Is absolute value an algebraic operation?

Question

What is an absolute value operation in terms of math classification?

We study absolute value in algebra class along with other algebraic operations, but if I understand correctly, it doesn't seem to be one of algebraic/arithmetic operations: Addition, Subtraction, Multiplication, Division, Exponentiation, nth Root, Logarithm (en.wikipedia.org/wiki/Addition has a block of "Arithmetic Operations" which includes Exponentiation, nth Root and Logarithm, but doesn't include Absolute Value). Why not? It just seems to me that Absolute Value treated a bit differently from Exponentiation of Logarithm. Is it because Absolute Value less fundamental than Exponentiation in some way, or this inclusion mostly arbitrary?

And what it is then? Is it just a "standard mathematical function"? What are the others standard mathematical functions then? How can we define them?

You have the elementary functions: https://en.wikipedia.org/wiki/Elementary_function but I'm not sure what else to tell you. — morrowmh, Jul 30 '22 at 05:09
@Michael It just seems to me that Absolute Value treated a bit differently from Exponentiation of Logarithm. For example https://en.wikipedia.org/wiki/Addition has a block of "Arithmetic Operations" which includes Exponentiation, nth Root and Logarithm, but doesn't include Absolute Value. Is it because Absolute Value less fundamental than Exponentiation in some way, or this inclusion mostly arbitrary? — simd, Jul 30 '22 at 06:32
For the title question, "Is absolute value an algebraic operation?", see https://math.stackexchange.com/questions/2813115/ — GEdgar, Jul 30 '22 at 06:56
@user3537411 "includes Exponentiation, nth Root" $;$ You can combine the two to get $;|x| = \sqrt{x^2},$. — dxiv, Jul 30 '22 at 07:17

score 1 · Answer 1 · answered Jul 30 '22 at 05:52

The absolute value (modulus) is a function that takes in any real number and makes it its non-negative value without regard to its sign. The definition of the absolute value of any real number $a$ is the number $|a| = a$ when $a \geq 0$ or $|a| = -a$ when $a < 0$.

As for it being a "standard mathematical function," I'm honestly not sure what you mean. If you're referring to the normal functions you learn in grade school, like roots, quadratics, polynomials, exponentials, logarithms, etc., then I suppose it can be a standard function since the graph $y = |x|$ being a "v" shape should be pretty easy to remember.

score 1 · Accepted Answer · answered Mar 22 '23 at 17:11

Is it because Absolute Value less fundamental than Exponentiation in some way, or this inclusion mostly arbitrary?

Yes, since in real analysis, absolute value can be viewed as a composition of Principal square root and Exponentiation: $$|x|=\sqrt{x^2}.$$

Also in real analysis, absolute value can be construed to involve Multiplication and the Sign function: $$|x|=x\cdot\operatorname{sgn}(x).$$

Both $|x|$ and $\operatorname{sgn}(x)$ are piecewise algebraic functions, so are elementary functions.

Is absolute value an algebraic operation?

2 Answers2