Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

What is the difference between norm and modulus?

  1. Aug 27, 2007 #1


    User Avatar

    norm is defined to be the length of the vector and we put we denote it by ||a||.

    However, modulus |a| also means the length of a from the origin?

    So, what is the difference between the symbol || || and | |?
  2. jcsd
  3. Aug 27, 2007 #2


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    There is none. Some authors use one, others use the other.
  4. Aug 27, 2007 #3
    I was always taught that the norm is defined to be any length which satisfies the definition of a norm, but modulus is specifically the Euclidean norm. That could have just been the instructor/book, though I suppose.
  5. Aug 27, 2007 #4


    User Avatar
    Science Advisor
    Homework Helper

    The modulus is the norm on [itex]\mathbb{R} [/itex]. [itex] \left(\mathbb{R},\left| , \right| \right) [/itex] is a Banach space.
  6. Aug 31, 2007 #5


    User Avatar

    I think norm is the length of a point from the origin.

    whereas the modulus is more of a distance from one point to another point.

    norm is just a specific case of the distance from a point to its origin.
  7. Aug 31, 2007 #6

    D H

    Staff: Mentor

    The semi-standard usage is that modulus is specialized to the reals (absolute value), complex numbers (complex modulus), and quaternions. However, some write [itex]|x|[/itex] instead of [itex]||x||[/itex] to mean norm. Norm is a very generalized concept that covers everything from Euclidean distance to distance as measured on a road grid (taxicab norm) to the L-infinity norm, and beyond. Anything that qualifies as a distance can be used as a norm. Even things much more complex and abstract than vectors can have a norm. Well-defined norms exist for matrices, for example.
  8. Aug 31, 2007 #7

    Chris Hillman

    User Avatar
    Science Advisor

    Actually, both norm and modulus are potentially ambiguous terms in mathematics; the intended meaning depends upon context. Generally speaking, modulus often suggests algebraico-geometric-analytic origins (as in the theory of elliptic functions) while norm tends to suggest operator algebras and functional analysis. In the context of functional analysis, esp. Banach spaces, I agree with DH about what one can reasonably expect "modulus" to denote.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?