natural numbers

Natural numbers have two basic purposes

to answer "How many?" and "Which one?"

When answering "How many?"

we use a natural number as a "cardinal".

When answering "Which one?"

we want to use a natural number as an "ordinal" or "index"

exclusive

three people up to and excluding third person in the queue

inclusive

four people up to and including fourth person in the queue

unary notation

We can express any natural number using the successor operation and 0. For example, the number 5 would be written SSSSS0.

positional notation

For example, we can use base ten, also called decimal (or denary).

natural number type

there is not a genuine natural number type in java , although integer is good enough for many applications. It is not natural though since natural numbers are an infinite set and integers only go up to 2^32 -1

Addition

commutative so the order doesn't matter. associative so we don't need to write brackets. It has a neutral element 0, meaning that a + 0 \= a.

Multiplication

commutative so the order doesn't matter. associative so we don't need to write brackets. It has a neutral element 1, meaning that a × 1 \= a. It has an annihilating element 0, meaning that a × 0 \= 0.

Laws of commutative semiring

semi means no subtractions. we can derive mathematical facts from a small group of basic "axioms".