loading

Logout succeed

Logout succeed. See you again!

ebook img

Modular arithmetic PDF

pages79 Pages
release year2017
file size1.39 MB
languageEnglish

Preview Modular arithmetic

Modular arithmetic Secondary Mathematics Masterclass Gustavo Lau Introduction On what day were you born? Worksheet 1 Going round in circles Modulo 12 How to represent time? t 0 1 2 3 6 9 12 12 t 9 3 6 Modulo 12 Instead of 13 = 1, in modular arithmetic we write 13 ≡ 1 (mod 12) and read it “13 is congruent to 1 modulo 12” or, to abbreviate, “13 is 1 modulo 12”. Examples: 12 ≡ 0 (mod 12) 17 ≡ 5 (mod 12) 37 ≡ 1 (mod 12) -1 ≡ 11 (mod 12) In general, a ≡ b (mod n) if a-b is a multiple of n. Equivalently, a ≡ b (mod n) if a and b have the same remainder when divided by n (remainder modulo n). Clock addition table + 1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12 Modulo 12 In modular arithmetic we use the numbers 0-11 instead of the numbers 1-12. The reason is that 0-11 are the remainders modulo 12. In general, when we work modulo n we replace all the numbers by their remainders modulo n. Modulo 12 addition table + 0 1 2 3 4 5 6 7 8 9 10 11 0 0 1 2 3 4 5 6 7 8 9 10 11 1 1 2 3 4 5 6 7 8 9 10 11 0 2 2 3 4 5 6 7 8 9 10 11 0 1 3 3 4 5 6 7 8 9 10 11 0 1 2 4 4 5 6 7 8 9 10 11 0 1 2 3 5 5 6 7 8 9 10 11 0 1 2 3 4 6 6 7 8 9 10 11 0 1 2 3 4 5 7 7 8 9 10 11 0 1 2 3 4 5 6 8 8 9 10 11 0 1 2 3 4 5 6 7 9 9 10 11 0 1 2 3 4 5 6 7 8 10 10 11 0 1 2 3 4 5 6 7 8 9 11 11 0 1 2 3 4 5 6 7 8 9 10 Examples: 7 + 8 ≡ 3 (mod 12) 10 + 2 ≡ 0 (mod 12) 13 + 2 ≡ 3 (mod 12) -1 + 14 ≡ 1 (mod 12) Modulo 12 Can we use arithmetic modulo 12 to represent something else?

See more

The list of books you might like