Topics within Modular Arithmetic
Modular arithmetic is the base for almost all mathematical solutions to real world problems. Why? In simple words, your laptop, your phone, your wifi router (the source of ultimate bliss) and even your bank protections would not function if nobody had taken advantage of this simply beautiful sophistication of numbers.
While modern day computer scientists use it to make the web a safer, secure place, you use it without even knowing when planning your week: if today is Monday the 5th, you know that the next Monday will be on the 12th, the next on the 19th, and so on. Simply, any number that can be expressed as 7x+5 with integer x will be a Monday in that month. This is the essential idea behind modular arithmetic.
The main operation is called the modulus, abbreviated mod. The modulus essentially divides the first number by the second and returns the remainder. For example, 12 modulo 5, or 12 (mod 5) would be 2, because when dividing 12 by 5, the remainder is 2. To state the answer, use not the equal sign, but the modular congruence sign: ≡. Used in context,
12 ≡ 2 (mod 5)
Notice that the mod, or the base, is written in parentheses at the end of an equation. Like English, Spanish, and French, each modulo is like a language. Even though they use almost the same script (symbols, numbers, etc.) the characteristics of each “language” is highly dependent on the value of the modulo. As the modulo gets lower, the following happens:
To train your brain to do math in modular arithmetic doesn't tak long with the right practice. Head over to the next page for an introduction to modular arithmetic.