Modular figuresModular numbers refer to integers considered through their remainders: two integers belong to the same equivalence class if they yield the same remainder when divided by a given positive integer. This framework is known as modular arithmetic, commonly referred to as “clock arithmetic”.
A classic example comes from telling the time: 9 + 6 hours = 15, but on a 12‑hour clock this is shown as 3, because the numbers “wrap around” modulo 12. 12 / 9 = 1 + 3 Clock arithmetic, formulated as a mathematical system thousands of years ago, was not chosen by accident; its conceptual clarity has been obscured in our own time by the ubiquity of digital clocks. In believing that we understand nature, we have in many respects drifted away from it. A careful examination of the world around us reveals a multitude of structures that can be read from the clock face — and not merely as a sequence of ratios. Addition 9+8≡17≡5(mod12) (17-12 = 5) Multiplication 7⋅6=42≡2(mod10) (4 x 10 +2) Residue 7 5=2 (17 = 3 x 5 + 2) Congruence 38≡14(mod12) (3 x 12 + 2) Negative integers −7 4=1 ( 1,0 - 1,12 - 1,25 - 1,6 - 2,0 In one particular book, the significance of the expression 1 + 12 is presented rather clearly, yet it has been almost entirely misunderstood. Leonardo da Vinci was, in all likelihood, among those who understood it best.
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