CircleMaths: Lesson 3

The Necessary Evolution of Fractions and Decimals

Who Its For and What Its About

This maths lesson is primarily intended for two categories of people. It is for those mothers, fathers and other teachers who are unafraid to step beyond the narrow confines of the curriculum and explore maths not for the sake of marks, but rather to discover what light maths can shed on the unsolved mysteries of existence. Science has not answered everything. Several major mysteries remain. How does the human mind work? What is the link between mind and matter? What is the deep origin of time, space and mind? These remain mysteries. If you have read some of the other material here you will be aware that the search for answers to these questions has been advanced in the content of this web site. Which brings us to the second category of people who may be interested in covering this lesson. It is also for scientists, particularly those in the fields of particle physics, relativity, evolution, anthropology and for those involved with understanding the human mind, neurologists, psychologists, philosophers etc. The issues tackled in this web page go beyond the normal confines of science, questioning the mysteries of life which both science and religion across the millennia have attempted to explain. Those who can gain an understanding of the maths can then go beyond the numbers to delve into these issues and perhaps to use the understanding they gain to go further and solve them, or at least to push our understanding one step further towards that goal. But first the maths must be accessible to everyone, so these lessons are intended to allow anyone, whether parents or professional, to gain that understanding.

Circlemaths does not treat maths as a separate subject walled off from all others, created in the vacuum of an ivory tower. Rather it is connected on one side to the objective world through physics, and to the other side, the subjective world of mind, in its biological origins within the species. Thus we will also deal with factors which in fact permeate and eclipse mathematics, in particular the law of opposites which runs through physics, maths and the operation of the human mind. Although in this lesson we will explain concepts which are more familiar to quantum mechanics or relativity, the level of explanation is pitched at the “pie arithmetic” level (literally!), and is intended for everyone. Because it is a new topic little reliance is placed upon previous knowledge beyond a general background in ordinary arithmetic. The maths taught here is not part of any curriculum. It deals with concepts on the curriculum, however it explains them from their origins, which no school curriculum will attempt to do. Unfortunately maths has a reputation for being boring. It is mechanical and dull. This is because mathematics based on a straight number line alone lacks its circular origins. The interesting bits are simply “skipped” over and the subject remains piecemeal rather than being an organic whole. The result of this intellectual poverty must impact on the consciousness of those who learn it, just as filling in those missing steps which turns mathematics into something which reflects the nature of the human mind must have an impact on those who learn it. So, what is this lesson about in particular?

This lesson concerns that bugbear of all children, fractions and the decimal system. We are not rushing in with examples of how to add, multiply and divide fractions. These can be done later in other lessons. Circlemaths has an ability to make such computations easier. However this is not where we begin. Rather we will be taking in the “big picture”, presenting a road map of maths which learners can have at their disposal for future reference. Children with such a background know where they are and where they are going . We are looking at the origin of the fraction and decimal systems from the viewpoint of circular number. Our focus in this lesson is on the depth of understanding rather than the memorizing of methods and techniques to acquire results. As we will show later in this article, the rational counting system (fractions) will be evolved as a necessary result forced upon us by the circular properties of the system. As in previous articles it is re-emphasized that this means of developing integers, rationals and so forth from existing properties of the circle is in total opposition to the traditional mathematical procedure of inventing them via definition. If you follow this lesson through you will enter a world where everything is topsy-turvy, where big is small and small is big. We won’t apologize for the bizarre nature of this experience any more than those who teach relativity would feel the need to apologize for the fact that clocks slow down and stop at light speed. Its just the way things are. We begin with simple “pie arithmetic”, transcend it, then once again return to the “ordinary” view of things. It should be an interesting journey. We hope it will make you re-evaluate your conceptions of “big and small”, infinitesimal and infinite, the terms we use to describe our universe.

Fractions

They are formed from a numerator and a denominator. I always forget which one goes on top. It doesn’t matter – they are two numbers in a fixed ratio to one another. Let’s look at how they are first taught in the form of “pie arithmetic” (in other words, let’s examine our earliest conception of them):

Figure 1: earliest conception of fractions

Given a whole we break it into a number of equal parts. The number of parts we break it into is the lower number in the fraction. The number of these portions we take (or indicate) is the top number in the fraction. Thus they are in a fixed ratio to one another. We do not have to invent a more complicated definition. This concept will do. We will now adopt it into a circular framework in the obvious way:

Figure 2: Fractions as expressed in circle maths

It’s not too hard to see the relation between “pie fractions” and fractions in circlemaths. The top number of a fraction indicates the number in the circle, while the bottom number indicates the circle size. For example the 1/4 mark in the top right of the diagram comes to 1 and the ‘size’ of the circle is 4circle.

Does that mean that any number in any circle is a fraction? Taken entirely out of context one could only answer “yes”. Given a 3 in 6circle it is reasonable to say it represents 3/6. However a fraction, just like its integer counterpart, requires a universal side to it. A fraction is a fixed proportion of a whole, no matter what that whole is. It could be half a grain of sand or half a tonne of sand. It is a portion of a whole. Any given whole. In other words it is more than a fixed proportion of a given circle. It is a fixed proportion of all circles. A fixed proportion of the whole circular system. We’ll demonstrate by showing ½ as a proportion of the entire system:

Figure 3: one half as a fixed proportion of the whole system

In figure 3 (above) we have shown the one constant fraction:

1 in 2 circle = 1/2

2 in 4 circle = 2/4

3 in 6 circle = 3/6

where 1/2 = 2/4 = 3/6 is, as shown a single fixed proportion of the entire system. We only had room to show 2, 3, 4, 5 and 6circle, however you can imagine that the circles go on indefinitely to include the entire numerical system so that we also have:

4 in 8 circle = 4/8

5 in 10 circle = 5/10 and so on and on.

The Decimal System

Introduction To Decimals: Ghost Zeros

Firstly we will cover a few commonly accepted terminology rules regarding decimals, beginning with “ghost zeros”.

0.5

= 00.50

= 000.500

= 0000.5000

“Ghost zeros” can be added to the left or right without changing the value of the decimal. On the left of the decimal we have units, tens, hundreds and so on. Thus the ghost zeros on the left simply tell us we have:

0 0 0 . 5

no hundreds no tens no units

Likewise further ghost zeros to the left simply inform us that there are no thousands, tens of thousands, hundred thousands or millions, and so on indefinitely. In other words it is perfectly OK to add ghost zeros to the left as they simply act to verify “nothing at all” is there.

On the right of the decimal point we have tenths, hundredths and thousandths etc. Here the ghost zeros tell us we have:

0 . 5 0 0

5 tenths no hundredths no thousandths

Further ghost zeros to the right simply inform us that there are no ten-thousandths, hundred-thousandths or millionths, and so on indefinitely. Again, it is perfectly OK to add ghost zeros to the right as they simply affirm once again that there is “nothing at all” further present.

Most generally we can write that:

0.5 = ... 000.500 ...

by which we use the 3 dots on either side to mean that the zero’s can be considered as continuing on without any definite limit, as long as you care to imagine.

Normally we do not bother with ghost zeros and we simply write ½ = 0.5 and leave it at that. In circlemaths the ghost zeros have a distinct meaning as we will shortly show.

Introduction To Decimals Continued : “Recurring Decimals”

The calculator always provides us with a decimal answer. If we divide 1 by 6 we may see:

1/6 = 0.1666667

Using a larger calculator we obtain: 1/6 = 0.1666666666667

The calculator is just rounding off our answer for us. Actually the “7” is incorrect. The real answer is that the division just goes on and on like this:

1/6 = 0.166666666666666666666666666666666666666666666666 ...

You can see at the end I got tired of writing the 6’s so I just used the 3 dots to indicate that the recurring digit goes on indefinitely. This comes about because when we divide 1 by 6 we get to a place where the same operations keep happening.

0.166

6) 1.000

We reach a point in our division where the remainder is 4. Next step is “6’s into 40” which goes 6 times with a remainder of 4 again. On and on. As you can see there is no end point, the “7” which the calculator kindly adds on the end to round the answer is never obtained. The simplest way to indicate the recurring decimal is like this:

1/6 = 0.16 ...

where it is understood that the 1 is not recurring, but the 6 is. Sometimes whole chunks of numbers will recur. For example if you divide 1 by 7 on your calculator you will obtain:

1/7 = 0. 142857142857142857 on and on.

We put a line above the recurring portion and add 3 dots once again to indicate that it is this portion which is endlessly repeated:

...........

1/7 = 0. 142857 ...

All these notes regarding ghost zeros and rules for showing recurrence really fall under the umbrella of terminology. They don’t tell us a thing about the decimal system or how it arose. We’ll get onto that task immediately.

Sneak Preview

In the course of looking at a decimal we will be showcasing a set of “opposites”, large and small in this case. We’re going to take a bit of a sneak preview at the development, so the arguments here are only sketchy, and will be fully backed up later.

The decimal 1.5 is more correctly written as … 01.50 … which means something more like:

0000000000000000001.5000000000000000000000

except the rows of ‘ghost’ zeros continue without limit, running off the page in both directions. Let’s allow it to do so in our imagination. The zeros go off to the right (over your table and onto the floor, out the window or through your door, rushing endlessly off to the right). At the same time the zeros disappear off to the left (over the desk and onto the floor, off to the left for ever more!). As we know this world, though it looks flat, is in fact spherical, so the zeros, if they were so allowed, would go right around the world and join up! The result would look something like this:

Figure 4: A decimal fraction (1.5) with ghost zeros going right around the world

IF we use the decimal point to mark the start (and end) we can see we have a single whole number. We could re-write it as:

500000000000000000000 (a whole “world-full” of zeros goes here) 00000000000000000001

The number of zeros in the middle bit would be enormous. If we imagine them as being without limit we could rewrite the number as:

500000000000000000000 ... ... 00000000000000000001

where we have used the 3 dots again to indicate the zeros recur without limit to the left and to the right, filling in that middle portion. More succinctly we could rewrite it as:

500... ...001

which says the same thing but uses less paper. You’ll notice that we no longer have a “whole” number because it is split, not by the decimal point, but by the indefinitely great number of zeros in the middle of it. It splits into two sides. On the right is ...001 which is unchanged from what we are familiar with, being exactly equivalent to the value 1. On the left is the 500... which represents the fractional part (the .5), only here it appears as something like “five umpity million trippity zillion” or some such enormous value. What a turnaround! Interpreted from a circular point of view fractions would be enormous and numbers would be small! This would represent a complete reversal in how we view decimals... IF it were true!

Believe it or not, this is exactly where we are heading. Furthermore, nowhere in the circlemath system have we shown that a “one” can be divided into parts at all! We have never shown a circle with a half in it. Yet we want to return to the view at the end that between 0 and 1 we have halves, quarters and so forth. And we will do so while still maintaining the very opposite, that the jump from zero to one is an indivisible leap which is generated within the human mind.

That's our goal. Reverse everything, big with small. Then reverse it all again so it returns to normal. Should be fun. The “sneak preview” is over. Let’s do it...

The Origins of the Decimal System

We know we can convert ½ into 0.5 by dividing the 2 into the 1 (so a fraction is a decimal which has not yet had the division performed). This is how to convert between the two, but it does not tell us what a decimal actually is. To see what a decimal actually is we need to go back to the series of circles which created the base. Before we tackle that we need to briefly look at what regular arithmetic considers an insignificant point, the placing of “ghost zeros” in front of numbers. We have covered why it is ok to write 3 = 03 = 003 = 0003 etc. but in circlemaths we go further and point out the special significance these “ghost zeros” have.

Quick Refresher (and a little Expansion) on “Bases”

[as covered in Lesson 1]

Recall from lesson 1 (“bases”) that Denary means our normal “finger counting” base. When we write “10circle” or “base10” we mean the Denary 10 circle or base. If we write it with a gap, e.g. “10 circle” or “base 10”, we could be meaning any circle or any base because “every circle is a 10 circle and establishes a base”. For example we could be looking at a circle with only the numbers 0 1 and 2 in it, and we could call it the “10 circle which establishes base 10”. The diagram of that circle would be sufficient context to let anyone know what we intended by the “10”. In Denary terms that would mean “the 3 circle which establishes base 3”. Looking at the decimal system we are going to pretty much stick to finger counting Denary, that is base10, so the decimals we arrive at are more familiar.

Here is the Denary 10circle which establishes Denary (base10):

Figure 5: Denary 10circle which establishes Denary (base10)

Here is the base written out:

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 .............. 97 98 99 100 101 ...

ONE time around the circle returns us to NOUGHT. As far as the 10 circle is concerned we have lost count and we are simply back at the start. As far as the base is concerned we remember our count by saying we are at “ONE-NOUGHT” or “10”. Go around the circle again and we have made TWO entire loops of the circle and again we return to NOUGHT. As far as the circle is concerned we have lost count yet once again and are simply back at the start. However the base records it as “TWO-NOUGHT” or “20”. When we reach the number after 99 we have returned to the original shape of the Denary 10circle, only the shape is a “fractal” like creature made up of 10circles itself (see diagram below). We say we have returned to the original “TEN” circle shape, while at the same time we have gone a full cycle of cycles and have again returned to “NOUGHT” (for the tenth time). We have reached “TEN -NOUGHT” or “100”:

Figure 6: 100 in base10 (Denary) is made of 10 lots of 10circles.

It would take too much space to fill in the values in the above circle. But we can open the circles out and turn them into the 100circle of Denary like so (the continuous dots indicate the values we haven’t drawn in):

Figure 7: 100circle, Denary

This is not the base. This is just another finite circle. As you may recall, we can take ten more of these 100circles and wrap them into a circle, thereby returning to the original “TEN” pattern once again. If we unfold this huge set of circles to create a single circle from them we find we have created 1000circle. On we could go without end repeating this same process creating in turn 10,000circle, 100,000circle, 1,000,000circle. [This process explains the well known fact that thousands are 10 hundreds, hundreds are 10 tens, and tens are 10 units.] But none of these circles qualify as “base10”. The base is an indefinitely huge circle, without known limit, as depicted below:

Figure 8: 10, 100 and 1000circles leading up to the indefinite sized circle (or base)

By continuing the above process of increasing circle size by leaps of 10 indefinitely, we end up with a circle of unknown size, without limit. This is the base. To clarify the use of dots, in all our diagrams “3 dots in a row” means exactly what it does on the “straight counting line”, that the process goes on without known end, indefinitely (compare to standard usage of 1, 2, 3, ...). Groups of 3 red dots indicate that the recurring digits (zeros or nines in a number) go on within a number without known limit. In circles where some values have been omitted for space, a continuous series of dots have been put in. They indicate a finite number of missing values, for example the numbers between 3 and 50 in the 100circle of figure 8 above.

If you are still unsure of our usage of the dots, here are the conventions in detail for figure 8:

· The continuous blue dots in the 100circle indicate the gaps between 3 and 50, and between 50 and 97 . The continuous blue dots in the 1000circle point to the missing values between 3 and 500 and again between 500 and 997. In general, a series of 3 dots, whether red or blue, always means an indefinite number of values are not being shown, whereas a continuous line of dots indicates a finite number of missing values not shown.

· The red dots on the “500...” in the far right circle mean something like : 5000000000000000000000000000000000000000000000000000000000000000000

except we actually mean without any limit whatsoever. So even more zeros!

· The red dots in front of the values “...997, ...998” and “...999” in the far right circle mean that the numbers are like:

9999999999999999999999999999999999999999999999999999999999999999997

9999999999999999999999999999999999999999999999999999999999999999998

9999999999999999999999999999999999999999999999999999999999999999999

only once again we really mean without any limit whatsoever. (So even more nines!)

· Finally the base is also called “1000...” circle which again means something more like:

1000000000000000000000000000000000000000000000000000000000000000 circle, where again we really mean a circle without any known limit whatsoever on its size.

Because the circle is indefinitely great it is impossible by counting to get all the way around it. So the 10 of “indefinity circle” can never be reached. To examine the decimal system we need a way to traverse right around this “indefinitely great circle” without counting. For this we need to look at another way of producing the base, using ghost zeros.

Ghost Zeros

Here is the 10 circle of base100:

Figure 9: The 10 circle of base100

Note: The 10 circle we are using here to establish the base is not the Denary (finger counting) 10 circle although it may look like it. It’s a close cousin! Every circle is a 10 circle and can establish its own base. The circle we are using here has a Denary100 numbers in it (i.e. it has as many numbers in it as there are cents in a dollar). Once around the circle makes one dollar... rather it makes ONE loop around and returns to NOUGHT. It brings us to “ONE NOUGHT” or 10. But the 10 here is not the 10 of base10. It is the 10 of base100. The true compound ten of this base being equivalent to denary100, we must use lockbars to indicate that the “ten which follows 9” is a single digit below the true ten of the base.

Here is the base of the circle in figure 9 written out:

0 1 2 3 4 5 6 7 8 9 10 11 12 ... ...97 98 99 10 11 12 13 ...

Every number up to 99 is a single digit number below the 10 of base100. After 99 comes not 100 as we are used to, but the 10 of base100 which means ONE time round the 100circle and back to NOUGHT. The ONE and NOUGHT make the 10 of base100. The 10 on the other hand is to be considered as a single locked digit in base100. It is not a compound number.

How then should we draw the 100 circle of base10? It’s a subtle difference. The base we will now use is not base100 but our regular counting base, base10. The finger counting 10circle is the circle used to establish it. But we want to look at the 100 circle of that base. Here it is:

Figure 10: The 100 circle of base10

Looks pretty similar to the base100 circle doesn’t it? Let’s put them side by side to highlight the differences, which we will illuminate in red. Here are the two circles, the “10 circle of base100” and the “100 circle of base10”:

Figure 11: The two seemingly similar circles side by side

The only difference is that we have placed zeros in front of the numbers 0 up to 9 in the “100 circle of base10” so they read 00, 01, 02 up to 09. This serves to emphasize the fact that this circle is the 10 x 10 or 100 circle of base10. Here are their two slightly different counting lines side by side:

base100

0 1 2 3 4 5 6 7 8 9 10 11 12 ....................97 98 99 10 11 12 13 ...

base10

0 1 2 3 4 5 6 7 8 9 10 11 12 ....................97 98 99 100 101 102 103 ...

Spot the difference? The ghost zeros which are simply an artifact for regular arithmetic have a special significance in circlemaths. They tell us what circle we are in. The number of ghost zeros tells us whether we are in a 10 circle, a 100 circle, a 1000 circle or greater. We’ve seen what the hundred circle of Denary looks like. Let’s take a look at its one-thousand circle. Here is the thousand-circle of Denary:

Figure 12

One digit numbers means we are in a 10 circle, two indicates a one-hundred circle, three points to a thousand circle and so on. This rule is independent of base. Consequently the ghost zeros tell us which circle a number in a given base belongs to. Here are some examples from base10 (Denary):

7 belongs to the 10 circle

07 belongs to the 100 circle

007 belongs to the 1000 circle

000007 belongs to the million circle

37 belongs to the 100 circle (it has two digits)

037 belongs to the 1000 circle (it has three digits)

The Denary base can be envisioned as an indefinitely great circle using ghost zeros as shown in figure 13 (below). The circle size is indicated by use of the ghost zeros. It is not the 10 circle of base10, nor the 100 circle, nor the 1000 circle, not even the million circle. It is the “indefinitely-sized-circle” of base10. The number of ghost zeros (and, by the way, “ghost 9's”) goes on without limit. Once again we have adopted the convention of using 3 red dots to indicate indefinitely great recurrence. For example ...997 means a number with an unlimited series of 9’s in front of it, ending in 7. Likewise …001 has an unlimited number of zeros in front of its 1. Again we have used 3 blue dots to indicate an indefinitely great number of numbers (just as the ordinary counting line counts 1, 2, 3, ... and the 3 dots mean the count goes on “forever”).

Figure 13: an indefinitely great Denary circle

It counts out like this:

...000 ...001 ...002 ...003 and so on indefinitely, ending in the sequence ...997 ...998 ...999

We cannot fill in the vast number of values between …003 and …997 because it is an unlimited sequence. However, just as readily as we can know what that unlimited sequence begins with (the …001, …002, …003), so we can determine what it ends with (the …997, …998 and …999). We know this because we are not dealing with a straight counting line. We are dealing with a circle, albeit one with an indefinite extent.

We now have an alternative picture of how to view a base by using ghost zeros. We can view all of the above circles leading up to the base simultaneously by placing them inside one another, so they resemble the waves made when a stone is thrown into a pool. The following picture is very important. If you can follow it you have virtually grasped the essence behind the Denary decimal system:

Figure 14: Denary Base as Expanding Waves

Refer to figure 14 (directly above). All numbers in it are in our normal Denary base.

The center ring is the 10 circle of the base (base10). It has no missing values.

The next ring out is the 100 circle of base10. There was not quite enough room to fit all the numbers from 1 to 100 around the circle, so only a few were put in and a continuous row of dots joining these values was used to indicate the others not shown. Because it’s a 100 circle every number is composed of 2 digits. The zero is 00, and the 1,2 and 3 have become 01, 02 and 03 respectively.

The third ring out is the 1000 circle of base10. There was definitely not enough room to fit all the numbers from 1 to 1000 around the circle, so only a very few were put in and again a continuous row of dots was used to indicate the others not shown. Because it’s a 1000 circle every number is composed of 3 digits. The zero is 000, and the 1,2 and 3 became 001, 002 and 003 respectively.

All these circles are finite. You can imagine we could continue this same process as long as we wished. The outer purple circle is the circle of interest. It is the indefinitely great circle based on Denary. It represents base10.

Notice the 3 red dots after 300, 500 and 700 in the purple circle. These indicate sequences of zeros which continue without known limit. For example the value “500...” stands for something like 5000000000000000000000000 only with potentially more zeros on the end.

Also notice the 3 red dots before the 001, 002 and 003 in the purple circle. These represent an uncountable number of zeros before each of these numbers. For example ...001 means something more like 0000000000000000000000001 except the row of zeros again has no limit.

Likewise notice the 3 red dots before the 997, 998 and 999. These represent an uncountable number of nines before each of these numbers. For example ...997 means something more like 9999999999999999999999997 except the row of nines in this case is without known limit.

We have drawn the zero with 3 red dots on either side as …000… just to indicate it extends with an indefinitely great row of zeros on either side, looking something like this: 0000000000000000000000000 only longer.

Finally, note the series of 3 blue dots in a row between ...003 and 300..., between 300... and 500..., between 500... and 700... and between 700... and ...997. These indicate that we haven’t had room to draw in the uncountable billions of values in-between, just as we missed out values from the 100 and 1000 circles. Only this time there are not hundreds of values missing in between, but a limitless number of missing values.

Now we have a picture of how the base can be viewed using ghost numbers we can make sense of the decimal system, because there is a way to traverse the indefinitely great circle when it is drawn in this form! But first we will deal with the gap between the numbers in the diagram above. This diagram is the key to understanding the decimal system so we will continue to focus on it in detail for a while.

Negative Numbers in Decimal

Notice in Figure 14 above, that in the 10 circle the number found where “-1” should be is 9. It is backwards, anticlockwise, one step from zero. Then note the pattern:

-1 = 9 in 10 circle

-1 = 99 in 100 circle

-1 = 999 in 1000 circle

and generally we find –1 = …999 in the base circle.

Likewise

-2 = 8 in 10 circle

-2 = 98 in 100 circle

-2 = 998 in 1000 circle

and generally we find –2 = …998 in the base circle.

Although, because of its circular origins, this is not common knowledge, nevertheless these strange looking values work perfectly for arithmetic. For example here is what happens when we add “-3” with 8:

…9 9 9 9 9 7 (Use the fact that –3 = …997 in the circles)

+ …0 0 0 0 0 8 (write 8 with its ghost zeros)

…0 0 0 0 0 5

“But wait!” you cry. What has happened to the carry of 1? 7 and 8 are 15, not 5. The carry of 1 adds to the 9 and becomes 10. Put down the 0, carry the 1. The process repeats. Indefinitely. There is no end to that particular computation. The result is as stated: …000005 which is correct.

What is important here for us is not how to calculate using these values, but simply to note that in a base10 circular decimal sequence,

-3 = …997

-2 = …998

-1 = …999

0 = …000

+1 = …001

+2 = …002

+3 = …003

So an array of numbers like that shown to the right above is equivalent to the integer counting line!

The Gap

Circlemaths is the maths which relates not only to the external world, but also to the inner world of mind. When we jumped from the expanding circles to the indefinitely great circle we crossed a great divide. We leapt off the page and into the understanding in our mind. It is impossible to draw an indefinitely great circle. Such a circle can only exist in the understanding. We have crossed an absolute boundary from “finite circle on paper” to “circle without known size limit in the mind”. When absolute boundaries are crossed it becomes possible to see opposites as identities, just as mass turns into energy when the mass of uranium in an atomic bomb reaches critical size. And when opposites fuse, a circle is created. [View a Philosophical Ammendment here.]

Reference is made to the indefinitely sized circle of Denary (figure 14 above) in the following discussion:

An indefinite gap exists between 100..., 200..., 300..., etc. This gap is unbridgeable by counting. This has the effect of splitting the circle up.

We can envision a fixed fraction in figure 14 by looking at the following numbers on the circles it contains:

3/10 which is 3 out of 10 in 10 circle

= 30/100 which is 30 out of 100 in 100 circle

= 300/1000 which is 300 out of 1000 in 1000 circle

= 300... /1000... which is 300... out of 1000... in 1000... circle

(imagine a fraction line drawn through the 3, the 30, the 300 and the 300... on figure 14)

In other words, the outer circle is neatly split into 10 equal portions:

one tenth = from 000... up to 100...

two tenths = up to 200...

three tenths = up to 300...

(and so on right around the circle).

And when we say “split” we literally mean that the circle is understood to be split into these portions, because it is impossible (or should we say “impassible”!) to start from one fraction and count from it up or down to another. The gap between them is limitless, uncountable, unbridgeable, an absolute division. Here is diagram 14 redrawn this time omitting the 3 dots between the fractions and replacing them with gaps in the outer circle, just to illustrate the point. As we understand what we are looking at omitting the dots simplifies the picture. (The diagram has a line breaking the system into two down the middle, illustrating the fraction one half).

Figure 15: Denary 10, 100, 1000 circles culminating in base10

pictured with gaps which separate the fractions

Don’t forget that there are many more decimal fractions available in this system. For example, the 100 circle creates 100 decimal fractions in all, beginning at:

1/100 which is 1 out of 100 in 100 circle

= 10/1000 which is 10 out of 1000 in 1000 circle

= 100/10000 which is 100 out of 10000 in 10000 circle

= 100... /10000... which is 100... out of 10000... in 10000... circle

then sweeping through 2/100, 3/100 all the way up to 99/100. Similarly the 1000 circle has 1000 decimal fractions beginning at 1/1000. In fact, every number in a finite circle marks the beginning of a fraction! The final result looks a bit like a ruler which has been twisted into a circle. Think of how a ruler looks (roughly):

Figure 16: rough idea of a ruler

Now imagine the ruler was wrapped into a circle:

Figure 17: Ruler wrapped in a circle

Decimal fractions like 1/10, 2/10 etc. start emanating from the 10 circle. Hundredths begin at the 100 circle and so on. (Of course 50/100 carries beyond the 100 circle into the heart of the system one more step as the fraction 5/10, something 49/100 cannot do). In figure 18 we illustrate some other common decimal fractions in the Denary base, such as 1/8 = 0.125 and 3/4 = 0.75. Translated into the circular system these become:

1/4 = 25 / 100 = 25 in 100 circle

= 250 / 1000 = 250 in 1000 circle etc

= 250... / 1000... = 250... in 1000... circle

3/4 = 75 / 100 = 75 in 100 circle

= 750 / 1000 = 750 in 1000 circle etc

= 750... / 1000... = 750... in 1000... circle

1/8 = 125 / 1000 = 125 in 1000 circle

= 1250 / 10000 = 1250 in 10000 circle etc.

= 125... / 1000... = 125... in 1000... circle.

Figure 18: Denary “ripples” showing decimal fractions as dots on the outer rim

each separated by an impassible, indefinitely great counting gap

In figure 18 (above) the half way mark is drawn in with a blue line. Orange values and lines (culminating in an orange dot on the outer rim) mark both the one quarter and three quarter marks. The light green values, lines and dot mark the one eight decimal fraction mark. We have used gaps in the circle to depict an indefinitely great counting sequence between values rather than the traditional triple dots, simply for clarity.

Now if you don’t get the point, you soon will. In fact, you will get many points!

Between any two decimal fractions there is an unlimited, uncountable number of numbers.

Consider what lies on either side of the 300... mark:

29999999999... ...9999999999997

29999999999... ...9999999999998

29999999999... ...9999999999999

30000000000 ... ...0000000000000

30000000000 ... ...0000000000001

30000000000 ... ...0000000000002

30000000000 ... ...0000000000003

The “300... …000” mark is in bold. It is a 3 followed by an endless sequence of zeros. We just leave a gap in the middle and remain with the start of that enormous number on our left, and its end on our right. Just to have something to play with, let’s call it “three umpitybillion”. Then after it comes “three umpitybillion and one”, “three umpitybillion and two” and “three umpitybillion and three” as shown above. The number before “three umpitybillion” must start with a 2 and have all 9’s after it (it is just not possible to give that number a name, even a silly one). And the number before that must be one less, so it ends in 8, just as shown. The numbers on the right hand side actually form an indefinite counting sequence which produces an impassible gap. We find a similar impassable gap if we examine the “distance” between 1/10 (100/1000 which becomes 100... / 1000... on the outermost circle) and 1/8 (125 / 1000 which becomes 125... / 1000... on the outermost circle):

010000000000 ... ...0000000000000

010000000000 ... ...0000000000001

010000000000 ... ...0000000000002

010000000000 ... ...0000000000003

[the right hand counts on …00000000004, …0000000005 endlessly.

It is impossible to count up to the …99999999997 so this is an unbridgeable gap.]

124999999999 ... ...9999999999997

124999999999 ... ...9999999999998

124999999999 ... ...9999999999999

125000000000 ... ...0000000000000

Count up or down using the right hand side as long as you like, you can never cross the limitless gap which separates each number into two parts, fractional portion (on the left) and numeric portion (on the right). When we pushed the circle size to “indefinity” we created two “gaps” at right angles to each other. There is the gap which separates fraction from fraction, an impassable, limitless gap. Simultaneously every number in the outer circle breaks into two parts, with an impassable, limitless gap down its center as shown directly above. Effectively number is split into two categories: fractional and whole.

Between any two fractions exists an unbreachable numeric gap. The system splits into tenths (ten circle), hundredths (hundred circle), thousandths (thousand circle) and continues on into millionths, billionths and trillionths if so desired. Because they are separate from each other, and they splinter endlessly into smaller and smaller fractions, the result is like a bubble bursting, leaving droplets in the air. The outer circle appears to be breaking up into a series of vanishing points:

Figure 19: Denary system explodes into vanishing points

Between any given two fractions exist an indefinitely great number of smaller fractions. Thus the outer circle tends to dissolve into vanishing points.

However the exact opposite phenomena is simultaneously occurring. Instead of shrinking into “vanishing points” each and every “point” is expanding into an indefinitely great counting line segment. For what is it that produces the “impassable counting gap” between the fractions other than an array of numbers so great one cannot count them! These “gaps” furthermore, march off, like the integers, in both positive and negative directions. The example already given of the numbers surrounding the 300... mark demonstrates a single example of this “infinite” line. The fractional element lies on the left, the indefinitely great number line lies on the right hand column:

29999999999... ...9999999999997

29999999999... ...9999999999998 negative line to “infinity”

29999999999... ...9999999999999 (view the right hand column alone

to see the indefinite counting line)

30000000000 ... ...0000000000000

30000000000 ... ...0000000000001

30000000000 ... ...0000000000002 positive line to “infinity”

30000000000 ... ...0000000000003

[Note: the …999, …998, …997 sequence corresponds to the sequence –1, –2, –3 in circle terms]

We have, in effect, what conventional maths terms the “infinite straight counting line” replicated an uncountable number of times between every fraction, creating trillions upon trillions of impassable, uncountable “line segments” which constitute the “gap” between any two fractions. But such a gap is really a line, so the uncountable trillions of “vanishing” fractions instead expand to an incredible size (assuming we can assign size to number).

Geometrically we began with a point (zero circle) which became a series of circles (10, 100, 1000 circles etc) which then dissolved into a vanishing series of points (tinier and tinier fractions separated by gaps). But the “gaps” constituted straight line segments of uncountable size emanating in positive and negative directions from each fraction. So the “dissolving fractions” are turning into their opposite, they are expanding into indefinitely great counting lines. It appears we have the classic definition of a circle given in calculus…somewhat inverted. In calculus a circle is seen as an “infinite” number of very tiny straight line segments, where each straight line segment is itself made up of an “infinite” number of vanishing points. (Imagine placing billions of tiny wooden rulers next to each other right around the world. The wooden rulers are the “straight line segments”, the “indefinitely great circle” is the circle around the world, and the “vanishing points” are the centimeter and millimeter measurements on each ruler.) Only here we have an “infinite” number of relatively tiny line segments (actually indefinitely great in size!), where each line segment is itself made up of an “infinite” number of “vanishingly small” whole numbers!

Not to be forgotten here is that the conception of the base circle is not on paper, it is in the mind. Furthermore it is a dynamic conception which exists in time. It is based on visualizing ever-expanding circles and imagining the process continuing indefinitely. Every finite circle on the inner rings has 10 more numbers than the next larger circle out. Each of its numbers becomes a starter for a fraction. Between any two such fractions the next circle out will have 10 more numbers, two of which will now be part of those fractions. Out and out it goes, numbers becoming fractions, but in the process producing more numbers in-between, which themselves become starters for yet more fractions. Fractions and numbers are forever interchanging and turning into one another. Until, on the circle without any definite limit to its size, the gap between the two becomes unbridgeable and the fractions and numbers fall apart as separate entities, as we have seen.

Going Around the Indefinitely Great Circle

We said earlier it was impossible to count around the indefinitely great circle. And that is correct. But that does not rule out the possibility of traversing that circle completely. For example, I cannot count to 100,000,000,000,000. However nothing stops me from adding 100,000,000,000,000 to itself and obtaining 200,000,000,000,000. Adding is not the same as counting. It can jump in leaps and bounds where counting must proceed one by one.

We know we cannot count up to the half-way mark of the indefinitely great circle, which is the value 500… (meaning 5000000000etc.) However nothing stops us from directly adding two such values together to complete a full circle.

500…

????....

But what will the result be? We might be tempted to answer 1000… because normally when we jump from smaller to larger circles we catch the count that way. Normally on the count of 10 a 10 circle returns to 0 and “loses count”. But it is caught in the circle which is 10-fold greater than it, the 100 circle, as the compound value “10”. The 100 circle returns to 00 after 10 such counts and itself “loses count”. This problem is solved by the surrounding 1000 circle which catches it as 100. But in this case we cannot jump from the indefinitely great circle to the next circle up as there is no greater circle! So the count is lost.

Clearly we can add around the indefinitely great circle. The question arises, if we cannot catch the count in an outer “greater” circle with a “10”, then how do we catch the count each time we traverse this great circle?

500… one half

700… seven tenths

400… four tenths

???... do we just lose count?

Furthermore, because these are decimal fractions, they are a fixed, constant proportion of every circle in the system. So just as the half-way mark (1/2) passes through 5 in 10 circle, 50 in 100 circle, 500 in 1000 circle and so on, similarly when we go right around the indefinitely great circle back to its …zero… so every other circle in the entire system will likewise return back to zero. So it seems we lose count completely.

So here is our problem. By adding we can leapfrog around the entire system, right back to zero in all circles, thus losing count completely. Going ONE time round the system we lose count. If we look at a wave in a bowl of water we can see that the wave wants to move outwards all the time, until it hits the side. It does not stop when it hits the side, it keeps moving, but it changes direction and moves inwards instead. It reflects back. We cannot keep count of the number of times we loop around when we leapfrog our way around the entire system on some “indefinitely greater circle” however, instead of moving outwards to a greater circle, we can reverse direction and keep the count by moving inwards. One entire circuit round the entire system is one. Two times around is two. Three entire loops is recorded as three. The number of loops around the