Everything about Addition Of Natural Numbers totally explained
Addition of natural numbers is the most basic arithmetic operation. The operation
addition takes two
natural numbers, the augend and addend, and produces a single number, the sum. The set of natural numbers will be denoted by
N, and "0" will be used to denote the natural number which isn't the successor of any other natural number.
Notation and terms
The operation of
addition, commonly written as the
infix operator "+", is a
function +:
N ×
N →
N. For
natural numbers
a,
b, and
c, we write
»
Here,
a is the
augend,
b is the
addend, and
c is the
sum.
Definition
Assume that
N has been defined by the
Peano postulates. We let
S(
a) denote the
successor of a.
Addition is defined inductively by fixing the augend. In other words, we let
a be any arbitrary, but fixed natural number, and we then make the following definitions:
- a + 0 = a [A1]
- a + S(b) = S(a + b) [A2]
By the recursion theorem, this defines a unique function "
a +":
N →
N. In words, it says that adding zero to
a gives back
a, and that applying the successor function to the addend has the effect of applying the successor function to the sum.
Since
a was an arbitrary natural number, we can "put together" all these functions into a single binary operation
N ×
N →
N.
Properties
The following are three immediate and important properties of addition which can be deduced from the definition.
Associativity: for all natural numbers a, b, and c, we've » (proof)
Commutativity: for all natural numbers a and b, we've » (proof)
Identity element: for all natural numbers a, we've » (proof)
Together, these three properties show that the set of natural numbers N under addition is a commutative monoid.
Further Information
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