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I'm trying to get to an abstract definition of a negative number that would fit in with the basic concept of addition/subtraction. There are questions here about multiplying and dividing negative numbers that really point to this basic question. The common analogy is of monetary debt.
If I start with a negative number, it's already on the negative half of the number line and will be flipped to the positive half: $-2 \times -3 : -3 \rightarrow -6 \rightarrow 6$. That is, multiplication by a negative is the same as two steps: multiplying by the thing as if it had no negative, then applying the negative sign.
How is a number defined? How is a number represented with symbols? An integer can be defined as the difference between two natural numbers. Natural numbers can be defined as the number of elements in a set. There are more mathematically robust definitions, normally using set theory, but they're complicated.
I understand that the factorial gives the number of arrangements. For example: the factorial of zero i.e. an empty set ( doesn't occur) is 1. As the empty set can be arranged only in 1 way - i.e....
$\begingroup$ This is really by convention of definition not intuition. Pivoting towards or away from zero also has its intuition and places where it is useful. floor(x)={-floor(-x) if x<0, floor(x) otherwise. $\endgroup$ –
Generally, the definition of prime numbers is all those natural numbers greater than 1, having only two divisiors [sic], the number itself and 1. But, can the negative integers also be thought of in the same way? Almost, but not quite.
$\begingroup$ Don't know if this has anything to do with the OP's reason for their question, but I'm looking for another term for "sign" because I want to store a value in a program that represents the "sign" of a numerical object (the object is numeric in nature but is not a built-in numerical type of the language).
Therefore, by Occam's razor (i.e., the simplicity clause) it is not necessary for $0$ to have a negative element. However, by definition, the given set must have a negative element for all the positive elements. Therefore, it makes no sense to conceive it as a positive number. Hence, $0$ is neither positive nor negative.
It is a positive number, but extremely small. To be easier for writing this value, scientists created a kind of notation, called scientific notation by using exponent/power of $10$. Therefore, the above number can be written such as $2.677931514 x 10^{-22}$ ($10$ power to negative $22$) So, do not get confused with a negative number of it ...
'Positive' and 'Negative' are defined only on the real number line, which is part of the system of complex numbers. So it makes sense to say, for example $1 -100i$ is positive and $-1 + 100i$ is negative, based upon their real number values. Although arbitrary, there is also some sense of a positive and negative imaginary numbers.