The distinction between the empty set 
and the number 
is similar to that between NULL and ZERO. For example, the set of real solutions (or informally "the solution") to 
is 
, but the solution to 
is 
.
In my mind there is no need for a concept like NULL in mathematics if you think of NULL as in NULL-pointers.
NULL in this sense is a technical necessity because you cannot un-define a variable: Once a variable has been assigned a value, a certain bit of memory is reserved for this variable and this memory is marked as re-usable only if the variable goes out of scope (simplified speaking).
You cannot say "The variable with this name doesn't exist anymore." without letting it go out of scope, because that would make language interpretation much more complicated without many benefits. Therefore, to indicate that the value of the variable has no meaning, one uses NULL.
What NULL stands for in the end depends upon the programming language: In some it is a special keyword, but in some it is also just a different name for the integer 
.
You can assign an arbitrary value to NULL in mathematics as mentioned in the other replies (
, 
, etc.) but as mathematics has nothing to do with memory allocation there is really no need for such a thing as NULL.
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To explain to a boss the difference between "zero" and "null":
"Zero" is a value. It is the unique, known quantity of zero, which is meaningful in arithmetic and other math.
"Null" is a non-value. It is a "placeholder" for a data value that is not known or not specified. It is only meaningful in this context; mathematical operations cannot be performed on null (the result of any such operation is undefined, and therefore also generally represented as null).
For example, as in the comments: "What is your yearly income?" is a question requiring a numeric answer. "0" is a perfectly valid answer for someone who does not work and has no investment income. If the user does not enter a value at all, they don't necessarily make no money; they just didn't want to tell your software how much (or little) they make. It's an unknown, not specified; therefore, to allow the software to continue, you specify the "null" placeholder for that data field within the software. That's technically valid from a data perspective; whether it's valid at the business level depends on whether an actual numeric value (even zero) is required in order to perform a mathematical operation (such as calculation of taxes, or comparison with thresholds determining benefits).
In computers, virtually any operation on a variable containing null will result either in null or in an error condition, because since one of the variable's values is not known, the result of the expression cannot be known. The equivalent of performing math on null would be if I asked you "What's five plus the number I'm thinking of right now?". It's impossible for you to give a definite answer because you don't know the number I'm thinking of. An operation on zero, except for dividing by it, is usually valid and will return another known, unique value.
Boss-speak is always tough...
Zero is a number so you can do things with it.
Null is a unicorn. It doesn't exist so you can't do anything at all with it.
I am reading maths for machine learning book (Chapter 1, page no 27-28) on particular and general solutions.
https://mml-book.github.io/book/mml-book.pdf
My question is
I understand intutively what is meant by particular solution and I understand mathematically the general solutions but I fail to understand it intutively. I also saw that solving for Ax =0 I get is null space of matrix A. What does it actually mean in application etc.