1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

[Algebra] Proving equations involving modulo functions.

  1. Mar 12, 2015 #1
    I would like to know some general properties of the modulo (remainder) function that I can use to rewrite expressions. For example, say we wanted to prove the following by rewriting the right-hand-side:

    $$ \Big{\lfloor} \frac{n}{d} \Big{\rfloor} = \frac{n - n \pmod d}{d} $$

    I have no idea how to prove this statement formally. Intuitively, ##n \pmod d## is the remainder of whatever ##\big{\lfloor} \frac{n}{d} \big{\rfloor}## is, and therefore

    $$ n \pmod d = n - d \Big{\lfloor} \frac{n}{d} \Big{\rfloor}$$

    Which, when we insert it into the right-hand side of what I'm trying to prove, results in:

    $$ \frac{n - (n - d \big{\lfloor} \frac{n}{d} \big{\rfloor})}{d} $$

    Which, thankfully, boils down to:


    $$ \Big{\lfloor} \frac{n}{d} \Big{\rfloor} $$

    What I want to know is: How do I formalize the beginning of this proof? Is there some general property or definition of the modulo function which I can invoke to justify that step?
     
  2. jcsd
  3. Mar 12, 2015 #2

    RUber

    User Avatar
    Homework Helper

    a = n mod d is defined as a number between 0 and d such that there exists an integer c such that n = cd+a.
    When you take d*floor(n/d), you are essentially saying that floor(n/d) = c. That is, there is no integer larger than c such that cd is less than or equal to n.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: [Algebra] Proving equations involving modulo functions.
  1. Algebra of functions (Replies: 3)

  2. Algebra of functions (Replies: 10)

  3. Modulo Equation (Replies: 7)

Loading...