What Is Similarity?

We may learn the etymology of the word from The Words of Mathematics by S. Schwartzman:

similar (adjective), similarity (noun), similitude (noun): from Latin similis "like, resembling, similar." The Indo-European root sem-, which appears also in native English same, meant "one," so two similar things look as if they 're "one and the same" in respect to a certain property. In algebra, the terms that contain the same powers of the variables involved are said to be like or similar terms; terms that are not similar are called dissimilar. In geometry, two figures are said to be similar if they have (one and) the same shape, though not necessarily the same size. The symbol "~" that we use to indicate similarity is due to the German mathematician Gottfried Wilhelm Leibniz (1646-1716).

So, in mathematics, the word "similarity" may be encountered in several different contexts:

1. In geometry, similar figures have the same shape but may differ in size. (See, for example, pages on nature of π, homothety transformation, or simply google the site for word "similarity".)

2. In algebra, polynomials or equations may have similar terms.

   Two terms ax^m and bxⁿ are similar if m = n. Similar terms are easily added or subtracted based on the distributive law. For example, the sum ax^m + bx^m could be simplified to (a + b)x^m. "To simplify an equation" often requires finding and adding groups of similar terms.

3. Matrices with the same eigenvalues are often called similar.

   Two matrices A and B are similar when there is a non-singular matrix P such that A = P^-1BP. Such matrices have the same eigenvalues and the characteristic polynomial, naturally.

4. Two or more arguments may be similar so that only the first one is given and the rest, to avoid repetition, are said to apply similarly. There are numerous examples of the argument by similarity; just google the site for word "similarly". To argue by similarity is similar (or even the same) as arguing by analogy.

References

1. E. J. Borowski & J. M. Borwein, The Harper Collins Dictionary of Mathematics, Harper Perennial, 1991
2. H. S. M. Coxeter, Introduction to Geometry, John Wiley & Sons, 1961
3. J. Daintith, R. D. Nelson (eds), The Penguin Dictionary of Mathematics, Penguin Books, 1989
4. S. Schwartzman, The Words of Mathematics, MAA, 1994
5. P. Zeitz, The Art and Craft of Problem Solving, John Wiley & Sons, 1999

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