How do the notions of oddness and evenness apply to both functions and numbers?

If Even and Odd functions share nothing in common with Even and Odd numbers, then why were Even and Odd adopted for functions? Why not use other adjectives?

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James Stewart, Calculus 7th ed. 2011. This isn't the Early Transcendentals version.

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    $\begingroup$ $x^n$ is even when $n$ is even and odd when $n$ is odd $\endgroup$ – Randall Jan 11 at 3:10

For an even function $f(-x)=f(x).$ Now you know that $(-x)^n=x^n$ if $n$ is even.

For an odd function $g(-x)=-g(x)$. Now you also know that $(-x)^n=-x^n$ if $n$ is odd.


They share lots in common!

  1. The product of two even functions is even, the product of an even function and an odd function is odd, and the product of an odd and an even function is odd. This is exactly how even and odd numbers add, and is a nice motivating example of an isomorphism.

  2. The Taylor series (at $x = 0$) of an even function has only even terms $x^{2n}$, and the Taylor series of an odd function has only odd terms $x^{2n+1}$. If you don't know what a Taylor series is, we can restrict our attention to polynomials: a polynomial is even iff it has only even terms, and a polynomial is odd iff it has only odd terms.

  • $\begingroup$ Unfortunately, the sum of two odd functions is odd :( $\endgroup$ – Joe Jan 23 at 21:35
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    $\begingroup$ @Joe: sure, that's fine. If you think about the polynomial example you'll see that when you multiply polynomials (or power series more generally) the exponents are adding. $\endgroup$ – Qiaochu Yuan Jan 23 at 21:59

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