April 2024 – According to The Kidd

INFINITIES – 4. No Vacancies But Rooms Still Available At Hotel Infinity

This is the fourth in a series of posts exploring infinity. There will be one infinity larger than another infinity, infinite perimeters, infinite regress, infinity paradoxes, an infinite set of bizarre things called cyclic numbers, and other infinity-related concepts.

A lot has been written about Hotel Infinity. Hopefully you will learn something new here. Here’s the set-up. Imagine a hotel containing fifty rooms, and there is a single guest in each room. If a new guest comes along and wishes a room she will be out of luck. Now imagine a hotel with an infinite number of rooms, with a guest in each room. Lo and behold a new guest can still be accommodated while still giving every guest their own room. It was German logician David Hilbert who came up with the idea of Hotel Infinity as a way of demonstrating the surprising and counter-intuitive properties of infinite sets (one of the people Hilbert worked with at Göttingen University in 1915 was a young man you may have heard of named Albert Einstein).

DAVID HILBERT
By Unknown author, derivative work Lämpel – Reid, Constance (1970) Hilbert, Berlin, Heidelberg: Springer Berlin Heidelberg Imprint Springer, p. 230 ISBN: 978-3-662-27132-2., Public Domain, https://commons.wikimedia.org/w/index.php?curid=130481740

Infinities can be quite perplexing. For example, the part may be the same size as the whole! This is demonstrated by the fact that there are just as many natural numbers(1, 2, 3 . . . ) as there are even numbers (2, 4, 6 . . .) because you can match 1 up with 2, 2 up with 4, 3 up with 6 and so on. The idea of Hotel Infinity was first introduced by Hilbert in his 1924 lecture “Über das Unendliche” (“On The Infinite”), and Hilbert’s idea was first popularized by George Gamow in his excellent and quite accessible book ‘One, Two, Three . . . Infinity’ published in 1947 (page 17). The celebrated Polish science fiction writer Stanislaw Lem (1921 – 2006) also wrote a story called ‘The Extraordinary Hotel, or the Thousand and First Journey of Ion the Quiet’. It was first published in 1968 as part of the anthology ‘Stories About Sets’ edited by N. Ya. Vilenkin. It takes place at the Hotel Cosmos which has an infinite number of rooms.

STANISLAW LEM IN 1966
By Courtesy of Lem’s secretary, Wojciech Zemek. Resize and digital processing by Masur. – Stanislaw Lem 2.jpg, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=915059

Here’s how the apparently impossible can be achieved. Hotel Infinity is a hotel with an infinite number of rooms, each room is occupied by a single guest, and none of the guests is willing to share their room with anyone else. A new guest arrives asking for a room for the night and the Hotel Manager is happy to oblige. In order to accommodate the new guest, the Hotel Manager can simply move Guest 1 from Room 1 into Room 2, move Guest 2 from Room 2 into Room 3 and so on. Then place the new guest into Room 1. In Hotel Infinity there will always be new rooms to move guests into.

Similarly, one can accommodate, say, five new guests. Move Guest 1 from Room 1 into Room 6 (1 plus 5), move Guest 2 from Room 2 into Room 7 (2 plus 5) and so on. Therefore one can accommodate g new guests by moving Guest 1 into Room 1 + g, move Guest 2 into Room 2 + g and so on. The Hotel Manager can, therefore, accommodate any finite number of new guests since g can be as large as you like.

What is more amazing is that one can also accommodate an infinite number of new guests. Simply move Guest 1 from Room 1 into Room 2, move Guest 2 from Room 2 into Room 4, move Guest 3 from Room 3 into Room 6 and so on. All the guests are now in the even numbered rooms and all the odd numbered rooms are empty. Since there are an infinite number of odd numbers, one can accommodate an infinite number of new guests.

What about an infinite number of infinite sets of new guests? Easy. First of all, do what you did in the previous case and free up all the odd numbered rooms again. Now take a look at the array in my illustration at the beginning of this post depicting an infinite number of guests in each of Bus 1, Bus 2, Bus 3 and so on, in fact an infinite number of buses. By following the arrows you can see that one can select all of the passengers on all of the buses one at a time and place them in those odd numbered rooms without overlooking any of the bus passengers. These infinities are called countable or denumerable infinities.

CANTOR’S DIAGONAL DIAGRAM IN HIS ARGUMENT FOR THE EXISTENCE OF UNCOUNTABLE SETS
By Cronholm144 – Own work, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=2206106

THE MOST EXTRAORDINARY THING OF ALL

Despite all these ingenious methods of accommodating larger and larger sets of infinities in Hotel Infinity, sometimes you can’t accommodate everyone new. The infinities we’ve been talking about are countable infinities. However it has been proven with mathematical rigour (over a century ago, by Georg Cantor), that some infinities are uncountable, and are larger than countable infinities, as bizarre as that sounds. The number of Natural Numbers (one, two, three etc.), designated aleph null, is countably infinite. However the number of Real Numbers, designated aleph one, is not countable, so there are more Real Numbers than there are Natural Numbers even though the Natural Numbers are infinite. If the size of the set of new guests was the same size as the size of the Real Number set then some of the guests would be out of luck.

NEXT POST – Pythagorean Infinity

POSTSCRIPT: More Advanced Accommodation Strategies:

There is a second way of accommodating an infinite number of infinite sets – The Prime Powers Method. Start with powers of two. Before the buses arrive rearrange the occupants who are already in the hotel. Put the guest from Room 1 into the room numbered two to the power 1 (Room 2). Put the guest from Room 2 into the room numbered two to the power 2 (Room 4). Put the guest from Room 3 into the room numbered two to the power 3 (Room 8). Continue so that all infinite guests are now in the infinite rooms numbered with the infinite powers of two (two is the first prime number). When Bus 1 arrives place its infinite occupants into the rooms numbered with the infinite powers of three (three is the second prime number). Place the infinite occupants of Bus 2 into the rooms numbered with the infinite powers of five (the third prime), and place the infinite occupants of Bus 3 into the rooms numbered with the infinite powers of seven (the fourth prime). Continue doing this forever, for as Euclid first proved about 2300 years ago the number of primes is infinite. It is because we’re using primes that none of these infinite prime power sets overlap. Furthermore, this way you also end up with an infinite number of unoccupied rooms.

We used the Prime Powers Method to accommodate two layers of infinity (an infinite number of infinite sets). There are at least four other different ways to accommodate these two layers of infinity: The Prime Factorizations Method, The Interleaving Method, The Triangular Number Method and The Arbitrary Enumeration Method. The nice thing is that one can also use those four methods to accommodate three, four, five, and in fact an infinite number of levels of infinity.

The Prime Factorizations Method, for example, involves infinite powers of prime numbers. All you need to do is put a new guest who was sitting in seat s of bus b into the room designated by the product of 2 (the first prime) to the power s and 3 (the second prime) to the power b. For example a new guest sitting in seat 4 of bus 5 would go into room 3888 (16 x 243). The value of b would be zero for the people already in the hotel. Since every number has a unique prime factorization, this solution even leaves some rooms empty. Here is a link to more details – https://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel .

INFINITIES – 3. Infinitesimal Ch’i Ch’iao T’u

Illustration copyright MURRAY YOUNG 2024

This is the third of a series of posts exploring infinity. There will be one infinity larger than another infinity, infinite perimeters, infinite regress, infinity paradoxes, an infinite set of bizarre things called cyclic numbers, and other infinity-related concepts.

Ch’i Ch’iao T’u is the Chinese name for a tangram set which consists of seven shapes called tans which can be used to create all manner of ingenious shapes. The seven pieces consist of two large triangles (each with an area of four square units), a medium sized triangle and a parallelogram and a square (each with an area of two square units) and two small triangles (each with an area of one square unit). One ingenious thing about these seven shapes is that you can put them together to form a giant triangle, a giant square and a giant parallelogram (as well as a giant trapezoid, and a giant rectangle). Some people use tangram sets to teach geometry. In the illustration at the top of this post the shapes are used to form a giant right-angled triangle consisting of the five triangles, square and parallelogram that each tangram set contains. In that illustration each of those seven shapes are then subdivided again into smaller replicas of those same seven tans that we started with. One can keep doing this forever creating smaller and smaller tans infinitesimally, as illustrated. Here is a second illustration of mine demonstrating that same idea of infinite regression:

Tangrams were first invented some time during the reign of Chia-ch’ing (1796 – 1820) in China. The first tangram set was invented by someone using the manufactured name Yang-ch-chu-shih which in English means dim-witted recluse. The corresponding English word ‘tangram’ was invented by Thomas Hill in 1848. Tans can also be used to create the silhouettes of people, animals, plants, buildings, letters, numbers and miscellaneous objects. There have been periodic crazes in which people are challenged to make these shapes, and many others, with tangram sets. I have a book in my mathematical library with 1756 different shapes that can be made using a tangram set. Here is my illustration of a giant set of tans in the form of a square (surrounded by other shapes), the entire design consisting of 2318 tiny tans:

In the illustraion at the top of this post, on the left side, there is a geometric proof of the Pythagorean Theorem for isosceles right-angled triangles, illustrated using tans. Here is a fourth illustration of mine, this one showing a similar proof:

In 1942 Fu Traing Wang and Chuan-Chih Hsiung proved that there are only 13 convex tangram configurations. I used that idea to construct a giant rectangle using only those 13 configurations. Notice that the configuration pattern in the bottom half of this rectangle is a mirror image of the pattern of configurations in the top half:

Here is a detail from the above illustration showing how each configuration is constructed from a tan set:

Over the centuries people have constructed tangram sets out of wood, mother of pearl, precious metals and ivory. Some sets are finely carved works of art. Dozens of books have been published about tangrams, many with shapes illustrated with comic figures or detailed scenes. There are dishes in the shapes of tans, and tangram table sets which can be rearranged to accommodate different numbers of people. People have written stories revolving around tans, and Edgar Allen Poe owned a set carved from ivory. Napoleon may also have been a tangram puzzle lover. Much has been written about tangrams but don’t believe anything written by the famous American puzzle master Sam Loyd. He wrote a book about the history of tangrams but he made everything up out of whole cloth. Here is a final illustration of mine highlighting tans:

NEXT POST: No vacancies but rooms still available at Hotel Infinity.

INFINITIES – 2. There’s No Business Like Snow Business

This is the second of a series of posts exploring infinity. There will be one infinity larger than another infinity, infinite perimeters, infinite regress, infinity paradoxes, an infinite set of bizarre things called cyclic numbers, and other infinity-related concepts.

How could one possibly have a geometric shape that has an infinite perimeter but a finite area? That is exactly one of the properties of one of the first fractals ever analysed. I’m talking about the Snowflake Curve, discovered and analysed by Swedish mathematician Helge von Koch back in 1904. The outline around the outside of the shape illustrated above is the Snowflake Curve (this shape also consists of six smaller Snowflake Curves plus an area in the middle displaying a set of six-pointed stars which get infinitesimally smaller).

One can also tessellate the Euclidean two dimensional plane using this snowflake curve, that is, you can cover an infinite plane with snowflake shapes without leaving any spaces between snowflakes and without any overlapping figures, however, you need to use two different sizes of snowflake.

TESSELATING THE PLANE WITH A SNOWFLAKE CURVE
By David Eppstein – Own work, CC0, https://commons.wikimedia.org/w/index.php?curid=16959022

It is extraordinary to think that one can have something with an infinite perimeter and finite area. Here is something even more extraordinary. Consider the dimension of the snowflake curve. A point has zero dimensions. A line has one dimension – length. A square has two dimensions – length and width. A cube has three dimensions – length, width and height. The concept of a fractional dimension doesn’t seem to make sense. However, consider the concept of a dimension as a measure of complexity. It turns out that the dimension of the snowflake curve is the natural logarithm of 4 divided by the natural logarithm of 3, and this equals approximately 1.26186 . Extraordinary.

Like all good fractals, a snowflake curve is also infinitely self-similar. That is, if you zoom in to any point on the edge of a snowflake curve you see smaller and smaller exact replicas of the original curve. In calculus, the first derivative of a curve at any point really just takes the form of the tangent to the curve at that point. However, a snowflake curve has no tangents, and therefore no derivatives. That’s rather extraordinary as well. There are variations of the snowflake curve using angles other than sixty degrees, and there are even three dimensional versions of the snowflake curve. There’s also the Minkowski Sausage.

THE MINKOWSKI SAUSAGE
By AnonymousBosch – Own work, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=17891107

There’s also Gabriel’s horn (also known as Torricelli’s trumpet). This is a three dimensional entity whose volume is finite yet its surface area is infinite.

Here’s how you construct this curve, and if you follow these steps an infinite number of times you end up with a closed shape with an infinite perimeter but a finite area. Start with an equilateral triangle with each side, say, 27 units long (or any length you like). Now use this algorithm: erect a new equilateral triangle in the middle of each side of this shape and erase the bottom of the new triangle. That is, divide each side into three equal sections (each one is therefore 9 units long) and on the middle section of each side erect another equilateral triangle and erase its baseline. This new overall shape will have twelve sides each nine units long (a six pointed star). You have just completed the first iteration of the algorithm. Use the algorithm again, erect a new equilateral triangle in the middle of each side of this new shape and erase the bottom of each new triangle. This gives you a 48-sided shape and each side is 3 units long. If you follow the algorithm again you have a 192-sided shape and each side is one unit long. Simply do this forever. Take your time. The shape pictured above has only gone through six iterations.

Each step increases the perimeter by four thirds and the length of the curve after n iterations is four-thirds to the nth power. These steps generate a divergent infinite series so the limit after an infinite number of iterations is infinity. It also turns out that after an infinite number of iterations the area of the snowflake is exactly eight-fifths times the area of the original triangle, a finite value.

I have in my mathematical library a rather unusual book. It is a science fiction novel titled “The Curve of the Snowflake” by William Grey Walter, an American neurophysiologist, cybernetician and robotician who was raised in Britain from the age of five. He is known for his construction of some of the first electronic autonomous robots in the 1940’s. From the dust cover: “Not since H.G.Wells has a writer combined such a variety of intellectual and scientific brilliance with so clear and witty a talent for fiction. The Curve of the Snowflake is a novel of both delight and true importance. The snowflake curve is a paradox which can be described and constructed by science but not yet explained. It is the device of this novel.”

By the way, a figure with an infinite perimeter is called a teragon, a term coined by Benoit Mandelbrot who also coined the term fractal and was the first to analyse fractals at an advanced level. An infin-tile is also a teragon, one that I will talk a bit about in a later post. Here is another snowflake curve I drew a few years ago.

NEXT POST: Infinitely regressive tangrams

Post 1 – https://thekiddca.wordpress.com/2024/04/06/infinities-infinity-everywhere/

INFINITIES – Infinity Everywhere

This is the first in a series of posts exploring infinity. There will be one infinity larger than another infinity, infinite perimeters, infinite regress, infinity paradoxes, an infinite set of bizarre things called cyclic numbers, and other infinity-related concepts.

Infinities (and infinitesimals) are everywhere. Start counting (1 2 3 . . . ) and see how far you get. If the word “universe” means everything that exists, the universe must be infinitely large as well. It would also seem that Time is eternal, that is it consists of an infinite number of seconds or minutes or decades or whatever arbitrary unit of time you wish to use. Cosmologists are not all in agreement with these concepts, however, depending on how you define the words “exists”, “time” and “spacetime”.

Some infinities are different than others. If, for example, I start adding up the Natural Numbers (1 + 2 + 3 + 4 + . . . ), and did that forever, I would end up with an infinitely large number. This is a divergent infinite series. On the other hand, if I start adding up a series of ever smaller fractions ( 1/2 + 1/4 + 1/8 + 1/16 + . . . ) I could also do that forever, but in this case I would never get to the number one, let alone an infinitely large number. This is a convergent series, and involves the concept of being infinitesimally small.

Infinity is not a number but a condition of endlessness. It is a slippery concept. For example, infinity plus infinity equals infinity. The set of odd numbers is obviously infinite, and so is the set of even numbers, and if you combine the two sets you get the set of Natural Numbers which is also infinite. However the set of even numbers and Natural Numbers are the same size even though there seem to be only half as many even numbers as there are Natural Numbers. We know these two sets are the same size because we can match the Natural Number one up with the even number two, the Natural Number two with the even number four, three with six and so on, doubling each time, so we know that both sets have the same number of numbers in them.

What is even stranger is that some infinities are larger than others. I’ve understood the proof of that fact (it’s not a difficult proof) but I still can’t get my head around that idea. How can you have a number greater than the number of Natural Numbers? Georg Cantor proved it was possible, over a century ago. He showed that the number of Real Numbers is greater than the number of Natural Numbers, which means that the number of Transcendental Numbers (pi is a transcendental number, for example) is also greater than the number of Natural Numbers.

INFINITE SETS

Infinity is tricky. I can use infinite sets to prove that 1/2 = 0, and all the Natural Numbers (1, 2, 3 . . . ) not only add up to -1/12 but they also add up to -1/8 so I guess -1/12 = -1/8. The proofs use simple arithmetic and I have placed them at the end of this post if you want to take a look at them and try to figure out why they seem to be correct even though they “prove” something obviously false.

ACHILLES AND THE TURTLE

About 2500 years ago the Greek philosopher Zeno of Elea (490 BCE – 430 BCE) argued that if the great athlete Achilles was running a Kilometre long race against a turtle, and the turtle was given just a short head start of, say, ten metres, then Achilles could never catch up let alone pass the turtle. Zeno explains that if the turtle gets to start at some point A, ten metres ahead of the starting line, then as soon as the race starts Achilles will soon catch up and also reach point A but in the time it takes for him to do so, the turtle will have advanced to some further point, call it point B. Then as Achilles runs from A to B the turtle will have advanced once more to some further point, call it point C. As this same process unfolds over and over again Achilles never catches up to the turtle.

THE PLATONIC SOLIDS IN THE FORM OF DICE
By Tomruen – Own work, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=39985790

INFINITELY NESTED PLATONIC SOLIDS

There is a set of five fascinating three dimensional solids called the Platonic Solids. One of them, with six sides, is called a hexahedron which is just a mathematical name for a cube. If you find the centres of the six sides of a hexahedron and connect those centres using triangles you get another one of the Platonic Solids, an eight-sided figure called an octahedron and it looks like this:

AN OCTAHEDRON INSIDE A HEXAHEDRON
By 4C – Own work, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=1049163

Now, if you find the centres of the eight sides of that inner octahedron and connect those centres using squares you get another, smaller, hexahedron. Then you can fit an even smaller, second octahedron inside that smaller hexahedron, and keep forming smaller and smaller Platonic solids forever, alternating between hexahedra and octahedra. The shapes become infinitesimal. You can do the same thing with two other Platonic Solids – the twelve-sided dodecahedron and the twenty-sided icosahedron. I wonder what happens if you connect the centres of the sides of the last and simplest Platonic Solid, the four-sided tetrahedron, using triangles?

TETRAHEDRON
By The original uploader was Cyp at English Wikipedia. – en:User:Cyp/Poly.pov, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=38709

PI UNEXPECTEDLY

Not only does pi equal an infinitely non-terminating non-repeating decimal, it also shows up in the most unexpected places, such as Buffon’s Needle Problem – https://en.wikipedia.org/wiki/Buffon %27s_needle_problem . It can also be expressed as an infinite series in a variety of ways which seem to have nothing to do with circle ratios. For example:

pi = 4(1/1 – 1/3 + 1/5 – 1/7 + . . . )
pi = 2(2 x 2 x 4 x 4 x 6 x 6 x …) / (1 x 1 x 3 x 3 x 5 x 5 x …)

AN INFINITE CONTINUED FRACTION

Here is a geometric proof that the square root of two is equal to an infinite continued fraction. This was taken from the book Through The Mathescope by Stanley Ogilvy.

ASYMPTOTES

Some curves when graphed get closer and closer to their asymptotes (which are usually, but not always, straight lines) and so they go on forever. The two branches of a hyperbola never reach their asymptotes. The witch of Agnesi, a curve analysed at length by Maria Agnesi (1718 – 1799), approaches asymptotes along the positive and negative x axes but never reaches them, as illustrated here:

By Dicklyon – Own work, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=118532845

In trigonometry the tangent function becomes infinite when x equals π / 2, 3π/2, 5π/2 and so on, getting ever closer to their asymptote positive and negatively but never getting there. I have illustrated that phenomenon in this illustration:

SOME NUMBER SETS THAT GO ON FOREVER

Natural Numbers (1, 2, 3, 4 . . . )
Integers (. . . -4, -3, -2, -1, 0, 1, 2, 3, 4 . . .)
Odd Numbers (1, 3, 5, 7 . . .)
Even Numbers (2, 4, 6, 8 . . .)
Prime Numbers (2, 3, 5, 7 . . .)
Perfect Squares (1, 4, 9, 16 . . .)
Perfect Cubes (1, 8, 27, 64 . . .)
and fourth powers, fifth powers and so on
Triangular Numbers (1, 3, 6, 10 . . .)
and Pentagonal Numbers, Hexagonal Numbers and so on
Perfect Numbers (6, 28, 496, 8128 . . .)
Cyclic Numbers (142857, 0588235294117647 . . .)

SOME INFINITE DECIMAL EXPANSIONS

The decimal forms of all sorts of numbers also go on to infinity.

4 = 4.0000000000 . . .
1/2 = 0.5000000000 . . .
1/9 = 0.1111111111 . . .
1/3 = 0.3333333333 . . .
1/7 = 0.142857 142857 142857 . . .

Some go on forever without repeating and are the roots of a non-zero polynomial of finite degree with rational coefficients (i.e. algebraic numbers):

The square root two = 1.4142135623 . . .
The square root of three = 1.7320508075 . . .
The Golden Ratio / phi = 1.6180339887 . . . which is equal to one plus the square root of five all divided by two. It is the positive root of the equation x² – x – 1 .

Some numbers go on forever without repeating but are not the roots of a non-zero polynomial of finite degree with rational coefficients (i.e. transcendental numbers):

Pi = 3.1415926535 . . . which is the ratio of the circumference of a circle to its diameter.
e = 2.7182818284 . . . which is the base of the natural logarithms aka Euler’s Number
The Liouville constant = 0.1100010000 . . .

FINALLY, DIVISION BY ZERO

In Mathematics division by zero is said to be undefined. Some people have got the impression that one divided by zero equals infinity, but this implies that infinity is a number but it is not. To say that the limit of one divided by x approaches infinity as x approaches zero makes more sense, but that still doesn’t make it clear exactly why division by zero is undefined. It is undefined because if you accept it you end up with inconsistent results.

To illustrate, take the equation y = 1 / x. Graph this on a Cartesian coordinate system. What happens to y as x gets smaller and smaller, i.e. it approaches zero, but along the positive x-axis? Is the limit simply infinity? No it isn’t. Set x equal to 1 / 2 and y equals 2. Set x equal to 1 / 3 and y equals 3. Set x equal to 1 / 1000000 and y equals 1000000. In other words, the limit is positive infinity. Again have x get smaller and smaller as it approaches zero, but have it approach zero along the negative x-axis this time. What happens? Set x equal to -1 / 2 and y equals -2. Set x equal to -1 / 3 and y equals -3. Set x = -1 / 1000000 and y equals -1000000. In other words, the limit is negative infinity. So, even if you try to figure out what the limit of 1 / x is as x approaches zero, you get that the limit is simultaneously positive and negative infinity, which is mathematically meaningless, inconsistent and ridiculous. If you graph the function you have two simultaneous curves, one going up forever and the other going down forever, both of them asymptotic to the y-axis. So don’t divide by zero. Just don’t.

NEXT POST: A curve with an infinite perimeter and a finite area.

PROOF THAT 1/2 = 0, AND -1/12 = -1/8 (AND THEREFORE EVERY WHOLE NUMBER IS EQUAL TO EVERY OTHER WHOLE NUMBER)

INFINITE SET 1. Let me show you that 1/2 = 0

Let A = 1 – 1 + 1 – 1 + 1 – 1 + 1 – 1 . . .
1 – A = 1 – (1 – 1 + 1 – 1 + 1 – 1 + 1 – 1 . . . )
1 – A = 1 – 1 + 1 – 1 + 1 – 1 + 1 – 1 + 1 . . .
1 – A = A
1 = 2A
A = 1/2

But wait a minute:

A = 1 – 1 + 1 – 1 + 1 – 1 +1 – 1 . . .
A = (1 – 1) + (1 – 1) + (1 – 1) + (1 – 1) . . .
A = 0 + 0 + 0 + 0 . . .
So A is also equal to 0. Apparently 1/2 = 0.

INFINITE SET 2

Let B = 1 + 2 + 3 + 4 + . . .
Let S = 1 – 2 + 3 – 4 + 5 – 6 . . .
B – S = (1 – 1) + (+2 – (-2) ) + (+3 – (+3) ) + (+4 – (-4) ) + . . .
B – S = 0 + 4 + 0 + 8 + 0 + 12 + . . .
B – S = 4(1 + 2 + 3 + 4 + . . . )
B – S = 4B

Now we need to find out what S equals. Add S to itself but when you put the second S under the first shift it over so that the 1 in the second S is under the 2 in the first S, the 2 in the second S is under the 3 in the first S and so on.

2S = 1 + [(-2) + (+1)] + [(+3) + (– 2) ] + [(-4) + (+3)] + . . .
2S = 1 – 1 + 1 – 1 + 1 – 1 + . . .
But we found out above that A = 1 – 1 + 1 – 1 + 1 – 1 . . . = 1/2
So 2S = A = 1/2
S = 1/4
Now go back to the line B – S = 4B
B – 1/4 = 4B
B – 4B = 1/4
-3B = 1/4
-12B = 1
B = -1/12 = 1 + 2 + 3 + 4 + . . .

You can also use infinite series’ to prove that B is also equal to -1/8 if you like:

B = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + . . .
B = 1 + (2 + 3 + 4) + (5 + 6 + 7) + (8 + 9 + 10) + . . .
B = 1 + 9 + 18 + 27 + . . .
B = 1 = 9(1 + 2 + 3 + . . .)
B = 1 + 9B
B – 9B = 1
-8B = 1
B = -1/8
If -1/12 = -1/8
then 24(-1/12) = 24(-1/8)
-2 = -3
-2 + 2 = -3 + 2
0 = -1

You can now multiply both sides by -1 (so 0 = 1), or by -2 (so 0 = 2), or by -3 (so 0 = 3) and so on, so every Natural Number equals every other Natural Number and they are all equal to zero.