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Old 2003-12-28, 06:38   #1
Pax_Vitae
 
Dec 2003

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Default Prime Numbers: Nothing but Errors in Multiplication???

I’ve been wondering why people think Prime numbers are so special. To me they are only an inconsistency within the logic of how Multiplication works, meaning all that Multiplication is, is a short form for Addition with the “Same Number!” An example:

(2 * 3) = (2 + 2 + 2) = 6

Again:

(3 * 7) = (3 + 3 + 3 + 3 + 3 + 3 + 3) = 21

It’s much easier and convenient to write:

(3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3) as (3 * 10) it’s easier to read and calculate.

Multiplication’s greatest strength is also its greatest weakness, which is the fact that it’s the Same Number being used again and again in the process of Addition. Meaning the short from of (3 * 7) can be expressed in Addition as either (7 + 7 + 7) or (3 + 3 + 3 + 3 + 3 + 3 + 3). To me all that a Prime number is, is a number that has no short form! The short form of 6 is (3 * 2), the short from of 24 is either (2 * 12), (3 * 8), (4 * 6), etc. It’s because of the limits within Multiplication that it’s impossible to create 3 as the only numbers you can Multiply with are 1 and 2 which is (1 * 2) = 2 likewise you can’t make 2 from 1, (1 * 1) = 1, a problem that doesn’t exist when Addition is written in its long form.

This, to me, is an inconsistent property inherent in the logical formula of Multiplication. There are no such things as ‘Prime Numbers’ with addition, as all numbers come from 1 even the number 2 is just a short form of (1 + 1), 3 = (1 + 1 + 1), etc. So to then call numbers which have no short form in Multiplication ‘Primes’ seems to be giving them an importance that is only bestowed because of a fundamental weakness in Multiplication when dealing with Natural Numbers, seems illogical. I’m not saying this weakness doesn’t have its uses, cryptology being one of the most important from a world perspective. But this is a use that is possible because of the flawed nature of multiplication.

This is only a flaw in Multiplication, as Division can create any number. To create 7 we just do (21 / 3) = 7. Well you could say you’ve just created a Prime with the help of another Prime. Not true, 3 and 7 are Natural Numbers and only considered Primes because you can’t create them in Multiplication. In the first 10 numbers there are 5 Primes because of the limited number of Numbers available. Then the range 11 – 20 has 4, 21 – 30 and 31 – 40 each only have 2, as you obtain more numbers to Multiply with the frequency of Primes reduce. They don’t vanish completely as the fundamental flaw in Multiplication scales (or lessens depending on how you want to view it) with the number of Numbers available for its use.

So to sum it all up: Prime Numbers are only the correction to an Error inherent in Multiplication. While this error has its uses (cryptology) are these numbers really “Special” or just “Special” in relation to Multiplication?
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Old 2003-12-28, 09:13   #2
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http://www.maths.ex.ac.uk/~mwatkins/zeta/index.htm

There is some kind of very deep connection, as yet tantalizingly just out of reach, between prime numbers, the most famous unsolved problem in mathematics (Riemann hypothesis), and the laws of physics. The properties of prime numbers may well be related, in some fundamental way, to the properties of the universe itself.
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Old 2003-12-29, 16:35   #3
ewmayer
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To turn Pax's argument on its head: one could equally well claim that is addition rather than multiplication which is "flawed," because every natural can be constructed infinitely many ways as the sum or difference of 2 or more other naturals, whereas there is just one way to construct any natural number as the product of its prime factors. Thus, we can think of the primes as constituting a sort of multiplicative basis for the naturals (and with a slight extension, all the integers), and any natural N can only be constructed in one way as a product of a finite number of these basis elements. That doesn't seem at all "flawed" to me.

While it is true that in some very limited sense (e.g. addition of pure integers) multiplication can be viewed as superfluous, we can equally well define every add in term of a multiply, which makes addition seem superfluous. It all depends on your point of view: You can view the operation of multiplying a complex number z by a complex root of unity as a complex multiply, or as an addition to the angular part of z in complex polar coordinates.

Also, your argument about multiply-as-add works fine for pure integers, but how would you define (say) the operation x*sqrt(2) without resorting to the concept of multiplication? And how would you handle the concept of a multiplicative inverse, which leads naturally to the operation of division?

Last fiddled with by ewmayer on 2003-12-29 at 16:35
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Old 2003-12-30, 16:20   #4
Pax_Vitae
 
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Quote:
To turn Pax's argument on its head: one could equally well claim that is addition rather than multiplication which is "flawed," because every natural can be constructed infinitely many ways as the sum or difference of 2 or more other naturals, whereas there is just one way to construct any natural number as the product of its prime factors
True, but my point is that when Multiplication is limited to natural numbers it ends up with wholes in the Number Line. This isn’t necessarily a bad thing, just an aspect of how Multiplication is limited, when it’s limited to only Natural numbers.

Meaning 2 can be created with multiplication by (1.4142135624 * 1.4142135624) [I know this is not quite 2, but given more decimal places it could be made equal to exactly 2]

Quote:
Also, your argument about multiply-as-add works fine for pure integers
That’s all I’m interested in, because that’s what defines a Prime. I can create 7 by saying (2 * 3.5) but I can’t, as I’m limited to natural numbers. Primes only exist to fill in the wholes left in the number line when multiplication is also limited to natural numbers.

As a side note, when you do multiply numbers that have decimal places it’s a combination of Addition and Division. Example:

(4 * 3.5)

(3.5) is made up of the following: (1, 1, 1, .5 [which converts into a half]) so

(4 * 3.5) = 4 + 4 + 4 + (4 / 2 the half) = 14 or
(4 * 3.5) = 3.5 + 3.5 + 3.5 + 3.5 = 14

It’s still Addition, but also the Addition of a fractional amount, which requires a Division of the whole to calculate its value.

Pax Vitae
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Old 2005-11-04, 11:56   #5
mianfei
 
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Quote:
Originally Posted by ewmayer
how would you define (say) the operation x*sqrt(2) without resorting to the concept of multiplication? And how would you handle the concept of a multiplicative inverse, which leads naturally to the operation of division?
Well, it is very difficult to add irrational numbers without introducing error into their value, yet if one multiplies the whole numbers of the infinite series that are used to obtain the value of these irrational numbers, one does not have that problem.

However, of course, this is a contradiction insofar as infinite series only exist as infinite sums. So, if only for this contradiction, it hardly makes sense unless we reduce allnumbers to infinite series!
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Old 2005-11-05, 09:31   #6
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Quote:
Originally Posted by Pax_Vitae
Meaning 2 can be created with multiplication by (1.4142135624 * 1.4142135624) [I know this is not quite 2, but given more decimal places it could be made equal to exactly 2]
No, it couldn't -- not with finite arithmetic. The square root of 2 cannot be exactly expressed with any finite number of decimal places, so you could never get exactly 2 as the product.
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Old 2005-11-11, 11:37   #7
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Maybe Pax meant a number that can never repeat? I guess it's possible to argue that way with 1*1.41...+.4*1.41...+... but to be fair, addition and multiplication are VERY inconvenient to just get rid of. If I wanted to be technical bordering on being overrigorous, then I could define both addition and multiplication in terms of For loops(iterative functions) and increments for each decimal place, with carrying being used to handle single-digit overflow. The carrying involved could be thought of as a form of recursion (opposite of iteration). This is too much like microcoding/gate-level design, though! Imagine if an OS had to be implemented as shifts/XORs/NANDs/memory! That's why they invented multiplication, addition, exponention, and of course CPUs with rich instruction sets that do these in one logical step.

This reminds me of the controversy in Calculus about using limits versus using infinitesimals. 1/3 has a decimal expansion of 0.3 with a line over the 3 which means .3333333 and so on. In limit notation that would be a sum of 3/10+3/100+3/1000+... which is useful when working on a computer but isn't the only way to look at the sequence. Note that infinitesimals are not often used because of discreet methods using computers working better with limits most of the time. Also note that infinitesimals aren't very popular with instructors. Hehe Wonder why - must be something to do with all the engineering majors in their classroom. Nah, it couldn't be that. <sarcism>

Last fiddled with by nibble4bits on 2005-11-11 at 11:38
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Old 2005-11-11, 12:15   #8
xilman
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Gather round everyone!

The phrase "glutton for punishment" suddenly came to mind. I wonder why?


Paul
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Old 2005-11-11, 12:23   #9
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What is he talking about?

...aw heck, I think I'll just sit back and watch the show.

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Old 2005-11-11, 20:45   #10
ewmayer
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I scored the last family-sized tub of popcorn at the concession stand - the rest of you will have to content yourselves with day-old hot dogs or greasy nachos. Ah, dammit, some dickweed stuck gum to my seat...I hate that.
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Old 2005-11-12, 02:03   #11
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This form of, what I consider to be rather high level stuff, math gives me a headache, as does alot of higher math. It goes beyond me and makes me feel really really stupid. Seriously!
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