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Multiplication = repeated addition? •We think of multiplication as repeated addition •But what does it really mean to multiply by a number in modular arithmetic •Note: 2?7(mod5) Does that mean the following? 2×4?7×4(mod5) •Are we repeating the addition 2 times? Or 7 times? A: Yes B: No E: Maybe A: 2 times B: 7 times C: Both D
Multiplication Use a circle with modulus 10 2 4 10 5 3 10 Subtraction Use a circle with modulus 7 6 4 7 3 6 7 Division Use a circle with modulus 8 6 2 8 5 3 8 Modular division is not as straightforward as the other arithmetic operations Sometimes there is no answer to a modular division problem Find an example of a modular division problem
wrapping around How do we write modular arithmetic? Continuing the example above with modulus 5, we write: 2+1 = 3 (mod 5) = 3 2+2 = 4 (mod 5) = 4 2+3 = 5 (mod 5) = 0 2+4 = 6 (mod 5) = 1 Challenge question What is 134 (mod 5)? It might help us to think about modular arithmetic as the remainder when we divide by the modulus
You put 0 through 11 in a circle Then to figure out what the answer to a modular math question, you begin at 0 and count around the clock a certain amount of times The number you end up on is the answer Example: We want to calculate 32 (mod 12) We start at 0 and go all the way back to 0 This uses up 12 of the hours We have 32-12 =20 left
the division a m, can be written in modular remainder form as amodm (notice the omission of parentheses around the modular suf?x) and is sometimes referred to as the modular residue of a (mod m) Example 2: Use modular de?nitions to justify 11 3 (mod 14) Convert the congruence into parametric form
Doing arithmetic “mod 5” works like normal, except you have to ?gure out where you’d wrap around to with this special counting For example, 2 3 = 6 1 mod 5 (We use the special symbol to denote that we’re doing this special modular arithmetic, not the usual arithmetic with integers ) As another example, 2 4 = 2 3 mod 5 P1
MODULAR FORMS LECTURE 2: ELLIPTIC FUNCTIONS LARRY ROLEN, VANDERBILT UNIVERSITY, FALL 2020 Mankind is not a circle with a single center but an ellipse with two focal points of which facts are one and ideas the other Victor Hugo Before describing modular forms in more detail, we will discuss the related theory of elliptic functions