I want this to be answered by the members here whose first language is English:
How do you call this:
Do you call this the "absolute value of x" , rather "modulus of x" or something else?
Could you also specify from which country you're from (US, UK, Australia) please ?
My random thoughts:
"Modulus" comes from the Latin for "measurement", and was/is used to describe the length of a vector (especially in complex numbers, which have a "dual nature" as both vectors in the plane (with the usual vector addition), and as field elements (using the usual complex multiplication, or considered as "dilation-rotation" maps of the plane, that is:
in the basis: ).
In the last two centuries, it has become clear that both "absolute value" and "vector modulus" are both examples of what is called a metric, which is a more general way of measuring "distance" (typically from some distinguished point, but not always). Slightly more specific is the concept of "norm": every norm induces a metric, but only some metrics induce a norm (specifically, the metrics which are "compatible" with the scalar multiplication of a vector space, and do not depend on "where" the "distance" is measured (technically, these properties are called "homogeneity" and "translation invariance")).
The term "absolute value" usually refers to elements of a (linearly) ordered field, such a beastie can be pictured geometrically as a line in space, and we can choose a "positive side" and a "negative side", so that the absolute value represents how far from 0 we are.
Although the distinction between and can be useful in distinguishing scalar quantities from vector quantities, often this is simply understood from CONTEXT, for example it is often common to represent the scalar multiple of a vector v by the scalar a simply as av (although some texts insist on using boldface type for vectors, and normal typeface for scalars...other texts differentiate between them by using different alphabets, such as Greek for the scalars and Roman for the vectors).
The short answer is: "it depends on what "x" is". If x is a real number (or a rational number, or an algebraic number, or an integer), it is customary to call |x| the absolute value. If x is a complex number, the term modulus is typically used (but "degenerates" to the absolute value for complex numbers that lie on "the real line" typically the "x-axis", that is: numbers of the form a + 0i). If x is a vector, often what is mean is the norm of x. If x is an element of a (pointed) metric space with base point *, then |x| usually means d(x,*).
Finally, (and this is a distinct usage), if S is a set, then |S| typically means the number of elements (or cardinality, if there are infinitely many) in the set. A variant of this exists in group theory, where |G| is called the ORDER of G (and equals the cardinality of the underlying set...this also holds for other algebraic structures, but finite rings, fields, and algebras, etc. are less commonly-encountered animals), and sometimes |x| is then used for the ORDER of the group element x, that is to say, the least positive integer k such that:
xk = e, the identity of the group G
(although some authors use o(x) for this, perhaps feeling that the vertical strokes are over-loaded).
We could probably give you a less confusing answer if we knew more about where you encounter this symbol.
(from Les Etats-Unis (da "US"), by the way)
I'm from the UK (England). If I knew that x was real, I would tend to use either ' the magnitude of ', or ' the absolute value of ' .
If I knew that x was complex then I would use either ' the modulus of ' or, ' the magnitude of '.