New names for odd and even functions

The concept of “odd” and “even” functions is quite important in areas such as Fourier analysis. An odd function is one that has 2-fold rotational symmetry around the origin; examples include $y=x$ , $y=x^3$ , and $y=\sin x$ .

$y=x^3$ is an example of an “odd” function

$y=x^3$ is an example of an “odd” function

An even function is one that has mirror symmetry around the y-axis; examples include $y=x^2$ and $y=\cos x$ .

$y=x^2$ is an example of an “even” function

$y=x^2$ is an example of an “even” function

The names “odd” and “even” come from the exponents in the examples above. For instance, $y=x^3$ is called an odd function because 3 is an odd number.

However, “odd” and “even” are actually not very good names for these functions. The thing is, there are several useful identities related to these functions, such as

whenever you add an odd function to another odd function you get an odd function

and

whenever you multiply an odd function by an even function you get an odd function

and almost none of these line up with the well-known identities for odd and even numbers. In mathematics, the whole point of using the same name for two different concepts is to tell the reader that much of the intuition from one concept can be carried over to the other. But this does not happen here. When you add an odd number to another odd number you get an even number, not an odd one. When you multiply an odd number by an even number you get an even number, not an odd one. Moreover, some functions are both odd and even (namely, the $y=0$ function), but no numbers are both odd and even. The analogy between odd/even functions and odd/even numbers simply does not work at all.

It would be better, I think, if odd functions were renamed as “negative” functions, and even functions were renamed as “positive” functions. These names aim to build an analogy to negative and positive numbers, and this time the analogy actually works. When you add a negative number to another negative number you get a negative number. When you multiply a negative number by a positive number you get a negative number. And the y=0 function corresponds naturally to the number 0 which is both positive and negative (or neither, if you prefer).

Of course, there are disadvantages to calling these functions “negative” and “positive”. For instance, calling a function “positive” might give the misleading impression that it lies entirely above the $y=0$ line. Still, no names are perfect, and I reckon that this disadvantage notwithstanding, these names are better than the current ones.

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