On Tuesday, 7 April 2015 17:41:04 UTC+2, Dan Christensen wrote:
> On Tuesday, April 7, 2015 at 2:44:37 AM UTC-4, John Gabriel wrote:
>
> > I am still waiting for the ranters and screamers to show me how 1 -:- 3 = 1/3 using algebra.
>
> Easy.
He says easy but does not even read or understand the question correctly. The question asks the reader to show using repeated subtraction how it can be that 1 -:- 3 = 1/3. Tsk, tsk.
>
> Assuming we have multiplication, for y=/=0, we define x -:- y = z if and only if x = z * y.
Dumbo "assumes" we have multiplication, but the question makes it clear that nothing is to be assumed. One cannot assume we have multiplication, because multiplication is derived from the quotient. Axiom 4 below.
El Dumbo assumes again that z * y is well defined for all rational numbers, but fails to notice there is a difference when it comes to proper fractions, that is, the multiplication operator takes on a different meaning (it's polymorphic for those with brains). So he makes a LOT of assumptions, but what can you expect from an idiot who subscribes to Peano's juvenile axioms?...
>
> We have 1 = 1/3 * 3. By definition then, 1 -:- 3 = 1/3.
Dumbo does not understand definitions (much like dullrich does not understand the same) and assumes a definition is well formed solely by his whims.
>
> Oops, I forgot... you banned if and only if definitions some time ago. I guess you and your legions of followers (hee, hee!) are screwed.
>
A lie which is considered libel in the USA.
Gabrielean Axioms:
1. The difference (or subtraction) of two positive numbers, is that positive number which describes how much the larger number exceeds the smaller.
Let the numbers be 1 and 4.
4 - 1 = 3 or |1 - 4| = 3
2. The difference of equal numbers is zero.
Let the numbers be k and k.
|k - k| = 0
3. The sum (or addition) of two given positive numbers, is that positive number whose difference with either of the two given numbers produces the other number.
Let the numbers be 1 and 4.
1 + 4 = 5 because 5 - 4 = 1 and 5 - 1 = 4
4. The quotient (or division) of two positive numbers is that positive number, that measures either positive number in terms of the other.
Let the numbers be 2 and 3.
2/3+2/3+2/3 = 6/3 = (6-2-2)/(3-1-1) = 2/1 = 2
3/2+3/2=6/2= (6-3)/(2-1)= 3/1 = 3
5. If a unit is divided by a positive number into equal parts, then each of these parts of a unit, is called the reciprocal of that positive number.
Let the positive number be 4.
The reciprocal is 1/4 and 1/4+1/4+1/4+1/4 = 1
6. Division by zero is undefined, because 0 does not measure any magnitude.
Since the consequent number is always the sum of equal parts of a unit, it follows clearly that no such number exists that when summed can produce 1, that is, no matter how many zeroes you add, you never get 1.
7. The product (or multiplication) of two positive numbers is the quotient of either positive number with the reciprocal of the other.
Let the numbers be 2 and 3.
1/2+1/2+1/2+1/2+1/2+1/2=3
1/3+1/3+1/3+1/3+1/3+1/3=2
8. The difference of any number and zero is the number.
Let the number be k.
|k-0|=|0-k|
Observe that all the basic arithmetic operations are defined in terms of the primitive operator called difference.
These are the true axioms of arithmetic and the definition of the arithmetic operators.
Axioms for negative numbers are easy to define with some trivial modification.