CircleMaths: Lesson 2
The Necessity of the Evolution of the Integers
Regular maths with its set theory breaks number into distinct categories such as:
Whole numbers 1, 2, 3, ...
The integers ... –3, –2, –1, 0, +1, +2, +3 ...
The Rationals 1/1 1/2 1/3 .... (fractions and decimals)
In the next few lessons we will be extending the information on bases we have already arrived at in lesson 1 to show how these sets come about in the first place. We will trace the origins of the mathematical systems and show their relationship to one another. This is an excellent introduction to maths as it gives the big picture at the start.
As children we are normally given these sets merely by definition. Do we know if there are other such sets? (there are in fact). Did they grow from one another like branches from a tree or do they exist like houses built independently next to one another? Were we taught how they evolved from the counting line we were originally given? No. As it turns out their origins are not taught until at least MSc level at university. The origins of number are considered far too difficult for anyone without a degree to even consider understanding.
Worse, when you get to university you find that they are taught in exactly the same way the child was taught. The number types are kept as isolated sets, and they are produced by the clever “invention” of suitably complicated definitions from which the rules for working with, for example the integers, are pulled out like a rabbit from a hat.
So the problems with teaching the origins of maths are:
1. Highly complicated origins available only to trained mathematicians.
2. Number types are created out of thin air by a “fortuitous” choice of definition.
3. The definitions don’t correspond to normal usage, they are complicated and obscure.
4. Integers, rationals etc. remain as isolated sets having little to do with each other.
We do not follow this formula in circlemaths. Firstly we are able teach the origins of the various counting systems to children as they emerge from a single stem in a holistic way. Secondly we stick with common definitions. The number systems should unfold as a seed unfolds its stem which branch into leaves, always coming from an integral whole, not be “invented” by definition and tacked together as seperate pieces like slabs of wood nailed together on the outside of a building. We will show that the systems unfold from one another necessarily due to the properties of number’s circular origins. One system turns into the other as bud must turn into flower which in turn becomes seed. There is an organic system hidden inside mathematics when it is seen in a circular context.
The Necessity of the Evolution of the Integers
The information regarding integers is better detailed in “The Science of Mental Arithmetic” and in the maths article “The Associative, Distributive and Commutative Laws”. It will be addressed more fully in a later article in this series. For now the essential point is that the integer system evolves as a necessary consequence of the circular system. We do not define the integers as a separate class of number, rather they are necessarily produced as a result of the inherent circularity of the system. Given that we have introduced numbers as circles we note that a circle has two directions of travel around it, positive and negative. We can reach every number in any circle in two ways from zero, clockwise (positive) and anticlockwise (negative). This fact is a necessary consequence of the fact that number is circular. Here we see the directions in a 4circle:
Figure 1: Positive and Negative Directions exist
around a circle
Thus every number in a circle can be described as a positive number:
Figure 2: Positive
numbers
And likewise, every number in a circle can be described as a negative number:
Figure 3: Negative numbers
Wait a minute! That means that every number is both a whole number (like 3) AND a positive number (like +3 in the circle above) AND a negative number (like 3 = –1 directly above). That is correct. In 4circle the truth is that 3 = +3 = –1. We will show how this leads to the laws of integer arithmetic and indeed explains them very nicely, while teaching a universal truth found in all maths and science.
The integer system is no more nor less than that just given. It simply describes number in terms of movement in a positive or negative direction around a circle. It exists as a necessary result of the geometrical circularity of number, rather than having to be “invented” separately as a mathematical “field” by definition alone.
We are only covering the origin of the integers here rather than going into all its detail however we will demonstrate one of the integer laws of multiplication and introduce the principal of universal truth, actions which themselves are beyond the reach of maths taught the ordinary way.
A Minus By A Minus Is A Plus
You may know it, you may have forgotten it, but a calculator will verify it for you. Tap in –2 x –3 on any calculator and the result is always the same, +6. There are four sign laws for integer multiplication that children are taught at school without demonstration or proof. They are:
plus by plus = plus example +2 x +3 = +6
plus by minus = minus example +2 x –3 = -6
minus by plus = minus example -2 x +3 = -6
minus by minus = plus example -2 x –3 = +6
The hardest to demonstrate is the last, that minus by minus is a plus. Here is why. To multiply is to take a given number by addition a given number of times. For example consider adding 2 + 2 + 2 to make 6. Multiplication is an addition shortcut. We do not need to say 2 + 2 is 4 and 2 more makes 6. Instead we simply remember that 2 x 3 = 6, which means 3 lots of 2 adds up to 6. This is quite simple and quite easy to demonstrate on the straight counting line to infinity:
Figure 4:
Demonstrating +2 x +3 = +6 (3 lots of 2 is 6)
Now for the impossible demonstration on the straight counting line. Demonstrate that –2 x –3 = +6 with the same clarity. Teachers will give stories about holes in the road and piles of earth or balloons and ballast. But the challenge is to demonstrate it with maths as simply and clearly as that laid out for the easy case demonstrated above. Here is the problem the teacher faces:
Figure 5: The Probem
in Demonstrating -2 x -3 = +6
Two steps left of zero brings us to –2. Now, we have to add that value a total of “negative three” times (whatever that means!) to reach +6 as shown. Search your memory. You will not be able to recall being shown this clear demonstration during your schooling years (you may recall talk of “holes in the road” and “mountains”, or “air balloons” and “bags of sand”, or other such analogies, but not a clear mathematical demonstration). Why? Because the clear demonstration on the number line does not exist. It is actually an impossible demonstration. The only way to reach from the negative numbers back to the positives by addition is if the number line is drawn in a circle, so every number is both negative and positive, as we will now show.
For the minus by minus calculation (–2 x –3), select any circle. We choose 7circle. Find the numbers corresponding to “–2” which means “go backwards two steps from zero” and “–3” which means “go backwards three steps from zero”.
–2 is
5 and –3 is 4
in 7circle in 7circle
Figure 6
As shown, –2 points to the number 5 in 7circle. Similarly –3 points to the number 4. So the sum to demonstrate, –2 x –3 has become 5 x 4 in the 7circle. This means that we must take 5 a total of 4 times by addition (moving clockwise) around the circle. The test is that we need to see the answer comes to +6:
5 plus 5 more is 3 plus 5
more is 1 plus 5 more is 6
when
counted out around 7circle
Figure 7
As demonstrated on the number circle we can see that 4 lots of 5 adds to exactly 6 when counted out around a 7circle. And 6 is exactly 6 places from zero in the positive direction, so it is also +6. So –2 x –3 = +6 can be clearly numerically demonstrated around a 7circle.
BUT you may say, 6 = +6 = –1 around a 7circle. So we have equally demonstrated that –2 x –3 = –1 which is wrong! This is a valid point. It also serves to demonstrate that looking and questioning every issue opens out the truth. In a particular circle, 7circle, it is true that –2 x –3 = +6 and also that –2 x –3 = –1. We need to search for the universal truth. These terms, particular and universal, abound in philosophy and are difficult to comprehend. The child can now receive a clear example of these terms in the course of learning ordinary arithmetic. Let us consider another circle. Pick any other circle.
Let us choose 10circle. We want to demonstrate on it that –2 x –3 = +6 and that –2 x –3 is not –1 ! Again we find what numbers on the circle –2 and –3 point to. Around a 10circle “–2” means go backwards two steps from zero so it comes to 8, similarly “–3” means go backwards three steps from zero so it comes to 7, as shown:
Figure 8: –2 and –3 come to 8 and 7 in 10circle
Figure 9: 8 added to itself 7 times comes to 6 in 10circle
The sum to demonstrate, –2 x –3, has become 8 x 7 in the 10circle. This means that we must take 8 a total of 7 times by addition (moving clockwise) around the circle:
On the 10circle we can see that 7 lots of 8 adds to exactly 6. Again we obtain the answer that –2 x –3 comes to +6. However again the objection can be raised that in 10circle 6 is +6 but equally it is –4 if we measure backwards from zero. Lets look at what we have demonstrated so far:
–2 x –3 = 6 = +6 or –1 in 7circle
–2 x –3 = 6 = +6 or –4 in 10circle
If we had repeated the demonstration in 8circle, 9circle, 11circle and 12circle here are the results we would get:
–2 x –3 = 6 = +6 or –1 in 7circle
–2 x –3 = 6 = +6 or –2 in 8circle
–2 x –3 = 6 = +6 or –3 in 9circle
–2 x –3 = 6 = +6 or –4 in 10circle
–2 x –3 = 6 = +6 or –5 in 11circle
–2 x –3 = 6 = +6 or –6 in 12circle
The pattern is clear. It does not constitute proof because for proof we would need to show it true in all circles, not just presume the pattern continues. But as far as demonstration is concerned, we can see that although the negative value differs from circle to circle, the universal truth appears to be that –2 x –3 = +6 in all circles.
This teaches a very important distinction between particular and universal truths. This is a fundamental philosophical lesson worth gold to any education. Lets look further at that.
Maths And The Nature Of Reality
In 1916 the world was shocked when the atomic bomb was dropped on Hiroshima. At that time a little known theory was thrust into the public limelight when it was discovered that Einstein's theory turned out to exactly describe the energy release that occurred within the explosion. Albert Einstein’s theories described clocks slowing down, rulers becoming shorter and energy being equivalent to mass. But surely that could not be right! Time is a constant just as is length. And mass is NOT equivalent to energy! There must be a mistake! After all, energy is light, heat, motion, zing! that “get up and go” feeling. Whereas the more mass (the more massive a thing is, like a huge boulder), the harder it is to get it to move at all. Mass means inertia, sluggishness, heaviness and equates to non-motion or the inability to change the state of motion one is in (try and stop a rolling boulder!). Energy and mass are direct opposites, they could not be linked. Yet that is precisely the meaning of Einstein’s world famous equation, E = mc2. Ignore the c2 which is just an enormously large but otherwise constant number (the speed of light[r1] multiplied by itself), and it reads “E = m”, or more correctly “Energy = mass x a fantastically huge value”. Einstein’s theory predicted that a few grams of matter could vanish from the world and reappear in a release of pure energy. The energy released by a few grams would be “a few grams x a fantastically big number” which equals enough energy to account for the destructive power of the atomic bomb. His equation states that “mass equals energy x a constant”. In other words it does not transform into it, but rather matter is composed of trapped energy. Matter is energy. The solid things around you are actually pure energy which has become locked into a loop creating the material form. But release the energy in your little finger, and you would flatten an entire city! Scientists did, and still do, find it incomprehensible to consider that two such opposite concepts as energy and mass could actually be the same thing.
The same problem occurred when scientists studied light. Apparently it acts like material “bullets” if you set up your experiment one way, and like a wave spread across space if you set it up another. That’s like saying “look through one lens on a pair of binoculars and you see a giraffe, look through them both and you see a flock of geese”. Again light appears to be a particular, material object (a photon) while simultaneously being a wave of energy, spread out in space. The two ideas are opposite and incompatible, and the idea that opposite things can be identical is in variance with all the maths that most scientists were taught. There is a means of proving theorems in maths by showing that if an assumption leads to a contradiction, therefore the assumption is wrong. It is called “proof by contradiction”. It is a valid means of proof, however it does open itself to failure under certain circumstances when you note the assumption and conditioning built into it.
Light itself turns out to be an example of “electromagetic energy”. That is, it is both an electric wave and a magnetic wave. The two are meant to be different, but they turn into one another and only together do they make the single entity we call light. Quantum physics (the science of the small) declares the universe to be run on probability, or chance. A subatomic particle on your toe might turn up on Venus, simply by random chance. Chances are it will not though, and the multitude of probabilities turns into the certainty of the laws of physics when we consider many many particles. Again, opposites turn into one another.
Circlemaths was devised as a showcase for a philosophical way of looking at the world which has as one of its fundamental tenets that everything breaks into opposites. Can we see an example of that in the maths? We already have in the integers.
Any number can be reached around a circle in one of two directions from zero, and this produced the integers. But it meant that every number in a circle was “just a number” (a whole number), and a positive integer and a negative integer simultaneously.
Figure 10:
The whole number 2 is also the integer +2 and the integer –3
In the 5circle example above we see the whole number 2 without a sign, sitting in its correct place in the circle “in space”. You can arrive at it by making a journey from zero in the clockwise (positive) direction, creating the integer “+2”, a journey involving motion and therefore time. Equally you can reach it by moving 3 steps in the anticlockwise (negative) direction creating the integer “–3”, again a journey involving time.
Every number in every circle exists in these twin, contradictory forms, as whole number and as integer, as unsigned numbers located in space, and signed numbers located in time. In this example we have: 2 versus [ +2, –3 ]
as a pair of opposites which represent the same thing (the same location on the circle).
The claim that “everything exists as opposites” can be applied to itself. Therefore, everything exists just as it is, without an opposite. This truth we also see above. Because the two opposed sides are whole versus integer. On the integer side we see the “thing” broken into two opposites, as positive and negative integers. By contradiction, the same thing is viewed as a single entity without any opposites, as a single unsigned whole number.
Going beyond mathematics, we are looking at a fundamental pattern in the universe. Look and it is everywhere. The human species has male and female. An individual is both body and mind. And as we have already noted, the universe is chaotically ruled by the chance and simultaneously governed by perfect order, light breaks into opposed electrical and magnetic waves, and mass is energy.
More importantly, the act of perception itself automatically breaks everything into opposites. There is the perceiver (thought, non-material, without color or form) on the one hand, and on the other side there is what is perceived (the object, such as a baseball). The observer is one and single. What is observed breaks into a pair of opposites (just as the integers break into positive and negative). We have the “thing in attention” which is considered essential, and against it the background, which is considered inessential. Nevertheless the background is always there, and indeed cannot be absent (Without a background the baseball would have to swell in size until its edges went out of view, leaving a white blob, no longer recognisable as a baseball). We pay attention to the “thing-in-attention” and ignore the background. The two sides are opposites. We say one side is inconsequential, yet in truth both must exist. And must exist together, simultaneously, creating a single whole, the thing we see.
Circlemaths showcases this philosophical truth, which also turns out to be a truth not only of how the world exists, of how we perceive it, but equally of how human understanding operates. We will see yet another clear mathematical example of how opposites turn into one another and back again when we look at how the decimal system unfolds from the bases in the next article.
The End
Copyright Ó
Robert Taylor 25th April 2004
Next Article: The Evolution of the Decimal System
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