create using Joycian Style Googology. Based on the pattern twenty, thirty, fourty, fifty, etc. we say that (z)-ty = 10z. Thus ity would be 10i. Simple. This number would have a magnitude of exactly 10 but pointed in the positive imaginary direction (thus an angle of pi/2 or approximately 1.571). See some other simple googolisms with i: iteen, fzi, iplex, iminex, etc. This number is well outside the Shell-Thron Region or The Mandelbrot Set.

10+i

iteen

Magnitude = 10.050 | Angle = 0.100

= + 10 + 0 i

Another "joycian style" complex googolism we can create. Here we take the generalization of thirteen, fourteen, fifteen -> (x)-teen = 10+x to mean we can apply this to complex numbers. Thus an "iteen" would be 10+i. This number ends up having a magnitude only slightly larger than 10, of about 10.05. See ity.

e^i#8

octexion

Magnitude = 19.3261315107 | Angle = 0.09110100746

~ 19.2190337462 + 1.75573319985 i

At last the Expion Sequence is starting to rapidly grow. Only one problem ... 1.755 is a radian measure for an angle in the second quadrant. So the number is going to shrink again ... but first ... see nonexpion.

10^(3i+3)

i-illion (short)

Magnitude = 1000 | Angle = 0.625

~ + 811.21465284 + 584.748481843 i

If we attempt to compute the ith -illion we have to consider whether we mean according to the short or long scale. This version is the short scale version, assuming that z-illion = 10^(3z+3). So here we obtain 10^(3i+3). Since we can factor out 10^3, it means we will get a magnitude of exactly 1000, making it an appreciably large complex number. The angle is determined by 10^(3i) which yields an angle of 3*ln(10). This works out to 6.9077 which is a little more than 2pi, meaning it goes around once and lands in the first quadrant. Given it's size it's roughly on the order of a gooiolquadriplex.

10^i#5

gooiolquadruplex / iquintuplex

Magnitude = 3490.8580759607 | Angle = 0.395

~ + 3222.3493201896 + 1342.5926281541 i

The modulus of this complex number is 3490.85807596. This value is semi-large. At this point it is gauranteed that the next value in the gooiol sequence will be super-astronomical, containing approximately 3222 digits. The imaginary part presents something of a challenge. We need to multiply it by ln(10) and then find it's value mod 2 pi in order to figure out the angle of rotation. The next value in the sequence is the first that I can not compute directly with my TI-89, as it involves a floating point value that exceeds 9.999x10^999. Therefore to find the next value requires more deliberate calculation.

e^i#9

nonexion

Magnitude = 222,187,848 | Angle = 1.756

~ -40,856,898 + 218,399,070 i

o_0; Well we finally got to something sizable in magnitude, but ... unfortunately the real component is now massively negative ... that means ... we are headed right back to Complex Zero! See decexpion.

10^i#6

gooiolquintuplex / isextuplex

Magnitude = 2.23521956196x10^3222 | Angle = 0.106600418937

~ + 2.22253145843x10^3222 + 2.3782431723x10^3221 i

When computing this value in radian mode on a TI-89 it returns inf + inf*i. This however is simply due to overflow. This means we need to find other means to compute it. First we begin with:

imagisextiplex = 10^imagiquintiplex = 10^(3222.34932019...+1342.59262815...*i)

= 10^3222.34932019...*10^(1342.59262815...*i) =

10^3222.34932019...*e^(1342.59262815...*ln(10)*i) =

10^3222.34932019...*e^(3091.43377154...*i)

10^3222.34932019...*[cos(3091.43377154...)+sin(3091.43377154...)*i]

Thus to fully compute imagisextiplex we need to find the mod 2pi of 3091.43377154. This returns 0.106600407637. Plugging this back in we get:

10^3222.34932019...*[cos(0.106600407637)+sin(0.106600407637)*i]

= 10^3222.34932019...*[0.994323555028...+0.10639862742...*i]

= 2.22253146317x10^3222 + (2.37824293594x10^3221)*i

The exactitude of this value is somewhat suspect. This is because it strongly depends on the value of 0.106600407637. There is some uncertainty in this value. By other means I computed 0.106600409289. This gives the alternative value:

2.22253146277x10^3222 + (2.37824297266x10^3221)*i

The bold digits are where there is uncertainty. This suggests that these values are fairly accurate. The next value however ...

10^i#7

gooiolsextuplex / imagi-septuplex

Magnitude = 10^10^3222.347 | Angle = 4.292

~ ( - 0.408  - 0.913 i )x10^10^3222.347

Imagiseptuplex is a complicated number very far from complex zero, but whose exact value requires a massive calculation. To calculate it's modulus all we need to do is calculate 10 to the power of the real component of imagisextuplex. This tells us 10^i#7 is a distance of approximately 10^10^3222 from 0 on the complex plane. The problem is however is determining the direction! To find that we would have to find the remainder of 2.37x10^3221xln10 when dividing by 2pi. This requires us to actually compute the 3222 digit number (plus some extra decimal places) in order to get an accurate remainder. With this information we could determine the quadrant 10^i#7 is in. If it's in quadrant I or IV then the real part is positive, and the modulus of 10^i#8 will be extremely far from 0, at about 10^10^10^3222 units away. However if 10^i#7 is in quadrant II or III then the real part is negative, and the modulus of 10^i#8 will be extremely close to 0, at about 10^-10^10^3222 units close. Figuring out which quadrant 10^i#7 falls into is not completely beyond what we can compute. With modern computers it is possible. But this value can definitely not be found by ordinary means. It required specialized programs that allow for arbitrary floating point calculations (up to the limits of the hardware and available memory). 

The argument of this number was not known until recently (2024) when I calculated it using a high precision complex number calculator. It turns out the argument is approximately 4.292 and that it falls in the third quadrant. On account of this it means that the real component is massively negative which means imagi-octuplex is in fact very very small rather than very very large. This means it's no longer unsorted but can now be placed on the list.

10^i#9

gooioloctuplex / imagi-nonuplex

Magnitude ~ 1± | Angle ~ 0±

~ + 1 ± 10^(-10^10^3222) i 

At this point we can not know much about this number. Since imagi-octuplex is very small yet we do not know it's precise angle, both it's real and imaginary part are almost 0. Since the real part is close to 0, this means the radius would be close to 10^0 = 1. Since the imaginary part is very small, this means that the answer is a value either a little above or below 1 on the complex plane, depending on the sign of the imaginary part. Since we can not precisely know the real part of imagi-octuplex we can not know if this number is a little larger or smaller than 1, thus we can not effectively decide whether it belongs in the "Sub-Normal" Category or the "Large" Category, making it effectively unsortable. Despite this, we do know it's general location. A nearly-circular path can be traced around 1 to represent all the possible values of this number. 

10^i#10

gooiolnonuplex / imagi-decuplex

Magnitude ~ 10± | Angle ~ 0±

~ + 10 ± 10^(-10^10^3222) i

This number looks a lot like dekalogue, but it is topped off with an i. The exact value of this number is not known, but it has a "point of orbit" at 10+0i. The exact nature of this "orbit" is not known at this time, but it is likely extremely small.

10^i#11

gooioldecuplex / imagi-undecuplex

Magnitude ~ 10,000,000,000± | Angle ~ 0±

~ + 10,000,000,000 ± 10^(-10^10^3222) i

This value is believe to be in "orbit" around the value of 10^10 on the real line. It's called an orbit because there is a curve of possible positions for this value but where it is along this curve is not known, and more to the point, may never be known, as it requires a calculation involving 10^3222 digits of precision, which is beyond any computers means to actually compute. There is believed to be an eventual "divergence" off the real line, but it's not known precisely when this happens. See imagi-duodecuplex for more information. 

10^i#12

gooiol-undecuplex / imagi-duodecuplex

Magnitude ~ (10^10^10)± | Angle ~ 0±

~ + 10^10^10 ± 10^(-10^10^3222) i

This is an extremely weird value. It is believed to be in orbit around a trialogue (10^10^10), but it is suspected that the sequence diverges significantly off the real line shortly after this point. See imagi-tredecuplex.

10^i#13

gooiol-duodecuplex / imagi-tredecuplex

Magnitude ~ (10^10^10^10)± | Angle ~ 0±

~ + 10^10^10^10 ± 10^(-10^10^3222) i

At this point the magnitude of the numbers is beginning to threaten the stability of the very very small imaginary component. In the next step it will finally be the case that the magnitude will overcome the very small value of sine and produce a large imaginary value instead. See imagi-quattuordecuplex.

10^i#14

gooiol-tredecuplex / imagi-quattuordecuplex

Magnitude ~ (10^10^10^10^10)± | Angle ~ 0±

~ + 10^10^10^10^10 ± 10^(10^10^10^10-10^10^3222) i

Absolutely positively insane number! When looked at from the scale of a pentalogue, it will appear to be indistinguishable from it. However if you zoom in sufficiently you'd discover that it's imaginary component was frightfully large, roughly on the order of a pentalogue, though approximately 10^10^3222 orders of magnitude smaller (which is why it appears so close to pentalogue when zoomed out). Despite this, the imaginary value becomes the angle in the next step (roughly) and so the result is the angle of the next number in the sequence is not knowable. At the same time, we still know it's going to be very very large ...

10^i#15

gooiol-quattuordecuplex / imagi-quindecuplex

Magnitude ~ 10^10^10^10^10^10 | Angle ~ ±10^10^10^10^10

~ (cos( ?.??? ) + sin( ?.??? ) i ) x 10^10^10^10^10^10

Still here? So at this point the angle or phase of this number is approximately 10^10^10^10^10. Think of it as if we spun a spinner googologically fast. Where would it land? That's basically why we can't know what direction this number points in. And since we don't know the direction we can not determine if the next member is small or large ...

10^i#16

gooiol-quindecuplex / imagi-sexdecuplex

Magnitude ~ 10^(±10^^6) | Angle ~ ±10^^6

~ (cos( ?.??? ) + sin( ?.??? ) i ) x 10^(±10^^6)

At last we reach a point where, because we can definitely not determine the phase of the previous member of the sequence, the next one can either be massively large or massively small, and there is no way to tell which. Even if we did know, the direction would be even more difficult to find than the last one because now we'd need to know where the "spinner" landed after traveling 10^^6 radians (approximately). If this value turned out to be near-zero, then we will get a sequence, 0±ze, 1±ze, 10±ze, (10^10)±ze , (10^10^10)±ze, etc. where ±ze here indicates a vanishingly small complex value. Eventually we get back to around the same size as 10^i#15 where we diverge from this sequence, and depending on our luck, extend it by a few additional steps until it collapses again. This cycle repeates indefinitely becoming more and more "twisted" as all of the previous cycles get folding into the precise calculations (each requiring googological precision). That leads us to ... 

10^i#100

graniol / imagi-centuplex

Magnitude < 10^^92 | Angle ~ ????????

~ ( cos(????????) + sin(????????) i ) x 10^????????

After we reach the 100th member of the gooiol sequence, we reach a point where basically nothing can be known about this number. Even if there is a point in the sequence where it reaches 1, because it's not exactly 1, the imaginary part will slowly grow as plexes keeps being applied. Eventually it would get large enough causing the imaginary part to be googologically large again. This again leads to rotations and uncertainty about whether we have a large or small number. This process is likely to repeat a few times before reaching 10^i#100. By the time we get here, there is no way to know what part of the sequence it is even at. So this is a completely mysterious and unknown complex number ... or it is. We can actually bound this value on the complex plane. We can note that if we have 10^z where |z| = 1, that the magnitude of the result must be equal to 10^Re(z), and since the largest possible real component is 1, this means the largest magnitude must be 10. This can only happen if all the magnitude is committed to the positive real direction. It follows then that 10^10^z where |z| = 1 is maximal when 10^z is maximal, which implies the largest magnitude is 10^10. In general the maximal magnitude of 10^i#n is 10^1#n which may also be written as 10^^n. Thus, even though we don't know the exact location of a graniol, we know it must be within an orbit of a giggol units (10^^100) of complex zero! The upshot of this is, even though we can't know where it is, this doesn't mean it can be arbitrarily large. It can be bounded from above. The smallest possible value of such an expression is far far more complex, however we can create a simple argument to create a weak lower bound. Note that by the same argument 10^z where |z|=1 can not be smaller than 0.1 and this can only happen when z=-1. For 10^10^z we can note that because 10^z lies within an orbit of 10, the "lowest value" it can have is -10. In practice this is not even remotely the case as it only has a magnitude of 10 at a single point where z=1. None the less by this argument, 10^10^z most definitely can not be smaller than 10^-10. By taking the negative of the maximum of the previous bound we can create a bound for the next member. Furthermore we note that 10^i#100 is always bound by 10^^n or 1/(10^^n). This means that a graniol, despite us knowing virtually nothing about it, must have a magnitude smaller than or equal to a giggol and larger than 1/giggol = 10^(-10^^99). So this value can not be arbitrarily (infinitely) small either! 

Actually we know a little more of this number due to recent breakthroughs in computing the gooiol sequence. We know that for n >= 9, | 10^i#n | < 10^^(n-8). This implies that this number can not have a magnitude exceeding 10^^92. So we actually know for a fact that this number is smaller than a giggol.

10^i#(10^1#100)

imagi-gigguplex

Magnitude < 10^^10^^100 | Angle ~ ?.???

~ ( cos(?.???) + sin(?.???) i ) x 10^???????

This is the stuff of nightmares. This computation would take a giggol steps, would involve numbers of digits of precision around a giggolplex, and is just completely beyond our ability to ever know ... and yet we know it's smaller than a giggolplex at least :/

This number is an example of something I will call googologically twisted. That is, it is so complicated as to require googologically large computations thus making it completely beyond reach to determine its size or it's direction.

10^i#100#2

granioldex

Magnitude = [UNDEFINED] | Angle = [UNDEFINED]

~ [UNDEFINED] + [UNDEFINED] i

Couldn't we just use Hyper-E Notation, and Extensible-E in general, to get to really high members of the gooiol sequence, the same way we normally use ExE to get to ridiculously high members of the googol sequence? Well ... no, it doesn't work that way. One might naively assume 10^i#100#2 would yields something of the form 10^10^...^10^10^i with some ridiculous number of 10s, but remember how Hyper-E works. We would get 10^i#100#2 = 10^i#(10^i#100). The problem is the number in the parenthesis (graniol) is not an integer or even real number, but a complex number, and we don't have a well defined meaning to imaginary power tower heights with i stacked on top. Basically this mathematical expression is simply undefined. We would need to define the complex function f(z) = 10^i#(z) first. Once this was the case then 10^i#100#2 would be defined as a consequence of that, since we can just set z = 10^i#100, which we at least know is a well defined complex number. Currently there is no clear way to define this. So it's not just that this number is "unknown" like a graniol or imagi-gigguplex, but that it's simply undefined. It also shouldn't be thought of as a member of the gooiol sequence at all, which is only defined at integer values.

10^i#100#100

greaiol

Magnitude = [UNDEFINED] | Angle = [UNDEFINED]

~ [UNDEFINED] + [UNDEFINED] i

This number is likewise undefined, however, it's worth noting that, if f(z) = 10^i#(z) was defined then it would be possible to nest it. So this number would simply end up being f^100(100) in this case. Welp ... we are well past the limit of reasonable ... so with that ... 

10^i#100#100#2

greaiolthrex

Magnitude = [UNDEFINED] | Angle = [UNDEFINED]

~ [UNDEFINED] + [UNDEFINED] i

This number is like undefined squared! Even if f(z) = 10^i#(z) was defined, we still could not calculate this!!! Instead we have 10^i#100#100#2 = 10^i#100#greaiol, and since greaiol is itself a complex number (assuming it was defined) we now would have to know what 10^i#100#(z) meant. So we would need to define a new complex function g(z) = 10^i#100#(z). You get the idea. Each new level would require the invention of a new function ... just to get through Hyper-E ... o_0; 

Welp ... we are well past the limit of reasonable ... so with that ... 

... we will end our Journey through Complex Googology ...