On Numbers

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kutto
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On Numbers

Post by kutto » Fri Nov 08, 2013 12:25 pm

Counting is simple. We use numbers to count. But even the system of number symbols we use is arbitrary arrangement of symbols we are taught to interpret. Even the basis and meaning of very familiar symbols is not as straight forward as they first seem to be.

In general everyday use we use the Base 10 number system. This means we have 10 symbols for numbers (0,1,2,3,4,5,6,7,8,9) and with those symbols alone we can express any whole positive number even though we are using only ten unique symbols to do so. We do this because we understand that the position of the number in the sequence of numbers is related to the order of magnitude of the number represents. The base is raised to a power representing the ‘ordinal position’ in the number.

The ordinal or order of magnitude position is simply the number of digits to the left of the last number. The last number is not using the base as a multiplier at all. Eg 7 is just seven.

For example:

2000 is 2 x 10 to the power of (3) or 2 x (10 x 10 x 10) = Two Thousand

2 is raised to the power of 3 because it is in the third order of magnitude position

27 = Twenty Seven or (2 x Base (10)* Plus 7) = ((2 x 10) +7) That is the basic counting arithmetic that is conveyed by the symbolic representation 27.

*Note 10(1) is 10.

Or 427 = Four Hundred and Twenty Seven (4 x (10x10) + 2 x (10) + 7)

With two more symbols we can express all ‘negative’ whole and decimal numbers. Namely ‘.’ Decimal symbol

Ie -2.7 for Minus two point seven.

Of course it is you say, why are you showing me how to count? I know how to count. Of course you do but I want to demonstrate a couple of things relating to familiarity of symbols and numbers are a very common use of something that is clearly symbolic.

The use of Base 10 is completely arbitrary. There is nothing special about Base 10. We use it because we use it.

OK You say – I get that – So?

Well lets consider aspects of the Base 10 number system and lets change them and see what it might do to the everyday world.

Lets consider that instead of Base 10 we use Base 8.

Well for a start the arbitrary roundedness of Base 10 numbers starts to look odd. Because 10 years is a decade because we use base 10 and the numbers are rounded in tens. If we lived in a Base 8 world a Decade (which would be more properly called an Octade) would be 8 years. Because the base is now eight. Two times 8 would be 16 – Two ‘Octades’ would be 16 years.

100 is only ‘rounded’ because it is Base 10 squared or 10 x 10. There is nothing significant about the number one hundred except that is the square of base 10. That’s it. – In base 8 'a century' would be 64 Years or 8 x 8. 64 does not look ‘round’ because I am still representing it in base 10 numbers.

So you say – that’s just playing with numbers – 100 Years would still be 100 years but it would now be represented in base 8. It would but it would no longer be a ‘rounded’ number. Like 64 - the square of the Base 8 number system.

My point is 100 is 100 because it is 10 x 10 (In base 10) and it looks neat.

So then lets extend this and show that 64 in base 8 is a nice round number. The problem we have in doing this is extreme familiarity with number symbols and the use of Base 10. But that’s the point I want to make so lets get going.

All I am going to do is use the first 8 symbols of the decimal system and then use them in a base 8 arrangement.

So then the eight symbols we are using are.

0,1,2,3,4,5,6,7

So far so good. 5 is still five. 3 is still three. But where have 8 and 9 gone. They are the ninth and tenth symbols used in the decimal system and we are not going to use them at all.

So then how do we write eight in the base 8 system. Exactly the same way we write ten in the decimal system. Ten is (10 to the power of 1) + 0 or just 10. Note the 0 is really just to ‘move’ the 1 to the first order of magnitude position so you can apply the 10 to the power of 2 rule.

Now back to eight in base 8. Exactly the same principal as base 10 but due to mixing of symbols it starts to look very strange. Eight in base 8 is ‘10’.

Expanding as above ‘1’ is in the first order of magnitude position.

I am now going to have to use words (using decimal or base 10 interpretation) to describe ‘numbers’ or things are going to get very confused indeed.

So the first position is eight to the power of one plus zero = eight.

Again back in base 10 form to explain the arithmetic. 8 to the power of 1 is 8 + 0 = 8. But remember we are not ‘using’ 8 and 9 but I have done this to explain the arithmetic in familiar terms.

So again ‘10’ is eight in Base 8 using Base 10 symbols as ‘numbers’ - Confusing? This is perfectly valid even though it is very hard to look at and not say that is ten. It is ten in Base 10 and eight in Base 8.

It is simply the base raised to a power (1) which the ‘1’ in ‘10’ represents. Then the ‘numbers’ represented by the base raised to the respective powers are added together to get the ‘total’.

Witness then the combination of symbol + principle and how when the underlying principle is adjusted (it is actually the same fundamental mathematical principle but we have changed something that is normally ‘out of view’ in the base 10 system) due to extreme familiarity with the symbols used. We really do not consider the systemic principle behind the Base number system because it is implicit when we are taught to count in the decimal system. It is actually not really explained at all. It is just taken as ten is a fundamental ‘base’ but it is not. It is a choice.

Lets look at some more numbers. What about twenty one (in Base 10) in Base 8.

It is ‘25’ or two x eight to the power of 1 + five

or in Base 10 (2 x 8 (1) + 5) or 2 times the base (8) which is 16 + 5. Note I have used (1) to denote to the power of 1.

Now you are thinking how could I see that and say it is twenty one. You would still say the words Twenty Five but the number represented would be 21 (In Base 10)

Hey? It is because the 'symbols' 10 and 20 are now (eight and sixteen) respectively - The word twenty would mean 16 or simply 2 x the base (8). Now of course I am mixing things up a little but as really twenty is a 'base 10' word but it highlights that you simply are so used to twenty being 2 x 10 that it seem preposterous that is could mean 2 x 8. It's not really at all you are just so used to seeing the world in arbitrary multiples of 10 and any number you think of is always composed of this basic arithmetic block of ten.

It is twenty one in Base 8. I can assure you if you had gone to school and been trained to count in Base 8 you would look at those symbols (remember that’s what they are) you would simply say twenty five (but mean (in base 10) - twenty one 'objects') with the same lack of effort as you would say ‘twenty five’ in Base 10 world and see twenty five 'objects'.

That’s crazy you say. No it’s perfectly valid but extremely unfamiliar. It is because Base 10 is implicit in the way you were taught to use number symbols. You say twenty five because implicitly you are converting the 2 to 20 and the 5 to five. But remember the 2 is the power of the base – it’s just base 10 is implicit and so familiar you do not think about it.

Back to the Base 8 ‘century’ then – Sixty Four or Eight Squared. How would that be written.

Exactly the same way as 100 is in base 10. In base 8 64 is ‘100’ or eight squared – remember it is the base (8) to the power of the ordinal position in the number sequence (which is 2 from the first number so 8 to the power of 2)

So you would have Happy 100th birthday on your card in Base 8 world when you were sixty four years old!

Imagine being told the world was about to change from base 10 to base 8 tomorrow and simply drop the 8 and 9 symbols. That is mathematically valid but clearly insane. Essentially no one would be able to ‘read’ numbers at all. Or at least very slowly with a way to ‘convert’ numbers. That is what you do now in your head but you just never think about it. You are so practiced at converting using 10 as a base. But it is arbitrary and mathematically no more significant than base 8 or base 12 or 14 for that matter. If you used base 12 or base 14 you need 12 and 14 basic symbols respectively. But the rules of ordinal position and power are exactly the same.

I am getting closer to my intended point which is what makes ‘100’ ‘special’. Or for that matter 10th, 20th or 30th birthdays. In base 8 you would ‘celebrate’ at 8, 16 and 24. When you turned 24 you would have ‘30’ on your Base 8 birthday card but only ‘twenty four’ actual candles…..

Twenty four would be a nice ‘round’ number symbolically – ‘30’

What is significant about 30 then or 40 – Indeed – you celebrate round birthdays based on someones historical choice of 10 as the Number base we use. It is simply the basic arithmetic building block of the number system.

Of course this happens to line up with a natural aging process but we associate roundness of numbers arbitrarily. We would just have more 'special' birthdays in a Base 8 world because 32 would be the new 40!

And you thought numbers were fixed and simple - right?

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TemporalDissonance
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Re: On Numbers

Post by TemporalDissonance » Fri Nov 08, 2013 6:10 pm

kutto thanks for pointing that out!

To add fuel to the fire, in ancient Mesopotamia, the Sumerians and Babylonians used a number system based on 60. This is where we got the 60 seconds, 60 minutes, etc convention… and not to mention the 12 months and 360 days and 12 zodiac signs. Having said that, using a number system based on 60 isn't as crazy as it sounds because much of our current mathematics came from them too. Fractions, algebra, the Pythagorean theorem to name a few.

azooo
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Re: On Numbers

Post by azooo » Fri Nov 08, 2013 6:17 pm

kutto wrote: The use of Base 10 is completely arbitrary. There is nothing special about Base 10. We use it because we use it.
It is not really arbitrary. It is most likely not a coincidence that you have 10 fingers!



And you thought numbers were fixed and simple - right?
Numbers are simple, their representation is not :) Yours is a good example. Let me just add that one does not limit oneself to the decimal representation. We could as well mark numbers with lines, i.e 1 = | 2 = ||, 3 = |||, or use the roman numbering system. No matter how complex this system would be it would describe the same (undescribable) notion of what we percieve as a number.

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KathleenBrugger
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Re: On Numbers

Post by KathleenBrugger » Fri Nov 08, 2013 6:23 pm

I love it Kutto! I think one of the reasons we use base 10 is because we have ten fingers (counting the thumbs). Although I guess if people had disregarded the thumb when they counted we'd have had your base-8 system! :D Did you know there was a rival counting scheme that was base-60; the ancient Babylonians used it. This is why we have 60 seconds in a minute and 360 degrees in a circle (http://en.wikipedia.org/wiki/Babylonian_numerals). [Wrote this before I saw the other posts!)

But your deeper point is very interesting...our minds are programmed so deeply. Our culture programs us with beliefs like "men don't wear skirts" and "homosexuality is wrong." These are sometimes hard to see, because they form the reality of everyone around us, but they are also relatively superficial and with questioning can be exposed as assumptions without basis in fact. But symbolic systems like numbers go deeper. They don't seem like a belief at all. You could go through the same process with letters. When we look at "f" it's hard to see it as an arbitrary shape representing a movement of the lips.

There's even deeper programming. The philosopher Immanuel Kant suggested that we are born with "categories of understanding" like space, time, causality, quantity, and quality. Evolutionary biologists now say that evolution has programmed our brains to filter all our perceptions through these categories to put together an image of reality in our minds. When we look out the window we might actually be looking at a multidimensional reality that goes far beyond the physical, but all we can see is the three dimensions of space plus time because that is how our minds have been programmed to see.
We are ALL Innocent by Reason of Insanity
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SandyJoy
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Re: On Numbers

Post by SandyJoy » Fri Nov 08, 2013 8:48 pm

Yes, Kathleen, and it goes even deeper than social programming, it goes to the foundation of our Real Identity.

Here is are a couple of little excerpts from a wonderful book that contains an enlightened message based on what Kutto is pointing out.

This is out of context, but it gives you the idea:

There is a remarkable and magnificent similarity between what is termed God (Fact, Reality, Truth) and the principle of mathematics. Entire philosophic systems have been built around this likeness. Nearly 2,600 years ago the Greek philosopher and mathematician Pythagoras said that no concept more nearly approached Divinity then the principle of arithmetic. He pointed out that early every quality attributed to God was clearly expressed for all to see and understand in the simplest fact of arithmetic. Indeed, there is inherent in nature a revealed as well as a hidden harmony that reflects itself under the image of simple mathematical laws.

This similarity aids in understanding more clearly who and what God actually is. It paints a picture considerably more accurate than humanity's anthropomorphic concept. Reader, the arithmetic illustration will help you understand Absolute Reality in such a manner as to free you from all your troubles which are the seeming consequence of ignorance concerning Reality, God.

Examine these parallels. God is spiritual and not material. The principle of arithmetic is not material either. It cannot be seen or weighed on a scale. It cannot be felt or picked up by the hand.

God is omnipresent. You can understand the all-presence of God by contemplating the omnipresence of arithmetic. The principle of arithmetic is everywhere, isn't it? Two plus two is four right where you are this minute. So is everything concerning the principle of arithmetic. It is all at the top of Mount Everest and under the waters of the sea. If one were on the moon or at the bottom of a well, the principle of arithmetic would be there too. There is no place in the universe one can go to escape the principle. It is everywhere, every bit, perfectly present; nowhere is it the least bit absent.

I am sure you cannot imagine a place where addition is present but subtraction is absent or where everything is present except the number five or where two plus two is four and a half. Just as one can easily understand that the entire principle of arithmetic is perfectly present everywhere, so can one comprehend the omnipresent perfection of God.---------------------



------------------ Consider Eternity in the light of arithmetic. When did the principle of arithmetic begin? Ponder this a moment and you will see clearly that it had no beginning at all. It has always existed as itself. It is as eternal as existence. And when will it end? What can happen to alter or destroy the principle of arithmetic? Does it get sick? Can it be maimed or crippled? Can it be killed in a war? Can an atom bomb destroy it? If this entire planet were blown to dust by hydrogen weapons, would the principle of arithmetic be affected at all? No. It would still exist untouched and perfect as ever. Two plus two would still be four. It would not be bent in the least.

In exactly this way the Principle of all existence (called God) is eternal, without beginning and without end, indestructible and continuous in being, no matter what the tortured meanderings of human activity. And since the principle continues, so continues all that principle is. As long as there is a principle of arithmetic, addition, subtraction and everything else arithmetic is, continues.

Well, God (Divine Principle, Reality) continues also; hence, all that God is continues. God, the substance and essence of all that is, remains perfect, unabated and untouched—everywhere present, nowhere absent.

Arithmetic is perfect, single, completeness. It is shown forth, made tangible or apparent, by the various signs and symbols of arithmetic.

Consider numbers for a minute. They exist because the principle of arithmetic exists. If there were no principle there would be no numbers. Because the principle is infinite, there is an infinity of numbers. We might say that numbers “live and move and have their being” in the principle of arithmetic because the principle is. The numbers are as eternal and immaterial, as indestructible and perfect as the principle. They are because principle is and for no other reason. So long as there is a principle, there will be the perfect signs and symbols that manifest it.


Even though there is an infinity of numbers, each of them is separate and distinct from the others. 3 is 3 and not 4. 4 is 4, distinctly individual and separate from 5. Yet the numbers in and of themselves are nothing. The value is the principle itself. The importance is principle itself. The primordial substance is principle, and the numbers are principle manifest as numbers. In or of themselves, numbers are just numbers being principle's self-evidence.

No number can exist outside the perfect principle. No number has a mind or will of its own to do what it chooses. No single number or group of them can be anything other than exactly what they are, which is Principle manifest as that particular aspect of principle. There is no life, truth, intelligence and substance in the numeral itself. The principle is all the life, truth, intelligence and substance there is. The numerals are what they are already, without having to do anything to get that way... without having to worry about sustenance and supply, food or money to stay that way. There is no separation at all between principle and numbers. It is all principle being itself.------------------------



----------------------- Well, so what? How does this illustration relate practically to this daily experience? “What good does all this business do me?” you ask.

All the people, places and things one sees, including the body called mine, me, Bill, or Anne, are all, like numbers, being what they are (actually) because a principle of Reality exists—call it a Principle of Existence, Being, Consciousness, God or whatever you like. The name is not important, but the fact of Being is.

It is not possible to deny that Being is. The mere fact that one is aware of existing is obvious proof that something exists; so, if something, anything, exists, then there is Existence—there is Being—there is the fact of Being. Since there is something being, there is reality concerning what is being. It is this Reality, this Principle of Being that is called God.

Reality is God. God is what is. God is the Principle of all being. Now, (note the following carefully, for it is the very heart of this presentation) awareness is being aware of these words, hence awareness is being. Just as God is the principle of all being, even so God is the principle of this very awareness aware of these words, right here, right now—this very awareness called Bill, Rachel, John or Betty.

Consider Awareness at this very instant. This same Awareness that reads these words may also look about and see a table or a chair, a bouquet there, or a little boy. If might look across the way and see a tree, a mountain, a star or a galaxy. Indeed, this Awareness, right here, right now, just might look up and see the Universe! Now I ask you, have you ever been separate from this Awareness? No indeed. Everything you see, have ever seen or ever will see comes as this very Awareness that sees these words. Is this not so? Why, this Awareness is life itself. When one says, “I am alive,” does he not mean, “I am aware”?----------------


If anyone is interested in more, you can get the whole book as a free PDF download here:

http://www.williamsamuel.com/TwoPlusTwoEnglish.pdf
You are not finished, until you play in that meadow and live there. You can, you know. But only you can take yourself there.

kutto
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Re: On Numbers

Post by kutto » Sat Nov 09, 2013 3:14 am

10 fingers - yes! We live in a Base 10 world because we have base 10 hands!! :D

Thanks for the link Sandy

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KathleenBrugger
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Re: On Numbers

Post by KathleenBrugger » Sat Nov 09, 2013 5:39 pm

Thanks Sandyjoy (it seems like I say that in almost every thread I participate in :D ) for sharing those passages about the divine nature of mathematics. Math has always been one of my favorite subjects and you gave me another reason why that is so. I will definitely check out the link you gave.

Just today I heard that the majority of mathematicians in the US are employed at the NSA. Another large contingent work at hedge funds, figuring new ways to squeeze profit out of financial transactions. How sad that our culture hasn't found any better way to employ such a noble profession.
We are ALL Innocent by Reason of Insanity
http://kathleenbrugger.blogspot.com/

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