Math Journey 4: Trips to Infinity
Uncle Bob
This math journey will involve several treks. The goal is to get a fuller perspective on the nature of infinity. What is it? Can we reach it? Can it be contained? Is there more than one infinity? To accomplish the goal we will make several trips employing different modes such as numbers, arithmetic, sequences, geometry, and maybe even going around in circles.
Trip #1: To Infinity via Numbers?
Pick a number. Any number. Now name a number that is larger than the number you picked. You are able to do this in every case – one reliable method is merely to add one (1) to the number. There is no largest number. If that is true, then what is infinity? Infinity historically has been regarded as both a metaphysical concept and a mathematical limit, but infinity is not a number. Let's see how close we can get to it.
The United States is trying to get to infinity by spending it: thebalance.com reports that President Biden’s budget for FY 2022 totals $6.011 trillion, eclipsing all other previous budgets. Written in full the number is
$6,011,000,000,000.
The national debt gets us even closer to infinity. From US Deb Clock, as of May 23, 2022 total public debt outstanding was more than $30,467,000,000,000. That's over $30 trillion.
Let's get away from the red ink for a bit and think about protecting our money. These days we commonly hear stories of thousands if not millions of people having private information stolen – this in spite of the security measures taken. Cryptology, the science of making and breaking codes, is a fast-growing and much studied field these days, and it is a necessity in e-commerce and other online transactions. Codes that protect account information can be made safer but not unbreakable. The degree of safety is related to cost and is therefore a business decision.
One simple code is a type of puzzle that many people solve every day – the cryptogram. This puzzle encodes a message by swapping out each letter for a different one in the alphabet. The new lineup is called the replacement alphabet, and it changes from one message to the next. How many different alphabets are there? A basic formula in mathematics tells us there are
26 x 25 x 24 x 23 x 22 x 21 x 20 x 19 x…x 4 x3 x 2 x 1
different orderings (permutations) of the 26 English letters. That's right, we multiply all the numbers from 1 through 26 together. Before we look at this number, we have to tell you that it over counts the usable substitution alphabets because some of the possible orderings have one or more letters remain in their original position. They can't be used because it would be like swapping an "I" for an "I." We need a count of new alphabets where all of the letters are in a new position. Mathematicians call these orderings "derangements." I know, it sounds crazy.
If we had just two letters in our alphabet, AB, there would be only one derangement, namely BA. In the case of three letters, ABC say, there are six possible permutations, including ACB, with A remaining first and BAC, with C last, but only two derangements BCA and CAB. Derangements become much more numerous with a larger supply of letters – 9 derangements for a 4-letter alphabet and 44 derangements for 5 letters.
We could solve for the exact count of derangements of our full alphabet, but we really want to know about how large the number is, and so we will save time (and money?) by taking a short cut. Don't worry – we wouldn't do this with your credit information! Interestingly, it turns out that for larger alphabets the derangements are approximately 37% of the total possible permutations. So we return to the enormous product of one through 26 and take 37% of it. For the English alphabet there are roughly
148, 000, 000, 000, 000, 000, 000, 000, 000 derangements.
That's 148 trillion trillion derangements. Well, that's a lot of security, but as I mentioned, puzzle people like me solve cryptograms with relative ease. And we have not come near infinity because I could type a 9 at the top of this page and fill the rest with zeroes to make a much larger number.
OK. Let's stop messing around and go for a large, large number. A googol? That's a 1 followed by a hundred zeroes – not big enough. A googolplex? That's a one followed by a googol of zeroes. That is a vast number.
Infinity? No. Take any newspaper and let one single printed letter stand for a googolplex. Do all of the letters in the entire paper represent at least a fraction of infinity? No. You'd be richer getting a penny change from a dollar.
So we have failed to reach infinity in this our first trip; however, there are other routes available to us, and we will explore them in three more trips.
Here's a cryptogram for you to solve. My one clue is that the replacement method has been referred to as “atbash.” Look it up.

MLD R'EV HZRW NB ZYX'H. GVOO NV DSZG BLF GSRMP LU NV.
Trip #2: To Infinity via Arithmetic
Sequences and Series: A mathematical sequence is an ordered list of items, and we'll limit our discussion to numbers. Some sequences are calculated, that is, each successive member is determined by some formula that uses either the ordinal of the member or the previous member in the list. For example, the halving sequence that begins {1/2, 1/4, 1/8, ...} can be formulated by halving at each step, or by the formula [(1/2)^ n] (one-half raised to the 1st, 2nd, 3rd, ... powers). This formula can be used to calculate the fourth member.
1/2 x 1/2 x 1/2 x 1/2 = 1/16
A mathematical series is a sequence of numbers that is keeping a running total of some other sequence. For the halving sequence, the series is
{1/2, (1/2) + (1/4), (1/2) + (1/4) + (1/8), ...} = {(1/2), (3/4), (7/8), …}
Halfway to the Wall. If I start at one end of a room and each minute proceed half of the remaining distance to the opposite end, then the sequence above tells me how far, as a fraction of the entire length, I must go at each minute, while the series above tells me the total fraction I have traveled. In an infinite number of hops, do I ever reach the wall?
The answer is no because I never go more than half the remaining distance with any hop. If we imagine that an infinite number of minutes can elapse, where would I be? I would be at the wall. The value of the fractions in the series get closer and closer to one, and that is one whole room length. For example, the fraction 1023 / 1024 is reached after only ten minutes, and that is over 99.9% of the way across.
One conclusion here is that even though a sequence has an infinite number of members, its series doesn't necessarily result in an infinite total. Let's see one that does.
The sequence {1/2, 1/2, 1/2, 1/2, ...} is associated with the series {0.5, 1, 1.5, 2, ...} and this series will reach an infinite total, given an infinite number of halves in the sequence. We can show this is true with symbols, but it's just as easy to reason that, with enough terms, the series will get larger than any number you can name, and that's the nature of infinity. [See Part 1 of this Journey]
Infinity via Counting. So far we've allowed our imaginations to entertain the possibility of sequences with an infinite membership. Let's formalize this. If infinity is larger than any number, and the counting sequence
{1, 2, 3, 4, 5, …}
can be extended to exceed any number, then it can be considered an infinite set. In the 1800's a German mathematician named Georg Cantor made attempts to comprehend infinity; in particular, he attempted to count it. He knew that the mathematical essence of counting was the act of matching objects to the numbers 1, 2, 3, ... until the objects were all counted. We know that our fingers aren't named "one", "two", et cetera, but we recite those names as we match fingers, numbers, and objects. It's called making a one-to-one correspondence. If I visit a first grade class out for recess in January, and I ask them all to hold up their left hands, and I see a mitten on every hand; and then I ask them to hold up the other hand, and again, a mitten covers every one, I know there is a one-to-one correspondence between right and left mittens. I haven't counted them yet, but I know the counts will be equal.
Cantor thought that the counting numbers could count anything, including infinite sets, because they are an infinite set as well. His strategy was to match other infinite sets to the counting numbers, à la mittens. That would establish the equality of one infinite set, counting-wise, with another. And that is when things began to get really strange.
It's Saturday and we are at the training school for volunteer firemen. Imagine an infinitely tall ladder reaching, like Jack's beanstalk, up through the clouds and into the heavens. We see a firefighter on each rung. We imagine it's a bit crowded. A station wagon pulls up with five more firefighters. We make room on the ladder for them by ordering all firefighters to climb up five rungs. Mission accomplished. Later, an Infiniti arrives with an infinite number of volunteers. We can accommodate them as well by ordering all on the ladder to climb up to an odd-numbered rung. The even-numbered rungs, infinite in number, are now available for the new arrivals.
Cantor thought similarly that parts of infinite sets count up the same. He matched all the even numbers with their halves and saw that they could be counted as equal in number to all the counting numbers:
The mathematical term for the count of a set is “cardinality.” The cardinality of the set {a,b,c} is three. The cardinality of {2, 3, 5, 7, 11] is five. Cantor discovered that the cardinalities of the counting numbers and the even numbers were the same infinity, as were those of any infinite set that could be “listed.” Then he found an infinite set with an even greater cardinality.
Cantor discovered that he could not make a complete list of all the real numbers, not even the real numbers between zero and one. This set includes fractions and the irrational numbers such as half the square root of 2 and one-fourth pi. Numbers like
0.5000000...

0.6666666...

0.7071068...

0.7853982...

…
Every time he imagined a complete listing he saw how he could construct a number that wasn't in the list. To add to the list above he simply picked digits differing in one of the places: a tenths digit different than 5; a hundredths digit different than 6; a thousandths digit not 7, a fourth digit not 3, and so on. And he saw that even if his list were infinite, he could form a new number with the same technique. Cantor theorized that there was an infinity greater than the one the counting numbers arrived at. Wow!
Something to think about: Would the series
S = 1/2 + 1/3 + 1/4 + 1/5 + ...
ever reach infinity?
The series S above does reach infinity, but at a snail's pace. We have shown that 
1/2 + 1/2 + 1/2 +… reaches an infinite total. By taking bunches of terms of S we can show that it eventually reaches that total. For example, clumping the 2nd and 3rd terms: 1/3 + 1/4 which is greater than 1/2; clumping terms 4, 5, 6, and 7: 1/5 + 1/6 + 1/7 + 1/8 also greater than 1/2; and so each cluster produces halves or greater. It just requires more and more terms, and there are an infinity of them to make use of.
In Trip #3 we look at infinity using a geometric perspective.
Trip #3: To Infinity via Geometry
In previous legs of the journey we learned that Georg Cantor used one-to-one correspondence to equate infinite sets of numbers with the infinite counting numbers. For a down to earth example of one-to-one, we visited the schoolyard during a winter recess and asked students to hold up their left hands. There was a mitten on each hand. Next they held up right hands, all covered by a mitten, and we concluded that the counts, i.e. the cardinalities, of left and right mittens, though uncounted, were equal.
Cantor discovered that many sets of numbers – the evens, the odds, the integers, and also the everyday fractions – could be matched up one-to-one with the counting numbers. Cantor also found that the irrational numbers and the reals seemed to be more numerous than counting numbers. Was there a hierarchy of infinities?
Geometry is another area rich in infinite sets. We are all taught that a line is made of an infinite set of points and it stretches infinitely in two directions. By a one-to-one matching, Cantor showed that the full line has no more points than a semicircle. In the figure above, each point on the line is matched with a point on the semicircle by a connecting ray from the center B, in the manner that A is matched with C. Each point on the semicircle has a mate on the line as well. If a correspondence can be found which leaves no point unmatched, then the sets must have equal cardinality.
Also in the category of "Believe It Or Else" we have Albert of Saxony (1316-1390) contemplating the infinite and proving that infinity is equal to a piece of itself. He worked with the infinite rod and the infinite shell, whereas I'll give you the ceramic version.
Imagine a potter with a copious supply of clay. The potter knows what all children do their first time working with clay – they love to roll out those long skinny worms. Our potter makes a worm, oh, about 9" long. She then reforms the worm into a ball. The worm and the ball, having been made of the same amount of clay, are equal in volume. Using more clay the potter makes a longer worm. She shapes it into a bigger ball. Suddenly, an idea hits her – thwack – in just the way that wet clay smacks down on her wheel. If she had an infinite supply of clay she could make a worm infinitely long, and it would be infinite in volume. If she then rolled it into a ball, the ball will be infinite in size. If you had a ball infinite in size, what would you have? Answer: the infinite worm would not take up all of space – the ball would.
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I created the following sketch to show the same principle in 2-D. A strip of paper is made to have the same area as a circle. As the strip stretches to an infinite length, it and the circle both grow infinite in area, but how do they compare in the space they take up on an infinite sheet of paper? The circle engulfs it all, the strip does not.
Cantor used another device to show that the points inside a square were no more numerous than the points on one of its sides. He imagined the points on a side having one coordinate, let’s say between zero and one, such as 0.1627384905…, and the points in the square having two coordinates ranging between (0,0) and (1,1). How did he match the sets. He feathered the single coordinate into a pair. Using the single coordinate above we create the pair (0.12340…,0.67895…), a point in the square which corresponds uniquely with the single coordinate. To find a mate for any point in the square we shuffle the two coordinates to make one for a unique point on the side. The pair (0.564738291, 0.1111111…) corresponds to 0.516141713181219…. Cantor’s comment: I see it but I don’t believe it!
The graphic artist Maurits Escher was fascinated by the concept of infinity and he looked for ways to model it in his works. Often working within a circle, he tessellated (filled) it with pieces growing infinitely smaller, either toward the edge or toward the center. I've recreated the framework for one of his pieces entitled "Butterflies." Not being an artist, I stopped at the frame.
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In our last leg of this journey, we will develop concepts of infinity within a circle.
Trip #4: A Trip to Infinity Inside a Circle
Cut a piece of wire 10 centimeters long and place a mark every centimeter for a scale. Bend the wire into a perfect circle which has circumference 10 cm. Pick a mark to be the start of our trip. Travel 3 cm along the wire in a counterclockwise direction and connect back to the starting point with a straight segment. Continue to move and connect back, and along the circle, you will have traveled 3, 6, 9, 12, and some more, and then finally 30 cm where the trip ends because 30 is a multiple of 10 as well as 3, and ten steps gets you to land back at the start. We have just constructed the star pattern (10, 3), one of UB's favorites, pictured above. An attractive star, but not an infinite trip as promised. The infinite trip is accomplished by taking advantage of two facts: increments of the square root of two (√2) can be marked as a scale on the wire, and √2 is irrational.
Now we repeat the process and create a (10, √2) star starting at point A, the first several segments of which are shown below.
The completed (hah!) star is an infinite trip around this circle. Why? Because if we ever returned to the starting mark, then a whole number n times √2 would equal a whole multiple of 10. In algebrese it would look like
n √2 = 10m
and that implies √2 = 10m/n which makes √2 a rational number, which it isn't.
We're taught in geometry that a circle has an infinite number of points, and so an infinite trip is possible, BUT here's a kicker, this infinite trip will never visit all of the circle's points and we can prove it.
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We will show that, beginning at a common point A, a (10, √3) star and a (10, √6) star, though visiting an infinite number of points on the circle, will never arrive at another common point. [Figure below] If they did so, then a whole number of steps (k) times √3 would equal another number of steps (m) times √6. That's impossible because if
k√3 = m√6,
then k/m, a rational, equals √6/√3, but that equals √2, an irrational. QED as they say.
In the next figure two star patterns are initiated at point A. Both take an infinite journey within the circle and they never share another point! For both trips the points visited are said to be "countably infinite," since they are in an order – first, second, third, and so on. Cantor found that an infinite set which appeared to be merely part of another infinite set could count up the same as the "larger" set. He also found that adding onto, or even doubling the items in an infinite set would not change the count. The mathematical establishment of the late nineteenth century was not equipped to deal with these incomprehensibilities. Many reacted scornfully to Cantor's new theories of sets. He took it all too personally and suffered from depression and, eventually, mental breakdown. Rather than gloss over these gritty bits of numerical bad news though, Cantor released his big bombshell: he had found sets which could not be matched with the infinite counting numbers. They were even larger! For example, the number of points in our circle is uncountable.
Infinite Ponderables for the Reader
Ponderable 1: The rationals all can be expressed as infinite decimals. One-half could be written as 0.5000000... and one-third 0.333333.... Let's pretend to make an infinite decimal. We will choose randomly each digit after the decimal point, one at a time, by rolling a ten-faced die, numbered zero through nine. Trust me that such dice exist. What are our chances of randomly creating an infinite decimal which is rational?
Ponderable 2: You may have doubts about whether those endlessly repeating decimals are all fractions (rationals). There is a method for converting repeaters to fractions, but I'll tell you the chocolate story instead. I found this example in Playing with Infinity by Rozsa Peter:
A certain candy manufacturer wraps a coupon inside every Biggie Bar, and they sell for a dollar. Collect ten coupons and get a free Biggie. Can you give the exact value of one Biggie Bar with its coupon? You may say it's worth a dollar because that's what you paid for it. Remember though, that the coupon represents one-tenth of a free bar. This might cause you to think that the package is worth $1.10 because ten coupons are worth a dollar (if you like chocolate). Alas, you are forgetting that one-tenth of another free bar comes, in effect, with one-tenth of yet another coupon, and that each fraction of a coupon represents tiny fractions of coupons to come. Complete redemption comes only with the exact answer – a rational number.
Solutions below
This concludes the four legs of a Journey, going at infinity using several approaches. There is much more to be said. Clifford Pickover, whom I recommend, is a skilled and entertaining writer on this and many other topics.
Solutions to Imponderables
1. It would be impossible to create an infinite rational decimal by generating the digits randomly. A rational must either terminate with an infinite string of zeros, or repeat a pattern of digits ad infinitum. The probability is zero. This makes yet another argument that the irrationals is a larger set than the countably infinite rationals.
2. What is a Biggie Bar with its coupon worth? Assume the bar is worth a dollar. The coupon is worth one-tenth a second (free) bar plus a tenth of that bar's coupon, which is one-tenth of one-tenth of a third bar plus one-hundredth of its coupon, which is one-tenth of one-tenth of one-tenth of another free bar, and so on. In dollars and cents the total is $1.1111111..., which is the fraction 10/9.
An easier way to figure this result is to find a kindly storekeeper and from him buy nine bars for nine dollars. Ask him nicely to give you the tenth bar for free. You can then unwrap all ten bars and hand over the requisite ten coupons. You have 10 bars costing $9, and so each Biggie Bar is worth 1 + 1/9 dollars.