However a distinction does need to be made between something which is merely a "number" and something which can be deemed a "large number". Louis Epstein originally made the criticism that numbers under a million don't really belong to a large numbers list. I would beg to differ: does not a thousand belong to a large numbers list, seeing as a million simply means great thousand. However Louis Epstein does make a valid criticism that one should make a distinction between a "number" and a "large number". Is zero a large number by any stretch of the imagination? It's the largest non-positive number, sure, but does that make it "large"?
When my "Large Numbers List" was originally released I played loose with my own rules and ultimately decided to begin the list at zero, only forbidding negative numbers. This was because I didn't want to exclude zero on the account that it is the default value for some googological functions, such as the Ackermann function for example.
However, I have recently come to a new policy in regards my large numbers list. For the purposes of this site a Large Number is any number between one and infinity, possibly including one, a proper small number is any number between zero and one, possibly including zero, and a small number is any real number less than one, including negative numbers. In order to make this distinction explicit I have split my original "Large Numbers List" into a "Small Number List" for reals numbers less than one, and a "Large Numbers List" for real numbers equal to or greater than one. Technically one is a borderline case, being neither small nor large, but I'd rather not give one it's own category, and so I have opted to include it amongst the large numbers, thereby giving the Large Number List a definite starting point.
The purpose of the present list is to discuss some real numbers that can't justifiably be called "large numbers". These are "small numbers" in the second sense, so negatives are included. If you want to get right to the action, go to the Large Numbers List Part I, and you'll start right off with one and then the sky is the limit ... literally ...
-infinity ~ -0.00000001
Note that there is really no equation that results in a negative infinity. Even -1/0 can't really be called negative infinity, because the expression in itself is undefined. It can only have meaning as the limit of some function, say -1/x. However the value of that limit depends on whether x is approaching zero from the left or the right. So the expression is still technically ambiguous. The best I can do in terms of a mathematical expression representing this number is a negative sign followed by the infamous lemniscate symbol. This symbol simply means a changing quantity whose value eventually drops below any given negative number, at least in calculus. In a technical sense this is not a number, but rather a "concept" for "falling without bound". However it is not difficult to treat negative infinity as a "definite quantity" defined as:
-∞ < x : Ax=R
In other words, by definition, negative infinity is a quantity such that negative infinity is less than x, for all x in the set of real numbers. It's smaller than any and every real number! The problem is ... by how much? If we add one to it, do we get a real number? No, because if we did then negative infinity would have to be a real number, which contradicts the above definition. But this would be true for any finite positive number, therefore negative infinity must be infinitely less than any real number. But then negative infinity is not unique in being less than the real numbers which means the original definition is flawed because the above property CAN NOT be used to define a definite quantity. So on some fundamental level the idea of treating infinities as any kind of ordinary numbers is flawed. I would go one step further and suggest that perhaps we should take the properties of ordinary numbers as axiomatic and therefore infinite number should be treated as impossibilities by definition. One of the most basic properties that googology is based on is that n < n+1 for all n. However -∞ = -∞+1, because -∞+1 must still be less than all real numbers, and therefore by the definition is the same as -∞. But this contradicts the fundamental axiom that n<n+1 for all n. Therefore -∞ is not an element of googology.
The only time that negative infinity shows up in googology is when we try to compute loglog1. Since log1=0, and log0 can be thought of as negative infinity it follows that loglog1 = log0 = -infinity.
-(10^^^100 & 10)
-(3^^^3 & 3)
The second largest number on my large number list is a triakulus. Here is the negative version. Negative triakulus is inconceivably less than the next entry :)
double logarithm of one point zero zero zero zero zero one
Curiously, if the argument of the double logarithm is larger than about 1.25 then the result is a small negative number, and if it's less than 1.25 but greater than 1 then the result is a large negative number. (See -1)
double logarithm of one point one
logarithm of one twelfth
logarithm of one eleventh
Negative one is kind of special among the negative numbers. If I was only going to mention a single negative number, this one would be it. It is the square of the imaginary unit: i^2= -1. It also pops up in this very strange equation:
Just a small note of passing interest: -1 = loglog1.25892541179... = log0.1
Although ordering negative numbers seems confusing at first, just remember that in this case a "larger negative" is smaller than a "smaller negative". In other words the order is reversed. In the case of negative numbers, the number closer to zero is always larger. This is in contrast to positive numbers where the number further from zero is always larger.
logarithm of one eighth
logarithm of one seventh
logarithm of one sixth
double logarithm of one point five
logarithm of one fifth
logarithm of one quarter
double logarithm of two
In otherwords, the corrected reduced estimate has to be cubed to get the rough estimate. Considering how large 10^10^19,727.7804 is, you have to imagine that shrinking to an unimaginably small dot amongst exactly that many dots, then imagine that as an unimaginably small dot amongst that many dots to get close to the rough estimate. So you can imagine, there is a big difference from factoring it in, from an ordinary perspective. Although we routinely ignore huge differences like this in googology (numbers are often so far apart that such differences are insignificant), such accuracy for smaller numbers is sometimes necessary to settle a close call. So the number loglog2, although negative, does in fact serve a practical purpose in googology.
logarithm of one third
double logarithm of three
logarithm of a half
double logarithm of four
logarithm of two thirds
double logarithm of five
double logarithm of six
double logarithm of seven
double logarithm of eight
double logarithm of nine
double logarithm of nine point nine
double logarithm of nine point nine nine nine nine nine nine
negative reciprocal of a tethrathoth
Properly Small Numbers
0 ~ 0.99999999
Negative numbers do not relate so much to the concept of quantity as to "position". An axis can have a "central point" at zero, with negatives to one side and positives to another. Quantity however can only travel in one direction from zero, namely, towards positive infinity.
The exclusion of negatives from this list much easier to justify than the exclusion of zero (although I've decided to include them none the less). This is because the negatives really serve almost no purpose in the large numbers field. In order for algorithms to terminate it is necessary to have a minimum value for every argument. This means we have to choose a least number allowable as an argument. Common choices for the minimum value are 1 and 0. If however we allow any integer value, including negatives, we kind of drop the bottom out so to speak and the result is either a function which does not terminate for all values, some values, or requires at least 3 rules (a base case, a less-than-base case, and a more-than-base case). None of these options provides any advantages over simply deciding on a minimum integer value for the function. This is why zero has some importance in the large number field, because it serves as a beginning value. Some of the uses of zero in googology are as the minimum value of an argument in the Ackermann function, and the minimum order of a separator in array notation. It is also the minimum arity of an array. In cantor's system of ordinals, it is the smallest possible ordinal.
Zero crops up when attempting to extend the hyper-operators to all integers. By definition a^^1=a. Since logaa^^n = a^^(n-1), for n>2, we can define this as a law for all integer values. Thus we obtain that a^^0 = logaa^^1 = logaa=1. Thus a^^0=1. Next let a^^(-1) = logaa^^0 = loga1 = 0. Thus we find that any positive integer>1, a, that a^^(-1)=0.
Zero is equal to the double logarithm of 10: loglog10=0. It is also equal to log1.
Zero sometimes leads to undefines as in 1/0 and log0. For this reason I have occasionally been wary of zero. However we rarely have such problems in googology since the functions we work with are usually integral and not continuous.
In some respects zero serves as a natural starting point for large numbers. We can think of it as the starting line, and any positive real becomes part of the race coarse. The end goal can be thought to be at infinity, although it is probably better to think of this as a race with no end goal! We can begin this race, but we can never finish it! Better get going then ...
reciprocal of Graham's Number
two divided by Graham's Number
Conway and Guy have suggested the name "googolminex" for the reciprocol of a googolplex (seen later). It's an example of an extremely small number with an actual name! One of the many consequences of being able to define very large numbers, is that we can also define very small ones. We simply have to take the reciprocal of some large number, and we get it's inverse: a number that is just as small as the original number was large! You can imagine this number as 0.0000000000000000000000000000000000000..................................000000001 where there a googol-1 zeroes after the decimal point. This number is tremendous when compared to the reciprocal of Graham's Number, and yet it is still mind-bogglingly "googol-scopic". If we were to continue with the multiples of a googolminex, such as two googolminex, three googolminex, etc. We would never even have a hope of reaching 1, let alone actually large numbers. So once again we must pick up the pace...
reciprocal of a googol
Planck Time (in seconds)
Planck Length (in meters)
electron diameter (in meters)
logarithm of two
The common logarithms of the primes are also useful because we can compute the logarithms of other positive integers by decomposing them into their prime factors and then using the laws of logarithms to figure out the value. As an example, we can compute the logarithm of 8 easily if we know the logarithm of 2:
There is also a curious feature of the decimal expansion of 1/3. It follows from the definition of 1/3 that 3(1/3) = 1. What happens if we multiply the decimal expansion by 3:
logarithm of three
log9 = log(3*3) = log3+log3 = 2log3 ~ 2(0.477) = 0.954
log27 = log(3^3) = 3log3 ~ 3(0.477) = 1.431
logarithm of four
logarithm of five
logarithm of six
logarithm of seven
logarithm of eight
logarithm of nine
logarithm of nine point nine
logarithm of nine point nine nine nine nine nine nine