How do you prove that eleven plus two equals one and nine plus five equals two?

Answer 1

By using modulo (also known as clock) arithmetic.

In this type of arithmetic, when the modulus is reached, the counting restarts back at zero - it is the same as the remainder when the number is divided by the modulus.

In the case of this question, the modulus is 12:

(11 + 2) MOD 12 = 13 MOD 12

13 ÷ 12 = 1 r 1

→ 13 MOD 12 = 1

→ (11 + 2) MOD 12= 1

(9 + 5) MOD 12 = 14 MOD 14

14 ÷ 12 = 1 r 2

→ 14 MOD 12 = 2

→ (9 + 5) MOD 12 = 2

Answer 2

You can consider number modulo 12. Two common counting systems based on modulo 12 arithmetic are the months of the year, and the 12-hour clock. But any algebraic structure which is based on modulo 12 addition will do.