Goal: to define division by zero. Zero is an unlimited value of negative quality. It negates the existence of something: "I have no idea...", means I possess zero ideas about... It describes, in general, my state of possessing ideas. I couldn't take the statement to mean anything specifically, because no specific number was quoted. If I had said, "I have no group of ten ideas.", then I am still speaking generally about a set of ten: zero sets of ten ideas. But, "I am perplexed.", infers a negative state of ideas exists----it has been displaced by confusion. Negation may be implied only in reference to another value. Maybe I am already ahead of myself. Our way of thinking is segmented into a tendency towards polarizing values. Even if our values include shades of grey, it is still a linear arrangement with two extremes at the poles. One of these poles may be the unknowable aspect to any given situation: unlimited value. The other pole may be the knowable aspect of limited value. Zero and infinity both share something in common: their magnitude is unknowable. But their sign is: infinity is the maximum limit to the existence of a thing and zero is the maximum limit to the non-existence of a thing. Even though I have used the word limit in both references I do not imply that they are knowable as things are normally known by the mind: as finite values. If there is an unknowable nature to ourselves, it can probably know the unknowable since it would be of like nature. And yet, we have somehow arbitrarily and simplistically defined the unknowable to some extent by imagining that a vague distinction could exist within the unknowable, namely: positive existence and negative non-existence. It is not wrong of us to do this, but it shows that we can distinguish the possibility of differences within a subject. Every relative value is finite. It is either positive or negative in its orientation. For a limited value to be distinguishable, it must be inferred that a field of unknowable magnitude and opposing sign surrounds it; otherwise, it would be engulfed by a field greater than itself and the number would be unknowable (but at least the sign of its surrounding field could be inferred): A positive number is surrounded by a sea of potential: negation, or non-existence: 1 = ......00000001.0000000..... This affords us the possibility of exercising parity, one-to-one relations among the digits of this number and the digits of another number of differing length. It comes in handy: 1 + 23 = 24 ....000023.00000.... + ....000001.00000.... ---------------------- ....000024.00000.... (A two in the ten's place plus a zero in the ten's place subtotals twenty, plus a three in the one's place plus a one in the one's place subtotals four, grand totaling twenty-four.) Alas, if: 1 = ......1111111(1).1111111...... How could we know it since it could also be an eleven: 11 = ......111111(11).1111111...... If: 0 = ......000000000.000000000...... Then infinity must equal: (& represents infinity) & = ......&&&&&&&&&.&&&&&&&&&...... Infinity is the maximum value that positivity may take; zero is the minimum---and vice versa: & = [unknowably maximum positive, unknowably minimum negative] = [+&, -0] 0 = [unknowably maximum negative, unknowably minimum positive] = [+0, -&] Infinity is made equal to zero when either one is negated: [+& = -0] [+0 = -&] Generally speaking, in terms of a number's field, division of one field by another can be defined. Since a field is nothing more than a sign, the sign of the answer's field times the sign of the divisor's field must equal the sign of the dividend's field. The only difference is the rules for multiplying field signs are the reverse for multiplying number signs: (dividend / divisor = quotient; dividend = divisor * quotient) (20 / 5 = 4; 20 = 5 * 4) [1] (-field / -field = -field; -field = -field * -field) [2] (-field / +field = +field; -field = +field * +field) [3] (+field / -field = +field; +field = -field * +field) [4] (+field / +field = -field; +field = +field * -field) [1] 0 / 0 = 0; 0 = 0 * 0 [2] 0 / & = &; 0 = & * & [3] & / 0 = &; & = 0 * & [4] & / & = 0; & = & * 0 The same rules of distributing negation within a whole expression still applies: -(a / b = c) = (-a / b = -c) = (a / -b = -c) -(& / & = &) = (0 / & = 0) = (& / 0 = 0) -[1] = [3] = [2] If relative numbers are present within both the numerator and denominator of a ratio's field, then the relative answer is still considered an exact one: (a / -b = -c) = (....0000a.0000.... / ....&&&&b.&&&&.... = ....&&&&c.&&&&....) But if a field answer is desired, or if the ratio is mixed with both relative and unknowable values, then both the field and relative aspects of the answer are both approximate: =: approximately equal |a| absolute value of a 0 < a <= & 0 <= -a < & 0 < b <= & 0 <= -b < & 0 < c <= & 0 <= -c < & [1] a / 0 =: 0; a =: 0 * 0 [2] a / & =: &; a =: & * & [3] -a / & =: 0; -a =: & * 0 [4] -a / 0 =: &; -a =: 0 * & [1] & / a =: &; & =: & * & [2] & / -a =: 0; & =: 0 * 0 [3] 0 / a =: 0; 0 =: & * 0 [4] 0 / -a =: &; 0 =: 0 * & [1] -a / -b =: &; a =: -b * & [2] -a / b =: 0; a =: -b * 0 [3] -a / & =: 0; -a =: b * 0 [4] -a / 0 =: &; -a =: 0 * & [1] a / b = c; a = b * c [2] a / -b = -c; a = -b * -c [3] -a / b = -c; -a = b * -c [4] -a / -b = c; -a = -b * c Modular division discards the quotient and just focuses on the remainder. It breaks magnitude down into smaller, more manageable pieces: 1 mod 3 -1 mod 3 ....., -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, ..... ....., 3/0, 1, 2, 3/0, 1, 2, 3/0, 1, 2, 3/0, 1, 2, 3/0, ..... ...., 0/-3, -2, -1, 0/-3, -2, -1, 0/-3, -2, -1, 0/-3, -2, -1, 0/-3, ..... -6 mod 3 = -3 mod 3 = 0 mod 3 = 3 mod 3 = 6 mod 3 = ... The range of magnitude has not only been shortened, but our references of minimum and maximum magitude have been shifted and cloned: every positive and negative multiple of ten is both a minimum zero modulo ten as well as a maximum