CircleMaths: Lesson 1

Counting Lines

General Introduction To CircleMaths

Why would you learn or teach circlemaths? If your interest is to pass grades at school then this is not the page for you. We won't fool you about that. Circlemaths is not on the curriculum. If the aim is to gain a pass in the regular clack clack clack mechanical mathematics then you are better with a regular maths book. (Mind you, learning circlemaths as an extra will probably help understand regular maths in fact.)

So what is circlemaths good for? Circlemaths is primarily a philosophical maths. It uses and exemplifies philosophical ideas which have come from some of the world’s greatest thinkers, Hegel, Einstein and others. It dovetails with relativity and makes sense of quantum mechanics, the foremost theories of modern physics. Dr Taylor (originator of circlemaths) began with a deep study of philosophy, culminating in understanding the most abstract philosopher, Hegel. The insights gained there were meant to be universal, to be applicable to any subject. He applied them to ordinary arithmetic, and the result is what you now see. So the real lessons in circlemaths go below the maths. And they are applicable to anything. As it turns out, the resultant maths not only showcased the philosophy, but also made sense of the physical universe. In particular it helps give an insight into the last dimension, the human mind, which you may notice much of this web site is devoted to. The answers to the big questions in life, the origin of the species as both a spiritual and material entity, the nature and mechanism of thought, our relation to the universe, can be better approached once a background in this maths is taken on board. If you want to teach your child in such a way that one day they might have a chance to understand their own inner nature, the intrinsic nature of the external universe, and the relation between the two, then this is the maths for you.

Brief Overview

This is the first in a series of lessons designed to impart an understanding of the methods and theory of Circlemaths. It begins with counting. Counting in different bases. The intention is that anyone can learn not only how to calculate in multibase, but also to comprehend the deeper theory. Unlike regular maths it is hoped that the deeper theory will be directly accessible to the reader, and that it may result in a change in the way they view the universe about them and themselves in general. For those teaching children, it is noted that the background philosophical approach is one of the most important lessons to be able to convey to the child. It is never considered that the teaching of the old, dry, dusty maths conveys a hidden message that everything is mechanical and dull. But it does. Learning circlemaths we hope will turn out to be an overturning and enlightening experience.

This article is broken into three sections. The first is a general introduction to bases. The second is a deeper view of the theory from a circlemaths perspective. Section II deals with fractions which are a direct result of the step into a base. By understanding it the reader will not only gain a deeper appreciation of how fractions fit into maths, but will also have had a first hand taste of a philosophy which begins to explain how the mind works and where we stand in relation to the universe. Section III concludes by looking even deeper into the philosophical origins of circlemaths.

The roots of the theory are attributed to a theory of mind and accordingly the initial development is radically different from the conventional approach. Instead of a single straight counting line we have several counting circles. Instead of relegating multibase arithmetic to the background it is elevated to the foreground.

Bases are introduced for the beginner, with this departure from the conventional approach, that they are described as being an algebraic generalisation of the circle. This seemingly insignificant difference results in number automatically breaking into two halves, fractions and non-fractions, as will be shown. This automaticity is noteworthy. Traditionally mathematicians have pulled the counting systems out of thin air like a rabbit out of a hat by the use of often quite complicated definitions, “creating” the integers, the rationals and all their relations simply “by definition”. Circlemaths does not follow nor condone this approach. Maths is viewed as being rooted in biology. The failure of conventional maths to reach back to its true biological origins is a reflection of its theoretical incompleteness with respect to the circularity which we believe exists in the neurological pathways of the brain. On the contrary, the circlemaths system is integral. Because circles have two directions, negative and positive numbers are evolved from them, with all the fundamental integer laws intact precisely due to this circular basis. Likewise the movement into an indefinitely large base means number is forced into two categories, fractions and whole numbers. The development of the rational counting system is a necessary evolution, not the result of a “fortuitous” definition.

These articles are designed to allow anyone, including those with no mathematical experience whatsoever, to understand circlemaths (there is no point in translating the Bible of science into Latin!). Should you find any part of it difficult to follow I would appreciate you letting me know by email and I will try to make it clearer. The routine development quickly takes one into depths as will be demonstrated near the end.

Section I

Problem:

Moving from counting objects to counting lines

The origins of number stem from the mind, no less than language. The child’s sensory system is arithmetical from the start. Counting begins with the first object recognised, the mother. We have the biblical trinity or the principle of “three in one” here. The mother herself (the first “one”), is distinguished from her background (we’re up to “two” now), which are both distinguished from the perceiving child (that makes “three” which appear in the single vision of “one mum”). We ignore the background as inconsequential (yet it must be present) just as we ignore ourselves when perceiving an object. Thing-in-attention against a doubled background (making a trinity altogether) is a recurring fundamental pattern. The child’s mind before perceiving in this way would be an unimaginable “zen-like” state without boundary, without object, where self, mother and background exist indistinguishable from one another. Children begin with objects. The mother is the first object, but after that everything is perceived as an external object. With the birth of the object in perception we have the birth of number as the “one”.

This is natural arithmetic, not an invention of “clever mathematicians” but the creation of natural processes predating humanity and common to us all. It is the basis of all mathematics. Here is the problem: the counting line is essentially a ruler, a device for measuring distance, not objects. The two systems do not match. Consider a child trying to match their understanding of counting objects against the counting line in the figures below:

Figure 1: Every number gets an object.

4 objects but a count of 3

We can “fix” this as shown below:

Figure 2: 3 objects and a count of 3.

But one number lacks an object.

In CircleMaths the zero has a special significance (as has the “one” as already mentioned). The zero represents the observer. Think of the child who enters a room and counts the people present, one, two and three. Actually there are four people in the room. Who did they forget to count? Themselves of course, the “invisible” and “zero-like” observer. External, “real” countable objects are counted with the one, two’s and three’s. But everything observed has an observer in our normal waking state, and this observer is represented by the zero in maths.

Treat the counting line as a ruler and you then have no problem. Except then it is a ruler, not a counting line. The child meets their first hurdle and we may not even notice it.

Solution:

The Counting Circle

Twist the line shown above into a circle and you resolve the issue, and step out of ordinary arithmetic and into the quantum world of circular counting lines, as shown in figure 3:

Figure 3

Creating 3 Circle

The “3” lands exactly on top of the zero. And this is correct for a circle, where the start is the finish. We start counting around the circle from zero, we count one, two and then we return and have the choice of calling it three or zero. What is the number before 1? It is zero. What is the number after 2? It is 3. Both possibilities exist in the circle.

Notice the 3-circle has exactly 3 numbers. It also has 3 journeys between numbers, from 0 to 1, from 1 to 2 and finally from 2 back to 0. However 3-circle lacks a 3! A good dictionary does not use the word it defines in the definition, nor does the symbol “3” appear in the 3-circle which is the meaning behind that symbol. (Ordinarily we would not write the “3” inside the zero as we have done above!).

Using pebbles we could obtain a counting line of circles:

Figure 4: a counting line of circles using pebbles

Underneath each circle is the symbol used to describe it (the 0, 1, 2, 3 and 4). We can use those symbols to construct the circle counting line:

Figure 5: a counting line of circles using numbers

Now we have moved from objects to counting line in a smooth transition. The model of the symbol “3” has “3” numbers in it, and “3” paths between those numbers. It does not use the symbol it is defining, which is “3”. This view of number is a radical change and is the heart of the circular system.

Problem:

The hurdle of memory for children.

Before children can learn the principles of long addition they must learn all the single digit additions by heart. That’s 1 + 1 = 2, 1 + 2 = 3, 1 + 3 = 4 and so on up to 9 + 9 = 18. Likewise they must learn all their times tables before they can learn the principles of multiplication and division. Earlier educators set about the task with chanting and repetition. It proved odious to the children and many failed to complete their education. This prompted modern educators to remove the hurdle, allowing them to look up answers on calculators and charts, focusing on the principles only. Unfortunately many children then failed to learn either tables or principles, and a “back to basics” reaction arose in the public eye. The educators are damned if they do, and damned if they don’t.

Solution:

Teach multibase arithmetic from the start

A child learning the violin should be given a tiny violin which works perfectly. Likewise a child learning arithmetic should be given either a tiny circle or a tiny base which works perfectly. One big argument against teaching bases is that “you can do everything in base ten so why bother teaching complicated bases”. The argument turns on its own head. If any base can do “everything” then why teach in a great big base like base “ten”? Why not use a tiny base to teach the principles of arithmetic? Give the child a “tiny violin”. The principles are the same, the arithmetic is still perfect in every respect, but the amount of rote memory required to get started is far less in a smaller base.

To illustrate, here are all the “dreaded times tables” which have to be learned in order to perform any multiplication in base 4. You can probably learn them all in one minute! The standard principles of long multiplication can be taught once these are known, the same standard principles which are used in our ordinary multiplication.

0 x anything = 0

1 x anything = anything (e.g. 1 x 3 = 3)

2 x 2 = 10 2 x 3 = 12

3 x 3 = 21

That's it! Do you think that is too much of a leap for a child? For those who haven’t been taught multibase arithmetic these results may seem strange, however they are entirely consistent and are part of “regular” maths. There is nothing new here. We will explain it later. For now, we merely want to illustrate how easy it is to learn the principles of multiplication in a smaller base. The principles we want to demonstrate will be the standard ones with units and tens columns and carries and so on. Here’s how to do a long multiplication in base four:

Example: (use ordinary addition rules, just substitute your answers for those given above)

1 3

x 2

3 2

The principles are the same you would normally apply:

1. 2 x 3 = 12 (rote fact, given above)

2. Put down the 2, carry the “1” into the tens column.

3. 2 x 1 = 2 (rote fact, multiplying anything by 1 as above)

4. 2 + the carry = 3

The answer is 100% correct, in base four. (We’ll explain why later). In fact we have jumped ahead of ourselves to show the way out of the teacher’s dilemma, as we are still dealing with counting, not multiplication. But the message is clear. There are only 5 basic multiplication facts to learn by rote, hardly anything compared to what the schools teach before you can start learning the principles of long multiplication and division.

Worldwide Prejudice Regarding Teaching Multibase Arithmetic:

Educational experts in their wisdom have declared mutibase arithmetic to be fit only for budding computer programmers, and have laid it down in the curriculum as a subject to be briefly covered by older and brighter children only. This is a mistake written in steel and enforced by the curriculum which every school and every teacher must blindly follow or suffer the consequences of funding withdrawal and closure. Parents likewise obey this precept eager for their children to succeed at school, assuming (incorrectly in this case) that the authorities know best. When we examine the reasons behind the decision to relegate bases to the sideline we find a single argument endlessly repeated.

It is claimed that equivalent results can be obtained in all bases so why not stick with the one “we all know” and avoid confusing children (forgetting the fact that children don’t yet know the base we “all” know). We have dealt with that argument above. It used to be claimed that teaching more than one language would confuse children whereas it is now acknowledged that the reverse is true, and children are standardly taught at least two languages as soon as possible.

The principles of ordinary maths are not truly mutibase. When a calculation is to be performed what generally happens is that the values are translated from the foreign base back to our “home” base. Then the calculation is performed using our finger counting base. The result is then translated back to the foreign base. Roughly speaking ordinary maths is able to translate foreign bases, but not “think” in them directly (the first step in language acquisition is translation, leading finally to the ability to think in the new language). The current maths system is unable to adequately deal with mutibase issues. To learn hexadecimal (Base 16) would require memorizing long tables such as 7x4=1C. There are 91 multiplication facts to memorize, not just the mere 38 that are currently taught, and they are a scramble of numbers and letters to boot! Add another base and the amount to memorize rises astronomically.

On the other hand Circlemaths is at home swimming in the sea of multibase arithmetic. Its methods apply to all bases and calculation occurs directly in the base concerned, with no translation required at all. It does not rely on letters to express larger bases. Working in the multibase dimension is greatly simplified. Circlemaths jumps between bases to produce answers in the desired final base. Much later we will see that the bases themselves are circular, and that they are derived from the circles (in fact, bases are merely the algebraic generalization of the circularity of number). The link between bases and circles is intimate and a proper understanding of circlemaths requires an understanding of bases and vice versa.

To obtain a full appreciation of number it is essential to understand number as it originally appears, in its multibase form. The scientific method teaches us to observe what exists and not to alter it according to our particular belief or taste. Multibase arithmetic is actually what exists and we ignore the facts present in reality at our own peril. By avoiding bases one might fit in with the curriculum and become “educated” but this will never produce what we need to break through to a new understanding of how the brain functions and how the universe around us operates.

Just as a knowledge of other languages assists one to understand one’s own, so knowledge of other bases helps one understand the processes of one’s “home” base. For example, can 1/3 be written as an exact decimal? We are taught it is 0.3333... a recurring decimal which never terminates, which might lead one to presume that we can cut an apple exactly in half, but not in thirds. But in base 9 “one third” is exactly 0.3 The problem of recurring decimals is linked to the base we use, not to the fact that it is impossible to exactly cut something (like a pie) into exactly 3 pieces! To understand division properly one needs to understand more than one base. We understand by comparing, and from a series of one (one base in this case) nothing can be learned.

For those readers interested in relating circlemaths to neurology, consider that although the answers total to the same value in different bases, nevertheless they look different to the human “eye”. To give an example, below we see a magic square. It is ‘magic’ because it adds to the same number (15) across each row, down each column and across each diagonal:

8	1	6
3	5	7
4	9	2

Figure 6: magic square in “Denary”

Here is the same magic square written in base three. You don’t have to understand base three to see patterns which just weren’t apparent in our ordinary base. This works with any magic square of any magnitude (in a circle the “100” would be a “00”, better still).

22	01	20
10	12	21
11	100	02

Figure 7: same magic square in Base3

Relate this to the process of perception, recalling the well known artistic “illusions” where we see either a vase or two faces, an old woman or a young one. What we see is conditioned by how we perceive. Consider the input to our brains being swept through not one but a multitude of bases (circles) until recognition occurs.

Finally, for the educator, parent or professional teacher, learning to calculate in different bases allows one to appreciate the task of coming to terms with arithmetic for the first time just as the child must, to see through the child’s eyes. So here it is, counting in multibase arithmetic.

Counting in Multibase Arithmetic

Here are some dots to count.

Figure 8: dots to count

If you got “8” you are correct. Now group them in lots of 3:

That's 2 lots of 3 and 2 left over. In base 3 we call it 22.

Figure 9: 22 dots in Base3

Now group them in lots of 4:

Figure 10: 20 dots in Base4

That's 2 lots of 4 and 0 left over. In base 4 we call it 20.

Now group them in lots of 5:

Figure 11: 13 dots in Base5

That's 1 lot of 5 and 3 left over. In base 5 we call it 13.

This time we’ll start with a few more dots. Count them. There are 13:

Figure 12: dots to count

Lets group them in lots of “6”:

Figure 13: 21 dots in Base6

That's 2 lots of 6 and 1 left over. In base 6 we call it 21.

Now group them in lots of ten:

Figure 14: 13 dots in Base10 (Denary)

That's 1 lot of ten and 3 left over. In base ten we call it 13.

Which is exactly what we initially called it when we counted it.

As we promised we will briefly revisit the base 4 multiplication given in our example above (which produced 13x2=32) and show how it makes sense. In base 4 it looks like this (using the fact that 13 x 2 = 13 + 13):

13 + 13 = 32 (base 4)

Figure 15: showing 13 + 13 = 32 in base 4

As you can see base 4 treats a group of 4 as a “10”. The “32” above has 3 “tens” which are really 3 groups of 4. In base 5 a group of 5 would be counted as “10”. So 32 in base 5 would be translated as 3 groups of 5 plus 2, which in denary comes to what we call 17. And so on. That is how to translate from one base to our own. Bases aren’t so hard to understand after all. They are how we “see” numbers. We tend to see things in groups or piles, and they form an excellent introduction to group theory.

The Idea of Ten

The true idea of “10” does not correspond to the number of fingers on the human hand. That is merely one example. As demonstrated in the bases above it appears 10 can be a group of any number. It is a number which can take on any value, just as the x’s and y's of algebra can take any value. Imagine the advantage to children picking up the concept of algebra for the first time, if they have already met a real number which can take on any value (for those into the Hegelian philosophy, 10 is a “universal”, as is the whole concept of algebra).

The idea of 10’s, 100’s and 1000’s is a return to an original pattern. To illustrate, suppose we use base 3. In base 3 a group of 3 is taken as its 10. Its pattern is a triangle:

Figure 16

A Group of 3

is the 10 of Base 3

The 100 of Base 3 will be 10 lots of 10. Again it should return to the primal “base 3” pattern of a triangle:

Figure 17

100 = 10 x 10

The Triangle Form Reappears in a Larger Size

The 1000 of Base 3 is 10 lots of 100 or 10 x 10 x 10. Again it should return to the primal “Base 3” pattern of a triangle, which it does in the form of a much larger triangle:

Figure 18

1000 = 10 x 10 x 10 or 10 x 100

The Triangle Form Reappears yet Again

Below is a picture showing the classic 5 pattern found on dice used to display 1000 in base 5. Each group of 5 dots is a “ten” in base 5, the larger groups are the 100’s, and the whole is grouped in a 5 pattern to create the 1000. You may have seen the beautiful images of fractals, mathematical drawings which swirl into infinity. They are made up of two sides. One side is the regularity of the base, which you can see above and below. The other side is a slight random irregularity, which pushes the regularity slightly skew into a myriad of slowly changing patterns. Here is the beautiful snowflake like pattern of base 5. Like a fractal it repeats indefinitely.

Figure 19: Fractal like pattern of Base5

Circularity as the fundamental concept of Returning Pattern

For Base 3 we get triangles, for base 4 we have squares, base 5 (above) gives a snowflake. But when we reach base 57 or base 986 the recognisable patterns have been used up. Who knows how 986 dots should be set out? However all these numbers have one thing in common and that is that their original form is circular. The return to the circle is the true “10”, and every circle can be conceived as a “10” circle and the founder of a base. The very concept of “returning to an original pattern” implies a returning to an original starting point, or circularity. Thus the idea of 10 as “a return to an original pattern” is the idea of a circle completed. To demonstrate by example, consider 3-circle.

Figure 20: 3-circle

Start at 0. Then count round the circle 1 and 2. The next step returns us back to the start. We have accomplished two things. We have gone round the circle ONE whole time. Secondly we have returned to ZERO. Spatially speaking we have returned to 0 as if we never left. But in temporal terms we can recall we have completed one circuit. We have a “1” (one circuit) and a “0” (back to the start). So once round the circle can be said to be a 1 and a 0 or a 10 in other words, taking number into both time and space.

We will pause here for emphasis. This is the 10 circle of base 3. Remember this fact as we will refer to it several times. When we refer to a “10 circle” this is what we mean.

Figure 21

The 10-circle of Base 3

[We do not call it “3-circle”]

We could count round and round the circle:

0 1 2 0 1 2 0 1 2 ... endlessly if we only viewed it spatially.

Taking into account our memory (time) we can count:

0 1 2 10

The next number is 11, meaning once round and on 1 more.

Then comes 12, meaning once round and on 2 more.

0 1 2 10 11 12

Next we move from 2 back to 0 to complete our 2nd compete circuit. Thus we call it 20, meaning twice round exactly.

0 1 2 10 11 12 20

Next step we add on 1 more and reach 21, meaning twice round and on 1 more.

Finally we reach 22, meaning twice round and on 2 more. We can draw these circular repeating forms just as we drew the “fractal-like” drawings above.

Figure 22

If you compare the above to the triangular drawings of base 3 you will see that this represents 10 lots of 10 or a unit of 100. Because we only have 3 numbers in the circle it looks more triangular, so we have drawn it with an enclosing circle to emphasize that we have once again returned to a much bigger version of the original 10 circle of base 3. Instead of units making up the positions in the circle we have whole 10 circles. It is a 10 circle made up of 10 circles.

Figure 23

Here are the numbers in it counted out:

0 1 2

10 11 12

20 21 22 100…

You might be wondering how we are going to create the straight line of the counting base. We are getting there... Have you seen the way that bubbles in a glass can pop and coalesce to create a single larger bubble? Circles behave in this bubble-like way too, and we can choose to draw all the numbers in the large enclosing circle, breaking open the smaller packets and turning it into a single circle. For example, the group of 100 can be broken into:

Figure 24

100-circle

Here are the numbers in it counted out:

0 1 2 10 11 12 20 21 22 100…

Now we can see the full “straight counting line” of base 3 is on its way. We can equally appreciate a very important fact: the “straight” counting line is an indefinitely great circle.

To create the base we simply continue this process indefinitely. Here is the next jump, from 100 to 1000 (10 x 10 x 10). The enclosing circle below is a 1000-circle. It replicates the pattern found in the ten circle of base 3 this time using 100 circles where the original 10-circle had units. It is 10 x 100 = 1000. As each 100 is also a larger version of the original 10 circle of base 3, being 10 x 10’s themselves, we can equally see that 10 x 10 x 10 = 1000. Using this small base one can actually point to the 10’s, the 10x10’s (hundreds) and the thousand units. In our own counting base we would need to use an A1 sheet of paper to accomplish the same task! Forget the educational benefits, consider the savings on paper!

Figure 25

The counting line of base 3 now goes:

0 1 2

10 11 12

20 21 22

100 101 102

110 111 112

120 121 122

200 201 202

210 211 212

220 221 222 1000...

Once again allow the circles to pop and coalesce into a single larger 1000-circle:

Figure 26

Here are the numbers in it counted out:

0 1 2 10 11 12 20 21 22 100 101 102 110 111 112 120 121 122 200 201... 1000

We didn't have enough room on the line to put in all the numbers so we just added 3 dots and put in the 1000 at the end. We could do this again, using three of these 1000-circles to make a single larger 10000-circle and so on and on. Soon we would need a sheet of paper that would go right round the Earth just to fit the numbers on! The Earth appears flat because it is such a large sphere, and likewise a partial view of an enormous circle appears like a straight counting line because of its size. Take it one more step, off the paper and into the imagination. Continue the process indefinitely. We won't specify the end. Now we can't specify the size of the popped out circle so we simply show our line like this:

0 1 2 10 11 12 ...

The “Straight” Counting Line

Of Base 3 to InDefinity

Philosophical Note: you can't draw a perfectly straight line just as you can't draw a perfectly circular circle. Put under a powerful enough microscope defects always show up. However the pure concept of straight and circular exist. They exist in the human mind. When we leapt off the paper into the imagination we crossed a barrier and leapt into the human mind, where a perfectly straight counting line can exist. So the above is in fact ONE of the many straight counting lines to “infinity”.

Creating Our Ordinary Base Ten

Our normal counting base is too big to show clearly all the steps, but because most people are familiar with it, we will produce the normal version of the “straight counting line to infinity”, at least showing as many steps as easily fit on the page.

Begin with the 10-circle of base “ten” (Because every base has a “ten” in it, and common usage links ten to our fingers, it will assist clarity if we refer to our finger counting base ten by its Latin name “Denary”.)

Begin with the 10-circle of Denary:

Figure 27

It counts 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. When we count from 9 to 0 we have turned ONE whole circle and returned to NOUGHT. Thus we get the memory (time) and space count of ONE NOUGHT or 10.

The next numbers are 11, 12, 13, .... up to 20, 21, 22, 23, .... and finally 97, 98 and 99 before we reach the 100 mark. At the count of 100 we have 10 lots of 10. This is shown in the circle below (not shown in great detail as it would need a larger piece of paper):

Figure 28: ten Denary 10-circles

The counting line so far goes:

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19

20 21 22 23 24 25 26 27 28 29

30 31 32 33 34 35 36 37 38 39

40 41 42 43 44 45 46 47 48 49

50 51 52 53 54 55 56 57 58 59

60 61 62 63 64 65 66 67 68 69

70 71 72 73 74 75 76 77 78 79

80 81 82 83 84 85 86 87 88 89

90 91 92 93 94 95 96 97 98 99 100

Whew! Its much bigger than our previous example. While this may be more suitable for our everyday needs, the principles in counting out the smaller base are exactly the same, but easier to demonstrate. One benefit of the larger base is that we can see the circular shape the ten circles make much easier. As in our base 5 example, we can pop the bubbles and coalesce them into one large circle of size 10 x 10 = 100:

Figure 29: the 100-circle of Denary

We will leave you to imagine the values in between (i.e. from 24 to 99). Here are as many of them as we can fit on one line counted out:

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23... 100

If we continue this process up to 1000 and then beyond to an undefined point, we have obtained the traditional “base ten” (Denary) straight counting line to “infinity”:

0 1 2 3 4 5 6 7 8 9 10 11 12...

NOTE: Although the counting line of the base remains circular it is indefinitely large. It follows that this indefinitely great circle can never be fully transversed. In this respect it is infinite. Arrival at the 10 of this circle is indefinitely postponed! As we cannot count ONCE around it and return to ZERO we cannot arrive at the 10 of this ‘infinite’ circle at all. Suppose, by some magical method, we could count right round the indefinitely great circle. Then we would get back to its zero, and we would be able to say ONCE round and back to ZERO makes a ONE-ZERO or 10. Then we would have a problem, because our counting line would look like this:

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 … 10

Oh oh… The problem is that we would then have two tens in our counting line, which would be very confusing to say the least! The fact is that we cannot count right round that huge circle at all so there is no second “10” on the end. The one and only true ten is therefore the 10 found after 9, and the base is named after that ten. See further comment under the heading “Base of Expression”.

Closing Comments

Binary

In true circular form, that which is most simple is that which is most difficult to understand. Binary is left to last as it is so simple that it lacks sufficient detail to follow what is happening. It nevertheless follows the same construction rules as any other base. Here they are:

Begin with the 10-circle of Binary:

Figure 30: 10-circle of binary

We would call this “2-circle” however if we want to establish Binary it is called 10‑circle. Begin at 0, count to 1. The very next count brings us full circle. We have accomplished ONE full circle and have returned to NOUGHT. We have ONE and NOUGHT or 10 in Binary.

The count so far is:

0 1 10

The next number brings us to 11. After that we have reached the value 100 for we have replicated the basic “10-circle” pattern of Binary, as shown:

Figure 31

The outer enclosing circle is drawn to assist one to see that the pattern is a circle, as opposed to a straight line (although perhaps this apparency has a hidden significance). The count now goes:

0 1

10 11 100

If we let the numbers in the circles pop and coalesce to create one larger circle, we obtain:

Figure 32

We might be tempted to call this a “4-circle” because it has 4 numbers in it, but it is really the 100-circle of Binary. The Binary 10 is equivalent to our normal “2”. So 10 x 10 = 100 is equivalent to our normal base’s 2 x 2 = 4. The count now goes:

0 1 10 11 100

If we take it one more step we repeat the basic 10-circle pattern of Binary this time with 100-circles, giving us 10 x 100 = 10 x 10 x 10 = the 1000-circle of Binary:

Figure 33

The count now goes:

0 1

10 11

100 101

110 111 1000

Or, if we allow the bubbles to pop and coalesce into a single 1000-circle we obtain:

Figure 34

The count now goes:

0 1 10 11 100 101 110 111 1000

If we continue this process on indefinitely we can write it as having no definite stopping point, wherein it becomes Binary the base:

0 1 10 11 100 101 110 111...

Binary is important because it is the first valid base. A smaller base cannot have a 10 for it would have no “1” in it, so Binary is the first, and it is entirely composed of 0’s and 1’s. This is why computers utilise it. The 0 corresponds to zero volts in a wire, and the 1 corresponds to a voltage present. A wire has a voltage across it or it doesn’t. On or off, 1 or 0.

The numbers 0 and 1 are special. A base can exist without a 9, an 8 or a 7. But it can't exist without a 0 and a 1. The 0 and 1 are the raw ingredients, the ‘eggs and flour’ of any base. The base is then like a cake. A cake is in a different class from its ingredients, as 0 and 1 stand on their own in a different class from all other numbers found in the bases.

Help With Writing Out the Bases

We can write Denary out in one line like this:

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 ...

or take advantage of its circular nature and write it like this:

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19

20 21 22 23. 24 25 26 27 28 29 and so on.

The circular theory relies on circles inside circles. For example the numbers inside a circle stand for circles themselves (the “2” in a 3-circle is a symbol which stands for a 2-circle). We have just seen that a base is an indefinitely great circle. Hence bases exist inside bases. Anyone who can write Denary out as above to 100 can write any base below it to 100. Here’s how:

Denary up to 100, showing Binary, Ternary and Quaternary up to 100: