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"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 19782005 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, Dzutse (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 greaterthanbase, 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, Nimaddition)
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 upsidedown):
(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)
TMATH
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
twentyone divided by one = 21/1
twentythree divided by one = 23/1
HONEST MULTIPLICATION
one times fifteen = 1*15
one times seventeen = 1*17
one multiplied by twentythree = 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 righttoleft baseeight Aramaean, rather than lefttoright baseten 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)
NIMADDITION
means addition in binary without carryover, 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, ...
NIMMULTIPLICATION
means multiplication in binary without carryover,
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 shiftanddisjunction
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 + tyone = 71
Or adding odd to even to get even:
sty + even = 70
ODD SUBTRACTION
Subtracting odd from odd and getting odd:
71  (tyone) = 7
or an even from an odd and getting an even:
61  (yone) = 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 = 1
19/95 = 1
26/65 = 2
49/98 = 4
64/16 =
65/26 =
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

