Lunar arithmetic

[lpetrich]

Where I found out about it:
Primes on the Moon (Lunar Arithmetic) - Numberphile - YouTube

 Lunar arithmetic was named because it seems "out of this world". It was earlier named "dismal arithmetic", as a pun on "decimal", but that name seemed too depressing.

Addition is done digit-by-digit, and is done with the maximum of the digit values. Multiplication is done much like in ordinary arithmetic, but with digit-by-digit multiplication being the minimum of those digits and overall addition being lunar.

As an example, I will consider 37+74 and 37*74.

In ordinary arithmetic,
37 + 74 = 30 + 7 + 70 + 4 = 100 + 11 = 111
37 * 74 = 30*70 + 30*4 + 7*70 + 7*4 = 2100 + 120 + 490 + 28 = 2000 + (1+1+4)*100 + (2+9+2)*10 + 8 = 2000 + 600 + 30 + 8 = 2738

In lunar arithmetic,
37 + 74 = 30 + 7 + 70 + 4 = max(3,7)*10 + max(7,4) = 77
37 * 74 = 30*70 + 30*4 + 7*70 + 7*4 = min(3,7)*100 + min(3,4)*10 + min(7,7)*10 + min(7,4) = 300 + 30 + 70 + 4 = 374

One can also define "lunar primes", primes whose only lunar-multiplication factors are themselves and 9.

A087097 - OEIS
19, 29, 39, 49, 59, 69, 79, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 109, 209, 219, 309, 319, 329, 409, 419, 429, 439, 509, 519, 529, 539, 549, 609, 619, 629, 639, 649, 659, 709, 719, 729, 739, 749, 759, 769, 809, 819, 829, 839, 849, 859, 869, 879, 901, 902, 903, 904, 905, 906, 907, 908, 909, 912, 913, 914, 915, 916, 917, 918, 919, 923, 924, 925, 926, 927, 928, 929, 934, 935, 936, 937, 938, 939, 945, 946, 947, 948, 949, 956, 957, 958, 959, 967, 968, 969, 978, 979, 989, ...

 

[lpetrich]

More generally, lunar arithmetic can be defined on sequences of members of some set of entities with full ordering, entities where minimum and maximum functions are well-defined.

The minimum and maximum functions, like addition and multiplication and boolean OR and AND, are commutative and associative, and thus orderless.

Both functions are idempotent: min(x,x) = max(x,x) = x

Minimum is distributive over maximum, and maximum is distributive over minimum.

The entities may have an overall minimum, or infimum, and/or an overall maximum, or supremum, but only one of them or none of them may exist.

For the minimum function, the identity is the supremum and the zero the infimum (also the annihilator or absorber). Likewise, for the maximum function, the identity is the infimum and the zero the supremum.

For lunar arithmetic, take
\( A = (a_0, a_1, a_2, \dots) \\ B = (b_0, b_1, b_2, \dots) \\ C = (c_0,c_1,c_2, \dots) \)
which is assumed to be padded with the infimum of the entities.

C = A+B:
\( c_i = \max(a_i,b_i) \)

C = A*B:
\( c_i = \sum_{j=0}^i \min(a_j,b_{i-j}) \)

Addition and multiplication are orderliness, and multiplication is distributive over addition, but in general, addition is not distributive over multiplication.
12+(34*56) = 12+34 = 34
(12+34)*(12+56) = 34+56 = 56

The additive identity and multiplicative zero is (all: infimum) and the multiplicative identity is (supremum, rest: infimum)