## Tuesday, 14 May 2013

### Cis Notation for Complex Numbers, Some History and Usage

The notation Cis(θ) is often sometimes used when writing the polar form of a complex number. For example the complex number r ei θ = r (cos(θ) + i sin(θ)) will be shortened to r Cis(θ).
As far as I can establish, the first use of this notation was by Irving Stringham in his Uniplanar Algebra, 1893. On page 71 I found:

I just received a note (Oct, 2013) from JD2718 who writes an educational blog where I had made some comments. He forwarded a comment from a reader who posted:
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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). * Jose Pujol
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I requested information on the use of this notation around the world on Twitter, and received several responses that indicate it might once have been more widely used. For instance Matt Jones @doc_jones1 replied that it was used, "Very occasionally in old highschool textbooks in the UK."
He added that "it's not used anywhere I know. I'll try to find an old text example for you if I can."

I just found a copy of Abstract Algebra: An Introduction to Groups, Rings and Fields By Clive Reis from 2011 which uses the Cis notation. In searching Google books I found dozens of college level textbooks using the notation.

Luis Biedma wrote from Argentina to say, "We use it here in Argentina sometimes." Ms Kat @hornmaths added that it is used in Australia as well. There were also reports of use in Turkey and Indonesia.
churiah hati safitri @churieheart wrote from Indonesia that "It is used in textbook of calculus 2 at univ. I think it is rather common here."
But Some places it seems it is more common in the lower levels. A comment to this post added "We use the cis notation in elementary math courses here in the philippines. Though we barely use it for math majors courses."

I am still  looking for examples from math books in different countries and different periods.  Would appreciate any contributions.