# Modular Arithmetic

Whenever someone asks what day of the week it will be in eight days, what they are really having you do is modular arithmetic! They are asking you to take your week divide the days by seven and take the remainder (in this case one) and add that to the current day of the week and determine what day it will be. Say it is a Tuesday, in eight days will be a Wednesday. This is the same as if someone had asked what day of the week it will be tomorrow or what day of the week it will be in 15 days. This is more of what modular arithmetic is, it allows for numbers to be comparable in different modulos.

Background:

The idea of modular arithmetic is the division algorithm stating that a=bq+r where a and b are integers where b is greater than zero and r is between 0 and b (Gallian 3). This is the idea that any number can be broken down to form this equation where a is the number that we are investigating, q is the modulo (or mod) which is the number that is being divided by, b is the largest whole number of times that q can go into a, and r is the remainder. The day of the week example has an a value of eight, q is seven since there are seven days in a week, b is one since seven goes into eight once, and the remainder has r equal to one. Based on modular arithmetic numbers can be equivalent if they have the same r value in the same modulo, this is called an equivalence class (Gallian 18). This is why one, eight, and fifteen days from Tuesday are all Wednesdays: One, eight, and fifteen are in the same equivalence class in mod 7. We will also see modular arithmetic in figuring out what month it will be if we are asking more than 12 months away.

Music is another example of modular arithmetic being used in daily life. Music has eight note scales. They are represented by the letters A through G. This means that every eighth note is the same letter repeated. The note is eight tones higher, but the letter is signified as the same. These notes together sound like the same pitch only in different octaves. Modular arithmetic surrounds us in ways (like this) that we do very naturally, yet this function is also used in a lot of advanced math practices. Many proof techniques require that modular arithmetic be understood so that more advanced topics can be proved true. For example, modular arithmetic turns a normal set of numbers into cyclic groups which allows for proofs referring to groups to apply to certain sets of numbers that would not form a group otherwise (Weisstein Cyclic). But as this is abstract algebra, I digress.

Modularity is a piece that takes a basic scale and arpeggio and shows how the different modulos affect the range of the piece. The first time through the scale is played regularly, then in mod 7, mod 6, mod 4, and mod 2 with B flat being the first note in each modulos. This shows how different notes become equivalent in their equivalence classes because in mod 6 and mod 2 B flat and A are in the same equivalence class, but in mod 4 and mod 8 they are not. This will change because in mod 6 and 2, B flat has the same remainder, but not in mod 4 or 8. Played by Liv Long

Modular the Beautiful is the same concept except that the piece is more recognizable, and the mod is more complicated. America the Beautiful is played as written the first time through, then mod 7 and then mod 4. It is intriguing because the notes are still near the correct relationships to hear the song, but they are not the same sound. This makes sense because the relationships between the notes are the same, just separated by less space than normal.

Played by Eric Lee