 index André Joyce Fan Club   |   Dominissimo   |   Googology   |   Hierogonometry   |   Jootsy Calculus   |   links   |   Laws   |   Merology   |   neologisms   |   non-Najunamarian   |   Pataphysics   |   Sequences   |   transfinites   |   trazillion   |   zerology   "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 Michael Joseph Halm copyright 1978-2005 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".  The name is alternatively said to derive from that ancient practisioner, the Chinese philosopher and that protopataphysicist, Dzu-tse (also known by the Latinized Jucius and the Anglicized  Jootsy).   With jootsy calculus, for example, two plus two not only can equal five, but an infinite number of other correct answers, depending on the viewer's 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 = integers 0   (ze + ro, Nim-addition) 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]) 2   (French [de + ux], German [zw + ei], Italian [do + ce], Portuguese [do + is]) 3   (French and Spanish [tr + es], German [dr + ei], Swahili [ta + tu], Yiddish [dr + ay, fir]) 4   (bases 4+, Swedish [fy + ra], English [fo + ur], German [vi + er], Norwegian [fi + re], Roman [ij + ij]) 5   (English [fi + ve], French [ci + nq],  Swahili [ta + no], Swedish [fy + ra], Yiddish [fi + nf]) 6   (Spanish [cuatro, se + is], Roman [ij + iv], French [quatre], German [se + is], Italian [si + ta], Norwegian [se + ks], Yiddish [ze + ks] 7 [French [se + pt], Italian [quattro], Portuguese [se + te], Spanish (sa + ba, Yiddish [zi + bn]) 8, [French [hu + it], na + ne, German [oc + ht], Italian [ot + to], Latin [quattuor], Portuguese [oi + to], Roman [vi + ij], Spanish [oc + ho)]) 9 [English [ni + ne], French [ne + uf], German [ne + un], (Japanese [ky + uu], Italian and Portuguese [no + ve], Yiddish, Yiddish [na + yn], [na + oi], [ti + sa] 10 (base 4, German [ze + hn], loglan [ne + ni], Spanish [di + ez], Swahili [ku + mi],  Yiddish [ts + en]) 11 (base 3) 12 (base 2) 13  (base 1) 14 (base 0) 8,918,432,389,728,768,545,100,000,000 (Goedel representation) = SSSS0 = g(2, 8, 2)g(2, 8, 3)g(2, 8, 5)g(2, 8, 7)g(2, 9, 11) decimals 0.5   (modulos 7/2, 7/4, 7/8) 1.5  (bases 8/3, 7/2, modulo 5/2) 2.5  (bases 8/5, 7/4) 3.5 (bases 8/7, 7/6) 4.5  (base 7/8) 5.5  (bases 7/10,  -1/2) 6.5 (bases 7/12, -1/4) 7.5 (bases 1/2, -1/6) 8.5 (base 7/16, -1/8) 9.5 (bases 7/18, -1/10) irrationals phi/2 (modulos (15 + G(2, 1/2, 5)/2, (15 + G(2, 1/2, 5)/4, (15 + G(2, 1/2, 5)/8) phi (modulo (7 + G(2, 1/2, 5)/2) imaginaries: 2i (base 2 - i/2) 3i (base (4 - i)/3) 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) T-MATH  2 teacups plus 2 teacups = 40 cups  6 teacups = 60 cups,  7 teacups = 70 cups,  8 teacups = 80 cups,  9 teacups = 90 cups BABYLONIAN MATH   The Babylonians practiced the oldest known jootsy calculus. We still use a modified form of their system for counting hours, minutes and seconds, as in 12:59:59 + 1= 1. It is interesting in that unlike ours, had no radix point to indicate fractions. A numerical expression therefore could have multiple meanings depending upon context, 2 + 2 = 4 could actually mean 2/60 + 2/60 = 4/60 = 1/15, 2/3600 + 2/3600 = 4/3600 = 4/3600 = 1/900, etc.     Even more interesting is Babylonian division (like jootsing from hours to minutes to seconds and back without colons). 11/12 = 55 = 11(5), 9/10 = 54 = 9(6), 5/6 = 50 = 5(10), 4/5 = 48 = 4(12), 7/10 = 42 = 7(6), 2/3 = 40 = 2(20), 3/5 = 36 = 3(12), 1/2 = 30 = 10(1 + 2), 2/5 = 24 = 2(12), 1/3 = 20, 3/10 = 18 = 3(6), 1/4 = 15 = (1 + 4)(4 - 1), 1/5 = 12, 1/6 = 10 = 2(6 - 1), 1/15 = 4 = 5 - 1, 1/20 = 3 = 1 + 2 + 0 Or multiplication: 30*30 = 15, 30*30*30 = 450 = 30*15, 30*30*30*30 = 225 = 15*15, 30*30*30*30*30 = 6750 = 30*15*15, 30*30*30*30*30*30 = 3375 = 15*15*15, 30*30*30*30*30*30*30 = 101250 = 30*15*15*15 CRYPTARITHMETIC    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.      Since this equation has only six symbols it could also have digits substituted for digits in any base six or above, only provided the three matching symbols are preserved, : 36 + 2(6) + 3 =  51         49 + 2(7) + 3                 64 + 2(8) + 3                       36 + 2(6) + 3 =  51       49 + 2(7) + 3                 64 + 2(8) + 3                      5(216) + 3(36) + 4(6)   5(343) + 3(49) + 4(7)  5(512) + 3(64) + 4(8)   = 1212                                = 1890                               = 2784   81 + 2(9) + 3                                     1331 + 2(121) + 3(11) 81 + 2(9) + 3                                     1331 + 2(121) + 3(11) 5(729) + 3(81) + 4(9)              5(14641) +3(121) + 4(11) = 3924                                                    = 73612    You can even add apples and oranges --         APPLES + ORANGES    TUVWXYZ   The most cryptic cryptarithms however are the ones that substitute digits for digits so that 2 + 2 = 4 could mean: 1 + 1 = 2, 2 + 2 = 4 or 4 + 4 = 8, and that's just in integral bases below 10.   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 A9E =                        90.933% BAD BAD cubed =  637B00E5D = 26,704,088,669 (base 10) BAD squared = 8852E9 = 8,934,121 (base 10) DAD = 1.511% 900D + 98.489% BAD DO9 = 1.027% 900D + 98.973% BAD E90 = 2.181% 900D + 97.819% BAD F1B = 2.591% 900D + 97.409% BAD F0E = 2.553% 900D + 97.447% BAD 900D + BAD = 8460 = 33,888 (base 10) 900D*BAD = 691E7C9 = 110,225,353 (base 10) 900D + 900D = 1201A = 73,754 (base 10) 900D   cubed = 2A39C581D395 = 50,149,516,458,133 (base 10) 900D  squared = 510EA0A9 = 1,359,913,129 (base 10) half BAD = 5D6.8 = 1494.5 (base 10) 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 octodecagonal 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*15 one times seventeen = 1*17 one multiplied by twenty-three = 1*23 HONEST SUBTRACTION eighteen minus two = 18 - 2 sixteen minus one = 16 - 1 twenty minus five = 20 - 5 MYSTERIOUS ADDITION      From Louis Hoffman's Puzzles Old and New comes such challenging stuff as: XI + X = XIX (11 + 10 = 19) XX + I = XIX (20 + 1 = 19) IV + VI = IX or XI (4 + 6 = 9 or 11) (hint: read upside down) MYSTERIOUS DIVISION IX/2 = IV (9/2 = 4 or 1 or 10) XII/2 = VII (12/2 = 7) XL/2 = VI (40/2 = 6 or 10 or 50) 8/2 = 3 or 0 MYSTERIOUS SUBTRACTION (or as Hoffman calls it subtraction extraordinary) has: "Take one from twenty and leave nineteen." XIX - I = XX (19 - 1 = 20) Similarly: IX - X= I (9 - 10 = 1) XIX - X = XI (19 - 10 = 11)                       XIX - XI = X (19 - 11 = 10) XIX - XX = I (19 - 20 = 1) XL - L = X (40 - 50 = 10) XLIX - XI = LX (49 - 11 = 50) XLIX - XIX = L (49 - 19 = 50) XLIX - XX = LI (49 - 20 = 51) XLIX - LIX = X (49 - 59 = 10) SIX - IX = S (6 - 9 = 1) This can also be even more mysteriously done for addition in right-to-left base-eight Aramaean, rather than left-to-right base-ten Roman, numerals. X - III = I (4 - 3 = 1) X - II = II (4 - 2 = 2) X - I = III (4 - 1 = 3) IX - X = I (5 - 4 = 1) IX - III = II (5 - 3 = 2) IX - II = III (5 - 2 = 3) XX - IX = III (8 - 5 = 3) XX - III = IX (8 - 3 = 5) SIX - IX = I (6 - 5 = 1) NIM-ADDITION means addition in binary without carry-over, as in the game, Nim. 1 + 1 = 0, 1 + 3 = 2, 1 + 5 = 4, 1 + 7 = 6, ... 2 + 2 = 0, 2 + 3 = 1, 2 + 6 = 4, 2 + 7 = 5, ... 3 + 1 = 2, 3 + 2 = 1, 3 + 3 = 0, 3 + 5 = 6, ... 4 + 4 = 0, 4 + 5 = 1, 4 + 6 = 2, 4 + 7 = 3, ... NIM-MULTIPLICATION means multiplication in binary without carry-over, 2*2 = 3, 2*3 = 1, 2*6 = 11, 2*7 = 9, 2*8 = 12, ... 3*2 = 1, 3*3 = 2, 3*6 = 13, 3*7 = 14, 3*8 = 4, ... 4*4 = 6, 4*5 = 2, 4*6 = 14, 4*7 = 10, 4*8 = 11, ... 5*4 = 2, 5*5 = 7, 5*6 = 8, 5*7 = 13, 5*8 = 3, ... NUMBRAL ADDITION means binary bitwise disjunction  1 + 1 = 1, 2 + 2 = 10 or 10 = 10 = 2, NUMBRAL MULTIPLICATION means binary bitwise shift-and-disjunction 2*2 = 100 or 10 = 110 = 6, 2*3 = 100 or 11 = 111 = 7, 3*2 = 110 or 10 = 6 ODD ADDITION   Similar to mysterious addition is adding an odd number (of letters) to an odd number and getting an odd number. 7 + -ty-one = 71   Or adding odd to even to get even: sty + even = 70 ODD SUBTRACTION Subtracting odd from odd and getting odd: 71 - (-ty-one) = 7 or an even from an odd and getting an even: 61 - (--y-one) = 60 or odd from even and getting an even: 400 - hundred = 4 even from even and getting an odd: 70 - ty = 7 PALINDROMIC EQUATIONS  are equations that read the same backward and forward: 12*42 = 24*21 12*63 = 36*21 12*84 = 48*21 13*62 = 26*31 13*93 = 39*31 14*82 = 28*41 23*64 = 46*32 23*96 = 69*32 24*63 = 36*42 24*84 = 48*42 26*93 = 39*62 34*86 = 68*43 36*84 = 48*63 46*96 = 69*64 112*422 = 224*211 112*633 = 336*211 112*844 = 448*211   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 (corresponding to 2776 in base 10)  UNPROPER FRACTIONS is where cancelation of digits actually works: 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 (without taking place value into account), TWO + TWO = 2O + 2T + 2W = 58 