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A mathematical notation query: Whence cis?

May 12, 2013 pm31 7:16 pm


I know a little about complex numbers and complex functions. Just a little. And along the way I picked up a nice piece of compact notation:  cos\theta + isin\theta can be written instead as cis\theta.

My question is a little one, but a historical one. When did this notation come into use, and where?  If I pick up a text in another language, will I also find cis?

7 Comments leave one →
  1. May 12, 2013 pm31 8:59 pm 8:59 pm

    I found the following on Jeff Richie’s page and he is awesomely accurate.
    Irving Stringham used cis β for cos β + i sin β in 1893 in Uniplanar Algebra (Cajori vol. 2, page 133).
    As for use in all countries, I will have to do a little research. It is not always popular even in this country, but I think it is pretty well known.

    The text is available on line
    The introduction of Cis notation occurs on page 71 (strangly in the volume online he uses Cis (theta) rather than Beta??? and this is the 1893 version.

    • May 12, 2013 pm31 9:14 pm 9:14 pm

      Had a feeling you’d be first! Wow, that was quick.

      I liked using it when I learned it. It felt cool to use a slick abbreviation. (There were all sorts of details about how professors wrote or spoke that felt cool to adopt. This was one of many)


  2. May 12, 2013 pm31 9:09 pm 9:09 pm

    Just one more quick note. Stringham uses Tensor and amplitude where it is more common today to use magnitude and argument (or angle) but he does include magnitude and argument in parentheses in his definition, then seems to use only Tensor and amplitude.
    Hope this is of some help to you.

    • May 12, 2013 pm31 10:10 pm 10:10 pm

      He sort of dedicates the work to a list of prominent 19th century mathematicians… but who were Grassman and Sylvester? I should be reading your This Day in Math more closely!

  3. May 13, 2013 am31 9:00 am 9:00 am

    Jonathan, Grassman is a good one to read up on. Some of his original work reads pretty hard, but he is a heavyweight in the history of Vectors although his credit came a little late.
    Crowe in his “History of Vector Analysis” says of him, ” “Grassmann’s Die lineale Ausdehnungslehre (Linear Extension Theory) demonstrated deep mathematical insights. It also in one sense contained much of the modern system of vector analysis. This, however, was embedded within a far broader system, which included n-dimensional spaces and as many as sixteen different products of his base entities (including his inner and outer products, which are respectively somewhat close to the our modern dot and cross products). Moreover, Grassmann justifies his system by philosophical discussions that may have put off many of his readers.”
    JJSylvester is a legend, He did important work on matrix theory, in particular, to study higher dimensional geometry. In 1851 he discovered the discriminant of a cubic equation. (similar to quadratic discriminant students learn, they may be surprised that one exists for a cubic) Earlier in his life, he tutored Florence Nightingale.
    You can find good bios of both of them all over the net.

    I have had some comments to my request for who uses Cis, and it seems it is across the world, although not necessarily popular. in all countries. Will pull the comments together in a few days and maybe post them on my blog and web page.

  4. May 13, 2013 pm31 12:09 pm 12:09 pm

    LaTeX knows the commands \cos and \sin (so \sin \theta instead of sin \theta). But LaTeX doesn’t know \cis. This is presumably because modern mathematicians don’t use the cis notation — by Euler’s identity, cis \theta = e^{i \theta}, and the exponential notation gets used instead. I think it would be very pedagogically challenging to explain to students just learning complex numbers (and in particular, students who haven’t taken calculus yet) why this really deserves the name of an exponential function, and (perhaps hardest) why the base should be the same e that shows up in other places.

  5. Jose Pujol permalink
    October 11, 2013 pm31 12:28 pm 12:28 pm

    See Hamilton, W., Elements of Quaternions, 1899. In a footnote on p. 262 (vol. I) Hamilton refers to the “recent abridgment cis for cos + i sin”. Hamilton died in 1865 and his Elements were first published in 1866. The 1899 edition is the second one and is based on the original text, which “has been faithfully preserved, except for a few places where trifling errors have been corrected”. Other changes made had the purpose of rendering the work more convenient. Therefore, it seems that the notation cis was introduced by Hamilton, who also introduced the scalar and vector products (among many other things).
    The book by Crowe Referred to by Pat Ballew has a good description of Hamilton’s contribution. The second edition of the Elements was reprinted by Chelsea Publ. Co. in 1969.

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