Proof: An Infinite Number of Irrational Numbers Can Be Found Between Any Two Rational Numbers On the Number Line.

number line

[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.]

Want Some Really Big Numbers? Here Are the Zoogol and the Zoogolplex.

zoogol

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.

On your nth birthday, you turn n – 1 years old.

birthday cake

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.

  • Student: Guess what? It’s my birthday!
  • Me: Congratulations! How old are you?
  • Student: I’m seventeen!
  • Me: Well, happy 18th birthday, then!
  • Student: Huh?
  • Me: Look, on that one day, 17 years ago, when you were born, that was your birthday. That day has a better claim on being your birthday than any other day, because it’s the day you were born. That was your first birthday. But you weren’t one year old yet. You turned one year old a year later, on your next birthday . . . your second birthday. A year later, on your third birthday, you turned two years old. Do I need to continue?
  • Student: So I’m 18? I can buy cigarettes without a fake ID, and vote, and stuff?
  • Me: No, not for another year, because you’re only 17 years old — but you have had 18 birthdays. Say, here come some of your friends. Use this bit yourself, if you want to, and have fun with it.
  • Student, to other students: Hey, guys, it’s my birthday! I’m 18 today!

…At least I try. Also, sometimes, the educational outcome is better than in this fictionalized example.

 

[Image source: http://www.decorationnako.tk/birthday-cake/]

A Plea for Consistency in the Use of Numerical Prefixes

consistency

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:

Penta Icositetra

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.

The Human Reaction, When Mathematics No Longer Seems to Make Sense: What Is This Sorcery?

Cubes 5

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.

  • Counting numbers: no problem, but what do I call the next one after [last one I knew at that time]? And the next one? And the next? Next? Next? [Repeat, until everyone within earshot flees.]
  • Zero exists: well, duh. That’s how much of whatever I’m snacking on is left, after I’ve eaten it all.
  • Arithmetic: oh, I’m glad to have words for this stuff I’ve been doing, but couldn’t talk about before.
  • Negative numbers: um, of course those must exist. No, I don’t want to hear them explained; I’ve got this already. What, you want me to demonstrate that I understand it? Ok, can I borrow a dollar? Oh, sure, I’ll return it at some point, but not until after I’ve spent it.
  • Multiple digits, the decimal point, decimal places, place value: got it; let’s move on, please. (I’ve never been patient with efforts to get me to review things, once I understand them, on the grounds that review, under such conditions, is a useless activity.)
  • Pi: love at first sight.
  • Fractions: that bar means you divide, so it all follows from that. Got it. Say, with these wonderful things, why, exactly, do we need decimals, again? Oh, yeah, pi — ok, we keep using decimals in order to help us better-understand the number pi. That makes sense.
  • “Improper” fractions: these are cool! I need never use “mixed numbers” again (or so I thought). Also, “improper” sounds much more fun than its logical opposite, and I never liked the term “mixed numbers,” nor the way those ugly things look.
  • Algebra: ok, you turned that little box we used before into an “x” — got it. Why didn’t we just use an “x” to begin with? Oh, and you can do the same stuff to both sides of equations, and that’s our primary tool to solve these cool puzzles. Ok. Got it.
  • Algebra I class: why am I here when I already know all this stuff?
  • Inequality symbols: I’m glad they made the little end point at the smaller number, and the larger side face the larger number, since that will be pretty much impossible to forget.
  • Scientific notation: well, I’m glad I get to skip writing all those zeroes now. If only I knew about this before learning number-names, up to, and beyond, a centillion. Oh well, knowing those names won’t hurt me.
  • Exponents: um, I did this already, with scientific notation. Do not torture me with review of stuff I already know!
  • Don’t divide by zero: why not? [Tries, with a calculator]: say, is this thing broken? [Tries dividing by smaller and smaller decimals, only slightly larger than zero]: ok, the value of the fraction “blows up” as the denominator approaches zero, so it can’t actually get all the way there. Got it.
  • Nonzero numbers raised to the power of zero equal one: say what? [Sits, bewildered, until thinking of it in terms of writing the number one, using scientific notation: 1 x 10º.] Ok, got it now, but that was weird, not instantly understanding it.
  • Sine and cosine functions: got it, and I’m glad to know what those buttons on the calculator do, now, but how does the calculator know the answers? It can’t possibly have answers memorized for every millionth of a degree.
  • Tangent: what is this madness that happens at ninety degrees? Oh, right, triangles can’t have two right angles. Function “blows up.” Got it.
  • Infinity: this is obviously linked to what happens when dividing by ever-smaller numbers, and taking the tangent of angles approaching a right angle. I don’t have to call it “blowing up” any more. Ok, cool.
  • Factoring polynomials: I have no patience for this activity, and you can’t stop me from simply throwing the quadratic formula at every second-order equation I see.
  • Geometry (of the type studied in high school): speed this up, and stop stating the obvious all the time!
  • Radicals: oh, I was wondering what an anti-exponent would look like.
  • Imaginary numbers: well, it’s only fair that the negative numbers should also get square roots. Got it. However, Ms. _____________, I’d like to know what the square root of i is, and I’d like to know this as quickly as possible. (It took this teacher and myself two or three days to find the answer to this question, but find it we did, in the days before calculators would help with problems like this.)
  • The phrase “mental math” . . . um, isn’t all math mental? Even if I’m using a calculator, my mind is telling my fingers which buttons to press on that gadget, so that’s still a mental activity. (I have not yielded from this position, and therefore do not use the now-despised “mental math” phrase, and, each time I have heard it, to date, my irritation with the term has increased.)
  • 0.99999… (if repeated forever) is exactly equal to one: I finally understood this, but it took attacks from several different directions to get there, with headaches resulting. The key to my eventual understanding it was to use fractions: ninths, specifically.
  • The number e, raised to the power of i‏π, equals -1: this is sorcery, as far as I can see. [Listens to, and attempts to read, explanations of this identity.] This still seems like sorcery!
  • What it means to take the derivative of an expression: am I just supposed to memorize this procedure? Is no one going to explain to me why this works?
  • Taking the derivative of a polynomial: ok, I can do this, but I don’t have the foggiest idea why I’m doing it, nor why these particular manipulations of one function give you a new function which is, at all points along the x-axis, the slope of the previous function. Memorizing a definition does not create comprehension.
  • Integral calculus: this gives me headaches.
  • Being handed a sheet of integration formulas, and told to memorize them: hey, this isn’t even slightly fun anymore. =(
  • Studying polyhedra: I finally found the “sweet spot” where I can handle some, but not all, of the puzzles, and I even get to try to find solutions in ways different from those used by others, without being chastised. Yay! Math is fun again! =)
  • Realizing, while starting to write this blog-post, that you can take the volume of a sphere, in terms of the radius, (4/3)πr³, take its derivative, and you get the surface area of the same sphere, 4πr²: what is this sorcery known as calculus, and how does it work, so it can stop looking like sorcery to me?

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.]

My Early Play with Informal Numbers, Such as Umpteen: A Look at Early Development of a Special Interest in a Young Person with Asperger’s

umpteen

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.

How to Get Fair Dice with Various Numbers of Possible Results, from Two to Thirty, Using Different Polyhedra

  • For a d2, number a cube’s faces with three ones and three twos.
  • For a d3, number a cube’s faces 1,1,2,2,3,3.
  • The standard d4 is a Platonic solid, the tetrahedron. Double-numbered (two ones, two twos, etc.) octahedra are sometimes used as d4s, also.
  • For a d5, an icosahedron can be renumbered with four each of the numbers one through 5. Double-numbering a pentagonal dipyramid or pentagonal trapezohedron also works.
  • The d6 is the familiar cube.
  • For a d7, one option is to roll an octahedron, but re-roll 8 each time it comes up.
  • For a d9, one option is to roll a d10, but reroll 10s.
  • For d10s, pentagonal dipyramids and pentagonal trapezohedra both work. There is also the option of double-numbering an icosahedron.
  • For a d11, one option is to roll a d12, but reroll 12s.
  • For a d12, the Platonic and rhombic dodecahedra both work.
  • For a d13, roll a d14, but reroll 14s.
  • For a d14, one option is to roll a d7 and a d2, then add 7 to the d7 result iff the d2 shows 2. Another is to roll a d15, but reroll 15s.
  • For a d15, simply double-number the thirty faces of a rhombic triacontahedron.
  • For a d16, roll a d2 and a d8 together, using the d8 result, but adding 8 to the d8 result iff the d2 result is 2.
  • For d17, roll a d18, but reroll 18s.
  • For d18, roll a d2 and a d9 together, using the d9 result, but adding 9 to the d9 result iff the d2 result is 2.
  • For d19, roll a d20, but reroll 19s.
  • For d20, icosahedra are used.
  • For d21, one option is to roll a d24, rerolling any result from 22-24. Another is roll a d7 and a d3 together, using the d7 result, but adding 7 to this d7 result iff the d3 result is 2, but adding 14 to the d7 result iff the d3 result is 3.
  • For d22, one option is to roll a d24, rerolling any 22s and 24s. Another is to roll a d2 and a d11 together, using the d11 result, but adding 11 to this d11 result iff the d2 result is 2.
  • For d23, use a d24, and re-roll 24s.
  • Options for the d24 include the triakis octahedron, the tetrakis cube, the deltoidal icositetrahedron, and the pentagonal icositetrahedron, all of which are Catalan solids (duals of the Archimedeans). Another d24 can be made by rolling a d2 and a d12 together, and using the d12 result, but adding 12 to this d12 result iff the d2 result is 2.
  • For a d25, roll two distinguishabale d5s, called d5a and d5b. The 1-25 random number is (d5a)+ (5)(d5b-1).
  • For a d26, roll a d13 and a d2, then add 13 to the d13 result if the d2 shows 2. Another is to roll a d15, but reroll 15s. The first option may require two different d2s, so they will have to be distinguishable, in that case, or one d2 must be rolled twice, each for a different purpose.
  • For a d27, roll a d9 and d3 together. The result is (d9) + (9)(d3-1).
  • For a d28, roll a d14 and d2 together, using the d14 result, but adding 14 to it iff the d2 result is 2. Another option:  roll a d30, but reroll results of 28 or 29. The first option may require two different d2s, so they will have to be distinguishable, in that case, or one d2 must be rolled twice, each for a different purpose.
  • For a d29, roll a d30, and reroll 30s.
  • The most common d30 is a rhombic triacontahedron. After the Platonic solids and the d10, these d30s are the most commonly available example of polyhedral dice.