Philostotle

n = (-0) * n/(-0) = (-0) * n divide every element by n n/n = 1 : first element divided by n (-0)/n = (-0): second element divided by n (n/(-0))/n = n/n = 1: third element divided by n (-0)/n = (-0) : forth element divided by n n/n = 1 : fifth element divided by n Therefore: 1 = (-0) * 1 = (-0) * 1 1 = 1 = 1 I do not think it breaks multiplication: as seen from the above equations it aids multiplication.......

Perhaps I begin to see your point. I should not be asking "is this a field"... How about ...."Is this a valid mathematical construct?" thus: can it be used in the creation of a field? Also it has been brought to my attention that perhaps I should work on redefining what it is to be equal: Thus have only one equality sign......I appreciate your time.... I agree and I am ready for you to continue with your equations....thanks!

Indeed it does...my SINCEREST apologies.....allow then the following edit to the op.... All elements except for (1) have a unique multiplicative inverse....1 has two unique multiplicative inverses...itself and (-0). I do agree with the equation you gave (-0) * n/(-0) = n To Studiot No list of formula is necessary.... There is an element 0 in R such that a + 0 = a for all a in R (that is, 0 is the additive identity). This is still a valid equation for all elements in R observe a + 0 = a : for all elements except 0 in R a + 0 |=| a : for all zero elements in R Thus all elements in any given field are addressed as requested.

Actually, the fact that multiplicative inverses are unique is still the case....1 and -0 both have a unique multiplicative inverses (1 just have more than one)....the term unique does not necessarily imply one and only one.....it implies only that it is different from all others. An apple is unique....and I may have many of them.... You CAN NOT use the NORMAL rules of equality...as given in the op....... Your math is clearly wrong...... n/(-0) = n multiple both sides by (-0) : (-0) * n/(-0) = n * (-0) (-0) * n = n * (-0) We therefore have n=n Divide both sides by n : n/n = n/n 1=1 *your mistake*.....n(-0) = n.....NOT......(-0) Thank you for your time...it is very much appreciated.

Perhaps I should have said.....is this construct a valid field..?....or some other such very specific thing.....perhaps you do not think it is..... and are in agreement with uncool on this matter. Either way .....thank you for your time.

0 + 0 |=| 0.........it still equals zero.....just under a new form of equivalencey There is nothing saying a given number can not have more than 1 multiplicative inverse....both (-0) and 1 both fit the order of defintion for such Please deduce any equations using the rules presented to show how (-0) = 1....it shouldn't be hard as you claim In short...I do not understand how anything useful has been taken away only added.... To Ferrum: perhaps...but division by zero leads to infinitesimals not infinities....thus you get into a can of worms with that one......thank you for your information and time!

Zero Element Equivalency Can this be considered a field? Can this be considered a solution for division by zero? Can this sufficiently create varying amounts of zero? Allow that there exists an integer zero element ( -0 ). 0 =/= (-0) |0| = |-0| 0 |=| (-0) Where |=| is defined as “Zero Element Equivalency”, where any two unique or similar additive identities are considered equal because they share the same absolute value and cardinality but may or may not possess different multiplicative properties. Allow that : 0: possess the additive identity property and possess the multiplicative property of zero. (-0): possess the additive identity property and possess the multiplicative identity property. The addition of any two additive identities is not expressible as a sum, except with |=|. 0 + 0 =/= 0 0 + 0 |=| 0 0 + ( -0 ) |=| 0 ( -0 ) + ( -0 ) |=| 0 Where n =/= 0: n + 0 = n = 0 + n n + ( -0 ) = n = ( -0 ) + n Multiplication of any two additive identities is not expressible as a product, except with |=|. 0 * 0 =/= 0 0 * 0 |=| 0 0 * ( -0 ) |=| 0 ( -0 ) * ( -0 ) |=| 0 Where n =/= 0: n * 0 = 0 = 0 * n n * ( -0 ) = n = ( -0 ) * n 1 * 0 = 0 = 0 * 1 1 * ( -0 ) = 1 = ( -0 ) * 1 The division of any two zero elements is not expressible as a quotient, except with |=|. 0 / ( -0 ) =/= 0 0 / ( -0 ) |=| 0 ( -0 ) / 0 |=| 0 ( -0 ) / ( -0 ) |=| 0 Where n =/= 0: 0 / n = 0 ( -0 ) / n = ( -0 ) n / 0 = n n / ( -0 ) = n Therefore the multiplicative inverse of 1 is defined as ( -0 ) 1 * ( -0 ) = 1 1/( -0 ) * ( -0 )/1 ) = 1 0 remains without a multiplicative inverse. Examples containing the distributive property: a( b + c) = a * b + a * c Where: a=1, b= 0, c=0 1( 0 + 0) = 1* 0 + 1* 0 1 * 0 = 1 * 0 + 1 * 0 Where: a=1, b=0, c=( -0 ) 1( 0 + ( -0 ) = 1 * 0 + 1 * ( -0 ) 0 + 1 = 0 + 1 Where: a=1 b=( -0 ) , c=( -0 ) 1( ( -0 ) + ( -0 ) ) = 1 * ( -0 ) + 1 * ( -0 ) 1 + 1 = 1 + 1 Therefore, non-zero elements divided by zero elements are defined. Therefore, the product of non-zero elements multiplied by zero elements is relative to which integer zero element is used in the binary expression of multiplication. The rules for exponents and logarithms exist without change. It continues that multiplication of any zero elements by any zero elements is not expressible as a product except with |=|. n^0 = 1 n^(-0) = 1 0^0 = 1 ( -0 )^0 = 1 0^( -0 ) = 1 ( -0 )^( -0 ) = 1 0^n = 0 ( -0 )^n = 1 0^(-n) = 1 ( -0 )^(-n) = 1 log0 |=| 0 log( -0 ) |=| 0