When a typical calculator is asked to find 555^555, it causes an overflow error, and returns an error message. Not Google’s calculator, though.
Infinity? 555^555 is absurdly large, but it isn’t infinite. Just for starters, 555^555 + 1 is bigger. Infinity is larger than any number. It’s as far from 555^555 as it is away from the number one — an infinite distance, on any number line. Hopefully, as Google continues getting smarter, this will get fixed.
[This theorem was proven long ago, in other ways, but this is my way to prove it.]
Let x and y be two rational numbers on the number line. Since both x and y are rational, both x and y can be written as fractions. All fractions, when written in decimal form, either terminate (such as ¼ = 0.25) or form a repeating pattern (such as 1/3 = 0.333…, repeating). To find an infinite number of irrational numbers between x and y on the number line, simply write both x and y in the form of decimals, and then follow the decimal expansion until the two digits no longer match, as is the case for 0.1111172 and 0.1111173, which match each other, up to the digit 7 in the millionths place, but no further. To generate an infinite number of irrational numbers between x and y, simply examine the part of the decimal expansion of the smaller of the two numbers, x or y, which does not match the other number, and randomly jumble up all of the digits (including trailing zeroes) of the smaller of the two numbers, x or y, after the match-point, changing these digits as well (in random ways), which will result in the creation of an irrational number between x and y. Since there is no limit to the number of decimal places a number may have, this may be done in an infinite number of ways.
[I have now fulfilled my ambition to use the phrase “randomly jumble up” in a formal proof.]
Portmanteaus of the words “zillion” and “googol” / “googleplex,” these two new number-names are offered for the use of anyone who feels a need for even more named numbers which are ridiculously larger than the number of particles known to exist. The vast majority of those particles are neutrinos, by the way, but fewer than a googol (let alone a zoogle) neutrinos are estimated to exist in the entire observable universe.
To write a zoogle out the long way, simply write a “1,” then follow it with a “mere” one million zeroes. To my knowledge, this has never been done, but it certainly is possible. To write out a zoogleplex the long way, you’d need to follow the “1” with a zoogol zeroes, but this is not possible, due to a lack of enough matter or space, in the entire universe, for such a task.
Pop quiz: which is larger: a googleplex, or a zoogle? Scroll down to find the answer, whenever you are ready.
(keep scrolling….)
A googolplex is vastly larger than a zoogol. However, all other numbers mentioned earlier in this post are dwarfed by a zoogolplex. However, even a zoogolplex is less than 1% of 1% of 1% of . . . 1% of infinity, no matter how long one makes the “of 1%” chain.
As a teacher, I have had variants of this conversation many times. The specific details, however, are fictional, for this changes, somewhat, each time it happens.
…At least I try. Also, sometimes, the educational outcome is better than in this fictionalized example.
[Image source: http://www.decorationnako.tk/birthday-cake/]
First, let’s face facts: the numerical prefixes currently in use, in English, are a horrible mess. Most of the ones used with polyhedra, for example, such as tetra- (4) and penta- (5), are derived from Greek. For polygons, however, a four-sided figure is usually called a quadrilateral, with “quad-” derived from Latin, just as it is in “quadrillion,” or “quadruplets.” Why use two prefixes for the number four? It would be more consistent (and therefore better), since four-faced polyhedra are called tetrahedra, for four-sided polygons to be called tetragons, just as we call five-sided polygons pentagons. Consistency improves comprehension, simply by reducing the number of prefixes one needs to understand, and can therefore aid in both teaching and learning. Inconsistency, though, has the opposite effect, and that benefits no one.
The Greek-based prefix for 5, “penta-,” has a Latin-based rival, also: “quint-,” as in quintuplets, or the number quintillion. It doesn’t make sense to use two different prefixes for the same thing, for both English, and mathematics, are complicated enough without adding unnecessary complications. The necessary complications are quite enough!
My preference is for Greek-based prefixes, for two reasons: (1) more of them are in use than their Latin counterparts, and (2) the Latin-speaking Romans appropriated ideas from the ancient Greeks, not the other way around.
Even the number one is not immune from this problem. For one, we use “mono-,” “uni-,” “un-,” “uni-,” “en-, “and “hen-,” all to mean “one,” and each has at least a slightly different derivation. Examples include “monomer,” “monologue,” “unicycle,” “undecillion,” “undecagon,” “endecagon,” and “hendecagon,” the last three of which all name the same polygon. (Also, these last three prefixes are for 11, actually, formed by combining a prefix for the one with the Greek-based “deca-” prefix for ten. Combinations of prefixes will be addressed later.) I call 11-sided polygons “hendecagons,” for both prefixes in that word are derived from Greek.
Prefixes for the number two are also unnecessarily numerous, as well as ambiguous. “Bi-” is used in “bicycle,” “binary,” and “billion,” but that’s a horrible idea, since “bi-” is also used, in some cases, for ½. This shows up, for example, in chemistry: the bulk of a carbonic acid molecule, if fully ionized, is called the carbonate ion. However, if it is only half-ionized, it is often called the bicarbonate ion, as in sodium bicarbonate, better-known as baking soda. In chemistry, “di-” is used for two, as in carbon dioxide, a molecule containing two oxygen atoms. “Do-” and “duo-” are also both used for the number two, with the first derived from Greek, and the second from Latin. When combined with the Greek prefix for ten, to make twelve, these prefixes appear in words such as “dodecagon,” “dodecahedron,” and “duodecimal.” I find the word “duodecimal” particular irritating, for it combines Greek and Latin prefixes in a single word. If one person had deliberately designed this entire system, with the goal of causing confusion, it would have taken a lot of work to invent a system more confusing than the one we actually use.
If, for ½, we only used “bi-,” that would be nice, but that isn’t what we do. Half a circle is a semicircle, and then half a sphere is a hemisphere. Since it originates from Greek, my preference is for “hemi-.”
At least three’s prefix is usually consistent, with “tri-” being all-but-universal. The only exception I know of appears when “tri-” is combined with “deca-,” to create a prefix for thirteen, and the Greek work for “and,” which is “kai,” often appears with it, as in triskaidecaphobia, the fear of the number thirteen — in this word, “tri-” is modified to “tris-.” However, a thirteen-sided polygon is simply called a “tridecagon,” with no “s” attached to “tri-,” and the “kai” omitted.
I don’t actually care if we use “kai,” or not, in numerical prefixes, but we should pick one or the other, and stick with it. It makes no sense that a fifteen-sided polygon is usually called a “pentadecagon,” while sometimes called a “pentakaidecagon.” Why do we not simply choose just one?
Six and seven are similarly troublesome. The numbers “sextillion” and “septillion,” as well as the month of September, all use Latin-derived prefixes for these numbers. I prefer the Greek-derived prefixes used with polygons and polyhedra: “hexa-,” and “hepta-.” With eight, though, as in the case of three, English-speakers lucked out, with “octopus,” “octillion,” “octagon,” and “octahedron” all starting with the same three letters.
With nine, however, our system falls apart again. In high school, geometry students are taught the Latin-prefix-containing word “nonagon” for a nine-sided polygon, and “November” contains yet another Latin-based prefix meaning nine. (It was named the ninth month, rather than the eleventh, because the start of each new year was marked with the first day of Spring in ancient times, rather than the first day of January.) A professional mathematician, however, is more likely to call a nonagon an “enneagon,” for “ennea-” is derived from Greek, making “enneagon” consistent with its “neighbors,” the octagon and the decagon. Ten is not a problem, though, for the Greek-based “deca-” was simply appropriated by the Latin-speaking Romans, who named their tenth month December — using a prefix close enough to “deca-” that it is unlikely to cause confusion.
One numbers exceed ten, though, a new problem is encountered, in addition to the issue of whether or not we use “kai.” Numbers such as 12 and 24 require us to combine prefixes, but there is no consistency in the order in which this is done. For example, a twelve-faced polyhedron is a “dodecahedron” — using a prefix for two, followed by a prefix for ten: the smaller number, and then the larger number. We continue this practice with words such as “pentadecagon,” already described above. Then, however, we have this thing, the dual of the snub cube:
The faces of this polyhedron are 24 pentagons, and it isn’t the only well-known polyhedron with 24 faces, so “pentagonal” is part of this polyhedron’s name, which makes sense. However, if its name followed the pattern in the paragraph above, that would make it a “pentagonal tetraicosahedron,” or perhaps a “pentagonal tetrakaiicosahedron” — the smaller “tetra-,” meaning “four,” would come before the larger “icosa,” meaning twenty. At least both these prefixes originated in the Greek language, but, for mysterious reasons, the prefixes are put in the reverse order, relative to the order used for the dodecahedron: it is called the “pentagonal icositetrahedron.” Polyhedral names are hard enough to learn without arbitrary switches between “smaller, then larger,” and its opposite, “larger, then smaller.” We should choose a method, one or the other, and then stick to it.
[Note: the rotating polyhedron above was created using Stella 4d, software you can buy, or try for free, at this website.]
In chemistry, naming-disputes (what to call a newly-synthesized element, for example) are settled by the IUPAC: the International Union of Pure and Applied Chemistry. I know of no organization with a corresponding role in the field of mathematics, but, if one were created, perhaps that would help get this mess cleaned up.
Unless you understand all of mathematics — and absolutely no one does — there is a point, for each of us, where mathematics no longer makes sense, at least at that moment. Subjectively, this can make the mathematics beyond this point, which always awaits exploration, appear to be some form of sorcery.
Mathematics isn’t supernatural, of course, but this is a reaction humans often have to that which they do not understand. Human reactions do not require logical purpose, and they don’t always make sense — but there is always a reason for them, even if that reason is sometimes simply that one is utterly bewildered.
In my case, this is the history of my own reactions, as I remember them, to various mathematical concepts. The order used is as close as I can remember to the sequence in which I encountered each idea. The list is, of necessity, incomplete.
Until and unless I experience the demystification of calculus, this blog will continue to be utterly useless as a resource in that subfield of mathematics. (You’ve been warned.) The primary reason this is so unlikely is that I haven’t finished studying (read: playing with) polyhedra yet, using non-calculus tools I already have at my disposal. If I knew I would live to be 200 years old, or older, I’d make learning calculus right now a priority, for I’m sure my current tools’ usefulness will become inadequate in a century or so, and learning calculus now, at age 47, would likely be easier than learning it later. As things are, though, it’s on the other side of the wall between that which I understand, and that which I do not: the stuff that, at least for now, looks like magic — to me.
Please don’t misunderstand, though: I don’t “believe in” magic, but use it simply as a label of convenience. It’s a name for the “box ,” in my mind, where ideas are stored, but only if I don’t understand those ideas on first exposure. They remain there until I understand them, whether by figuring the ideas out myself, or hearing them explained, and successfully understanding the explanation, at which point the ideas are no longer thought of, on any level, as “magic.”
To empty this box, the first thing I would need would be an infinite amount of time. Once I accepted the inevitability of the heat death of the universe, I was then able to accept the fact that my “box of magic” would never be completely emptied, for I will not get an infinite amount of time.
[Image credit: I made a rainbow-colored version of the compound of five cubes for the “magic box” picture at the top of this post, using Stella 4d, a program you may try here.]
As a young child (before I started school), my strong interest in mathematics was always there. No one knew I had Asperger’s at that time, but it is clear to me now, in retrospect, that I was a young “Aspie,” in the early stages of the development of a special interest.
I cannot remember a time without my math-fascination, to the point where I speculate that I was motivated to learn to talk, read, and write English simply to bring more of the mathematics in my head into forms which I could express, and also to gain the ability to research forms of mathematics, by reading about them, which were new to me: negative numbers, fractions, names for extremely large numbers, and so on. I would devour one concept, internalize it, so it could not be forgotten, and quickly move on to my next mathematical “snack.” The shift to geometry-specialization took many years longer; at first, my special interest was simply mathematics in general, to the extent that I could understand it.
I was too young, then, to even understand the difference between actual numbers, and informal numbers I heard others use in conversation, such as zillion, jillion, and especially umpteen, and, armed with this lack of understanding, I endeavored to figure out the properties of these informal numbers. Zillion and jillion were uncountably large: that much seemed clear, although I could never figure out which one was larger. Umpteen, however, seemed more accessible, due to the “-teen” prefix. It seemed perfectly reasonable to me to simplify umpteen to a more fundamental informal number, “ump,” simply by subtracting ten from umpteen, following the pattern I had noticed which connects thirteen to three, seventeen to seven, and so on. This led to the following:
1ϒ – 10 = ϒ (umpteen minus ten equals ump)
I wasn’t using upsilon as a symbol for the informal number “ump” at that age. Rather, I simply needed a symbol, today, to write this blog-post, so I chose one. The capital Greek letter upsilon seems like a good pick. I’m using it more like a digit, here, rather than a variable — although, when I first reasoned this out, over forty years ago, I had not yet learned to distinguish between digits, variables, and numbers, at least not using other peoples’ terms.
Occasionally, I would hear people use ump-based informal numbers (I grew up in Arkansas, you see) which clearly seemed larger than umpteen. One such “number” I heard was, of all things, “umpty-ump.” Well, just how large is umpty-ump? I reasoned that it had to be umpteen minus ten, with this difference then multiplied by eleven.
1ϒ – 10 = ϒ (umpteen minus ten equals ump)
10(ϒ) = ϒ0 (ten times ump equals umpty)
ϒ0 + ϒ = ϒϒ (umpty plus ump equals umpty-ump)
Factoring ump out of the third equation above yields the following:
ϒ(10 + 1) = ϒ(11)
Next, ump cancels on both sides, leaving the following, which is known to be true without the involvement of informal numbers:
10 + 1 = 11
Having figured this out, I would then explain it, at great length, to anyone who didn’t make their escape quickly enough. It never occurred to me, at that age, that there actually are people who do not share my intense interest in mathematics. (Confession: I still do not understand the reason for the shockingly small amount of interest, in mathematics, found in the minds of most people. Why doesn’t everyone find math fascinating, since, well, it is fascinating?)
What I didn’t yet realize is that I was actually figuring out important concepts, with this self-motivated mathematical play: place value in base-ten, doing calculations in my head, some basic algebra, and, of course, the fact that playing with numbers is ridiculously fun. (That last one is a fact, by the way — just in case there is any doubt.)
I did not distinguish play from work at that age, and considered any interruption absolutely unacceptable. This is what I would typically say, if anyone, including my parents, disturbed me while I was working these things out, but was not yet ready to discuss them: “I’m BUSY!”
Everyone who knew me then, I am guessing, remembers me shouting this, as often as I found it necessary.
In this graph, each number on the x-axis (from 2 to 30) is plotted against the sum of all its factors (including one, but excluding the number itself) on the y-axis. Numbers on the blue line y = 1 have no factors other than one and themselves, and are therefore prime numbers. Numbers on the green line y = x are equal to the sum of their factors (including one, but excluding themselves), and are therefore perfect numbers. Perfect numbers are much rarer than prime numbers in the entire set of natural numbers, as well as in this small sample.
If a number’s factor-sum, examined in this manner, is smaller than the number itself, such a number is called a “deficient number.” This applies to all numbers with points below the green line. Numbers which have points on the blue line are deficient numbers, as well as being prime numbers – and this is true for all prime numbers, no matter how large. The numbers represented by points between the green and blue lines are, therefore, both deficient and composite, and can also be called “non-prime deficient numbers.”
A few numbers on this graph, called “abundant numbers,” are represented by points above the green line, because their factor-sum is greater than the number itself. There are only five abundant numbers in this sample: 12, 18, 20, 24, and 30. As an example of how a number is determined to be abundant, consider the factors of 30: 1+2+3+5+6+10+15 = 42, which is, of course, greater than 30.
Of the 29 numbers examined in this sample, here is how they break down by category:
• Abundant numbers: 5 (~17.2% of the total)
• Perfect numbers: 2 (~6.9% of the total)
• Non-prime deficient numbers: 12 (~41.4% of the total)
• Prime numbers: 10 (~34.4% of the total)
These percentages only add up to 99.9%, due simply to rounding. Also, the total number of deficient numbers in this sample (both prime and composite) is 22, which is ~75.9% of the total sample of 29 numbers.
So what happens if this survey is extended far beyond the number 30, to analyze much larger (and therefore more meaningful) samples? Well, for one thing, the information on the graph above would quickly become too small to read, but that is only of trivial importance. More significantly, what would happen to the various percentages, for each category, given above? First, both prime and perfect numbers become more difficult to find, as larger and larger numbers are examined – so the percentages for these categories would shrink dramatically, especially the one for perfect numbers. With smaller percentages of prime and perfect numbers in much larger samples, the sum of the percentages for the other two categories (abundant and non-prime deficient numbers) would, of necessity, grow larger. That has to be true for this sum – but that says nothing about what would happen to its two individual components. My guess is that abundant numbers would become more common in larger samples . . . but since I have not yet examined the data, I’m only calling this a guess, not even a conjecture. As for what would happen to the percentage of non-prime deficient numbers when larger samples are analyzed, I don’t even (yet) have a guess.
This program is written in Just BASIC v1.01, which you may download for free at http://justbasic.com/download.html.
*** *** ***
10 print “For what number do you want the prime factorization”;
20 input n
25 c = 3
30 if n <> int(n) then end
40 if n < 4 then goto 450
50 if n/2 <> int(n/2) then goto 100
60 print ” 2″;
70 n = n/2
75 if n = 1 then goto 400
80 goto 50
100 if n/3 <> int(n/3) then goto 200
110 print ” 3″;
120 n = n/3
125 if n = 1 then goto 400
130 goto 100
200 c = c + 2
205 p = 0
210 for t = 3 to c step 2
220 if c/t = int(c/t) then goto 290
230 if n/t <> int(n/t) then goto 290
240 if p > 0 then goto 290
250 p = t
290 next t
295 if p = 0 then goto 200
300 print ” “;p;
310 n = n/p
320 if n = 1 then goto 400
330 if n/p = int(n/p) then goto 300
340 goto 200
400 print
410 goto 10
450 if n = 2 then print ” 2″;
460 if n = 3 then print ” 3″;
470 goto 400
500 end
RobertLovesPi.net 