Chapter 2 ("The Pool of Tears"): Alice tries to perform multiplication but produces some odd results: "Let me see: four times five is twelve, and four times six is thirteen, and four times seven is—oh dear! I shall never get to twenty at that rate!" This explores the representation of numbers using different bases and positional numeral systems: 4 × 5 = 12 in base 18 notation, 4 × 6 = 13 in base 21 notation, and 4 × 7 could be 14 in base 24 notation. Continuing this sequence, going up three bases each time, the result will continue to be less than 20 in the corresponding base notation. (After 4 × 12 = 19 in Base 39, the product would be 4 × 13 = 1A in Base 42, then 1B, 1C, 1D, and so on.)
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Because the base is increasing by 3, but the number by four, the answer will seemingly increase by a single number each multiplication, but since the base is increasing by three each time, the number based on the base, which is increasing by 1 each time, will never catch up and form a 20, but this only works when the 10 is introduced, so the system does not work for (4*n) where n equals any number between -3 and 2.
This rule works backward as well.
example of system working both ways
4 x -7 = -28, -28 in base -18 number system is 1A
4 x -6 = -24, -24 in base -15 number system is 19
4 x -5 = -20, -20 in base -12 number system is 18
4 x -4 = -16, -16 in base -9 number system is 17
4 x -3 = -12, -12 in base -6 number system is 20
4 x -2 = -8, -8 in base -3 number system is 22
4 x -1 = -4, -4 in base 0 number system is ∅
4 x 0 = 0, 0 in base 3 numeral system is 0
4 x 1 = 4, 4 in base 6 numeral system is 4
4 x 2 = 8, 8 in base 9 numeral system is 8
4 x 3 = 12, 12 in base 12 numeral system is 10
4 x 4 = 16, 16 in base 15 numeral system is 11
4 x 5 = 20, 20 in base 18 numeral system is 12
4 x 6 = 24, 24 in base 21 numeral system is 13
4 x 7 = 28, 28 in base 24 numeral system is 14
4 x 8 = 32, 32 in base 27 numeral system is 15
4 x 9 = 36, 36 in base 30 numeral system is 16
4 x 10(a) = 40, 40 in base 33 numeral system is 17
4 x 11(b) = 44, 44 in base 36 numeral system is 18
4 x 12(c) = 48, 48 in base 39 numeral system is 19
4 x 13(d) = 52, 52 in base 42 numeral system is 1A
4 x 14(e) = 56, 56 in base 45 numeral system is 1B
4 x 15(f) = 60, 60 in base 48 numeral system is 1C
