 index.htm André Joyce Fan Club   |   GAMES   |   Chi Xi Digamma   |   Googology   |   Hierogonometry   |   Jootsy Calculus   |   links   |   Merology   |   neologisms   |   Oogology   |   Pataphysics   |   Sequences   "In creative problem solving, it is frequently more important to look at a problem from different vantage points rather than run with the first solution that pops into your head." -- Eugene Raudsepp (Creative Growth Games) Jootsy Calculus in which 2 + 2 = 5, among other things translated by Razilee Purdue C1978-2003 Hierogamous Enterprises   Jootsy calculus is that branch of metamathematics in which calculations are not limited to just one alphanumeric system but allowed to joots, "jump out of the system" in which it began or had been previously "jootsed".   With jootsy calculus, for example, two plus two not only can equal five, but an infinite number of other correct answers, depending on pov ["point of view"] even including infinities. One, in fact, can use any mathematical representation included in the set of Generalized Orthographic Denotations.                        "All things are possible with GOD."    One can add two letters to two letters to get a number word in any number of languages, two letters to two letters to get a Roman numeral, two multiplicative factors to get a quotient, use a different base for the number system than the usual decimal, including "improper" representations using greater-than-base,  half, imaginary or even irrational place holders, modulo arithmetic that notes only remainders.   For example here are a few of the possible answers to what 2 + 2 = 00  (Loglan (ni + ni)) 0   (ze + ro) 0.5   (modulos 7/2, 7/4, 7/8) f/2 (modulos (15 + G(2, 1/2, 5)/2, (15 + G(2, 1/2, 5)/4, (15 + G(2, 1/2, 5)/8) 1   (French [on + ze], German [ei + ns], Yiddish [ey + ns], Italian [on + ce],  Japanese [ic + hí , ka + ta], modulos 3, 3/2, Swahili [mo + ja], Russian [od + in, od + na]) 1.5  (bases 8/3, 7/2, modulo 5/2) 1.5i (bases (8-2i)/3, -12.5+2i, -12-2i) f (modulo (7 + G(2, 1/2, 5)/2) 2   (French [de + ux], German [zw + ei], Italian [do + ce], Portuguese [do + is]) 2.5  (bases 8/5, 7/4) 2i (base 2-i/2) 3   (French, Spanish [tr + es], German [dr + ei], Swahili [ta + tu], Yiddish [dr + ay, fir]) 3.5 (bases 8/7, 7/6) 4   (bases 4+, Swedish [fy + ra], English [fo + ur], Roman [ij + ij], German [vi + er], Norwegian [fi + re]) 4.5  (base 7/8) 5   (English [fi + ve], French [ci + nq],  Swahili [ta + no], Swedish [fy + ra], Yiddish [fi + nf]) 5.5  (bases 7/10,  -1/2) 6   (Spanish [cuatro, se + is], Roman [ij + iv], French [quatre], German [se + is], Italian [si + ta], Norwegian [se + ks], Yiddish [ze + ks] 6.5 (bases 7/12, -1/4) 7 (Italian [quattro], sa + ba, French [se + pt], Portuguese [se + te], Yiddish [zi + bn]) 7.5 (bases 1/2, -1/6) 8, Yiddish [ak + ht], French [hu + it], na + ne, Spanish [oc + ho)], German [oc + ht], Italian [ot + to], Latin [quattuor], Portuguese [oi + to], Roman [vi + ij]) 8.5 (base 7/16, -1/8) 9 (Japanese [ky + uu], na + oi, Yiddish [na + yn], French [ne + uf], German [ne + un], English [ni + ne], Italian and Portuguese [no + ve], [ti + sa] 9.5 (bases 7/18, -1/10) 10 (base 4, Spanish [di + ez], Swahili [ku + mi], Loglan [ne + ni], Yiddish [ts + en], German [ze + hn]) . . .  imaginaries: 5i/2 (bases (2i - 8)/3, -93/8 + 2i, -87/8 - 2i) 3i (base (4- i)/3) 7i/2 (bases (8 - 2i)/5, -14 + 2i, -73/6 - 2i) 4i (base 1 - i/4) . . . inversions (read upside-down): (9 - 8)(9 - 8) 9 - 8 + 9 - 8 (9 - 8)(98 - 88) 91 - 81 + 91 - 81) (9 - 8)(908 - 808) (91 - 81)(91 - 81)) . . . negatives: -10 (base -4) -11 (base -5) BABYLONIAN MATH   The Babylonians practiced the oldest known jootsy calculus. We still use a modified form  for notatinhours, minutes and seconds. It interesting in that it jumped between base 10 and 60 and, unlike ours, had no place-holder such as zero (though they did use symbols for 10, 20, 30, 40, 50), A mathematical expression therefore could have multiple meanings depending upon the values assigned the terms, 2 + 2 = 2AG(2, w, 10)G(2, x, 60) + 2AG(2, y, 10)G(2, z, 60), so that: 2 + 2 = 4, but also -- 22, 122, 240, 14400, 7202, 7320, 86400, 432002, 432120, 439000, 51840000, 3110400000, 186624000000, 11197440000000, 67184640000000, etc.    Addition and multiplication become interesting when changing bases: 59 + 1 = 1; 6 x 10 = 1,   Most fascinating is division however in which we get equations like: 4/2 = 30, with the fractional "/60" understood. CRYPTARTHMS    These are mathematical cryptograms in which letters are substituted for digits. TWO + TWO = FOUR could therefore mean many other things:   In base 10 we have seven possible answers:     734        765          836           846            867        928          938   +734     +765       +836        +846          +867     +928       +938   1468     1530        1672        1692          1734     1856       1876 which could just as validly be other words than "four"-- BOAS, BOAR, BODY, BOER, BOGS, BOGY, BOIL, BOLE, BOLA, BOLD, BONS, BONY, BONA, BOND, BONE, BONG, COAL, COAX, COBS, CODE, CODA, COGS, COIF, COIL, COIN, COKE, COLA, COLD, COMA, COMB, COME, CONE, CONY, CONK, etc.   DIGITAL ROOTS   The digital root is the end product of "casting out nines" or repeatedly summing the digits of a number. 2 + 2 which is 4 is also the digital root of: 13, 22, 31, 40, 103, 112, 121, 130, 202, 211, 220, 301, 400, 1003, 1012, 1021, 1102, 1111, 1201, 1210, 1300, ... ETHICALCULUS   Ethicalculus is a subdiscipline of jootsy calculus dealing with the hexadecimal pseudowords* formed by the alphanumeric characters -- 0 1 5 9  a b c d e f, where a (base 16) = 10 (base 10), B {BASE 16} = 11{BASE 10}, C {BASE 16} = 12 {BASE 10}, D {BASE 16} = 13 {BASE 10}, E {BASE 16} = 14 [BASE 10}, F {BASE 16} = 15 [BASE 10}.   100% 900d (base 16) = 36877 (base 10)     100% bad (base 16) = 2989 (base 10) and countless other alphanumeric calculations using the basic relationship for pseudowords between good and bad, better numbers (above 900d) and worse numbers  (below BAD), many of which are four letters: abe =                        92.004% bad age =                        90.933% bad bad cubed =  637b00e5d = 26,704,088,669 (base 10) bad squared = 8852e9 = 8,934,121 (base 10) dad = 1.511% 9ood + 98.489% BAD do9 = 1.027% 9ood + 98.973% BAD e90 = 2.181% 9ood + 97.819% BAD fib = 2.591% 9ood + 97.409% BAD foe = 2.553% 9ood + 97.447% BAD 900d + BAD = 9bba = 39,866 (base 10) 900d - BAD = 8460 = 33,888 (base 10) 900d x BAD = 691e7c9 = 110,225,353 (base 10) 900D + 900D = 1201a = 73,754 (base 10) 9ood   cubed = 2a39c581d395 = 50,149,516,458,133 (base 10) 9OOD  squared = 510ea0a9 = 1,359,913,129 (base 10) half bad = 5d6.8 = 1494.5 (base 10) *different values would, of course, be given for ethicalculus based on the less familiar hexatrigintal or septemdecimal systems, where GOOD = 21853 and 149125, BAD = 509 and 1489 respectively. GNOMONICS   Square roots and cube roots are not the only roots any more than base ten is the only number system. The other polygonal roots just are not as well known. The triangular root, for example, is the length of the side of an equilateral triangle formed by successive gnomons (as in the additional series of linerally decreasing pile of "bricks"). Some of which are square roots of other numbers.                           [        ]                      [        ][        ]                  [        ][       ][        ]             [        ][        ][       ][        ] 2 + 2 =   the triangular root of 10 the dodecagonal root of 64 the 18-gonal root of 100 . . . Honest calculations  are, like the honest number four, honest in having the same letter count as the meaning of the expression.  HONEST ADDITION one plus twelve = 1 + 12 seven plus seven = 7 + 7 two plus nine = 2 + 9 two plus twelve = 2 + 12 HONEST DIVISION twenty-one divided by one = 21/1 twenty-three divided by one = 23/1 HONEST MULTIPLICATION one times fifteen = 1 x 15 one times seventeen = 1 x 17 one multiplied by twenty-three = 1 x 23 HONEST SUBTRACTION eighteen minus two = 18 - 2 sixteen minus one = 16 - 1 twenty minus five = 20 - 5 SEPTEMVIGINTARIES   These are based on Lee Sallows' system of transforming words and phrases into numbers via base 27, with A through Z evaluated at 1 through 26 and the space as 0, so that: two = 20(729) + 23(27) + 15 = 15216 (base 10) two + two = 30,432 (base 10) = antc two plus two = 20(5559060566555523) + 23(205891132094649) + 15(7625597484987) + 16(10460353203) + 12(387420489) + 21(14348907) + 19(531441) + 20(729) + 23(27) + 15 = 116031263657698329 (base 10) SEXATRIGINTARIES  In this base 36 system A = 10, B = 11, ..., Z = 35 so that we have: TWO + TWO = 236G (2776 in base 10) TALL DIVISION           (different from both short and long division, even giving different answers for same expression) 4/2 = I, V (dividing right and left Roman numerals),          IV (dividing Roman numeral top and bottom) = {1, 4, 5} 6/2 =  V, I, VI = {1, 5, 6} 7/3 = V, I, VII = {1, 5, 7} 8/2 = VI, II, VIII, 3 (Arabic numeral divided right and left),           0 (divided top and bottom) = {2, 6, 8}  UNPROPER FRACTIONS 16/64 = 16/64 = 1/4 19/95 = 19/95 = 1/5 26/65 = 26/65 = 2/5 49/98 = 49/98 = 4/8 64/16 = 64/16 = 4/1 65/26 = 65/26 = 5/2 WORDNUMS   A wordnum is the sum of the letter values in a word, as in SEPTEMVIRGINTINARIES. or base 27, TWO + TWO = 2O + 2T + 2W = 58 