Why is the symbol of the set of negative numbers a Z?

Answer

First, that set Z (double drawn with the slash) is not the set of negative numbers, but of the integers. So all integers are positive and negative.

The name Z comes from the first letter of the German word “Zahlen”, which simply means “numbers”.

In mathematics you have consecutive number sets that always include the previous one:

N : the natural numbers : 0, 1, 2, 3, 4 ….
Z : the integers : …. -3 -2 -1 0 1 2 3 …
so you see that every element of N is also an element of Z, but the reverse is not always the case : -4 is element of Z but not of N
Q : the rational numbers : these are all fractions of the form z1/z2, and this set is again greater than Z.
R : the real numbers. There are also numbers that cannot be written as a fraction, for example the number pi. But there are infinitely many more. So they are not in Q, but in R

You may hear of a larger set of C, which in turn includes R.

So : N ⊂ Z ⊂ Q⊂ R ⊂ C with “⊂” meaning : is a part of.

Each of those letters is written with a double bar somewhere, but I can’t type those special letters here.