Can you prove the definition of absolute value from the properties (over the reals) $|x|\geq 0$, $|x||y| = |xy|$, $|x+y| \leq |x|+|y|$?

Question

I had this homework problem:

Use the following definition of absolute value to prove the given statements: If $x$ is a real number, then the absolute of $x$, $|x|$, is defined by: $$|x|= \left\{ \begin{array}{} x & x \geq 0 \\ -x & x < 0 \\ \end{array} \right.$$ a) For any real number $x$, $|x| \geq 0$. Moreover, $x = 0$ implies $x = 0$.
b) For any two real numbers $x$ and $y$, $|x||y| = |xy|$.
c) For any two real numbers $x$ and $y$, $|x+y| \leq |x|+|y|$.

I spent almost 2 hours on this problem unable to make any headway because I thought it wanted me to prove the first definition given the 3 statements a, b, c. Only to realize that the intended interpretation of the problem (prove a, b, c given the definition) is absolutely trivial...

It still seems possible to prove the definition from a, b, c, but I can't justify spending anymore time on this. Anyone willing to give it a shot?

Answer 1

No, there are other functions satisfying $(a)\land (b)\land (c)$. Namely, $\lvert x\rvert^{1/2}$.

Answer 2

It is not true that the "usual" absolute value is the only map that fulfill these properties. First of all it is important where this function is defined, i.e. it is an absolute value on $\mathbb{Q},\mathbb{R},...$.

In order to make my point clearer let's suppose we want a map on the rationals fulfilling properties (a)-(c) then one may define $$ \vert x\vert = p^{-v_p(x)}$$

where $p$ is a prime number and $v_p(x) = v_p(a)-v_p(b)$ if $x=a/b$ and $a,b\in\mathbb{Z}$ and for integer $a$ we define $v_p(a)$ as the multiplicity of $p$ in the unique prime factor decomposition. One verifies easily that the definition of $\vert\cdot\vert$ is well-defined in the sense that it does not change when "expanding fractions".

The map defined above also fulfills (a)-(c) and is the so called $p$-adic absolute value on $\mathbb{Q}$ and is not even equivalent to the usual absolute value in the sense that there are sequences that are cauchy with respect to one of them without being cauchy with respect to the other.

If one now completes the rationals in exactly the same way in which we obtained the reals but interchanging the absolute value with our new absolute value one gets the $p$-adic numbers (the wikipedia page is very nice). I hope this makes clear that there are other non-trivial "solutions" the the problem: find a map obeying the properties (a)-(c):)