Definition:Addition
in the process of being refactored, or:
has been identified as a candidate for refactoring. In particular:
You know the routine
Until this has been finished, please leave {{refactor}} in the code.
Contents |
Definition
Naturally Ordered Semigroup
Let $\left({S, \circ, \preceq}\right)$ be a naturally ordered semigroup.
Then the operation $\circ$ in $\left({S, \circ, \preceq}\right)$ is called addition.
Peano Structure
Let $\left({P, 0, s}\right)$ be a Peano structure.
Then we define the binary operation $+$ in $P$ as follows:
- $\forall x, y \in P: \begin{cases} 0 + y & = y \\ s \left({x}\right) + y & = s \left({x + y}\right) \end{cases}$
This operation is called addition, and it exists and is unique.
Addition on Standard Number Systems
Natural Numbers
The addition operation in the domain of natural numbers $\N$ is defined in Definition by Induction of Natural Number Addition and written $+$.
As the Natural Numbers are a Naturally Ordered Semigroup, it is appropriate to write this set as $\left({\N, +, \le}\right)$.
From this, it can be seen that $+$ corresponds with the operation $\circ$ in the structure $\left({S, \circ, \preceq}\right)$.
Integers
The addition operation in the domain of integers $\Z$ is written $+$.
As the set of integers is the Inverse Completion of Natural Numbers, it follows that elements of $\Z$ are the isomorphic images of the elements of equivalence classes of $\N \times \N$ where two tuples are equivalent if the Unique Minus between the two elements of each tuple is the same.
Thus addition can be formally defined on $\Z$ as the operation induced on those equivalence classes as specified in the definition of integers.
That is, the integers being defined as all the Unique Minus congruence classes, integer addition can be defined directly as the operation induced by natural number addition on these congruence classes:
- $\forall \left({a, b}\right), \left({c, d}\right) \in \N \times \N: \left[\!\left[{a, b}\right]\!\right]_\boxminus + \left[\!\left[{c, d}\right]\!\right]_\boxminus = \left[\!\left[{a + c, b + d}\right]\!\right]_\boxminus$
Modulo Addition
The addition operation on $\R_z$, the set of set of all residue classes modulo $z$, is defined by the rule:
- $\left[\!\left[{a}\right]\!\right]_z +_z \left[\!\left[{b}\right]\!\right]_z = \left[\!\left[{a + b}\right]\!\right]_z$
Although the operation of addition modulo $z$ is denoted by the symbol $+_z$, if there is no danger of confusion, the symbol $+$ is often used instead.
More usually, though, the notation $a + b \left({\bmod\, z}\right)$ is used instead of $\left[\!\left[{a}\right]\!\right]_z +_z \left[\!\left[{b}\right]\!\right]_z$.
It means the same thing and, although obscuring the true meaning behind modulo arithmetic, is more streamlined and less unwieldy.
See modulo addition.
Rational Numbers
The addition operation in the domain of rational numbers $\Q$ is written $+$.
Let $\displaystyle a = \frac p q, b = \frac r s$ where $p, q \in \Z, r, s \in \Z - \left\{{0}\right\}$.
Then $a + b$ is defined as:
- $\displaystyle \frac p q + \frac r s = \frac {p s + r q} {q s}$
This definition follows from the definition of and proof of existence of the quotient field of any integral domain, of which the set of integers is one.
Real Numbers
The addition operation in the domain of real numbers $\R$ is written $+$.
From the definition, the real numbers are the set of all equivalence classes $\left[\!\left[{\left \langle {x_n} \right \rangle}\right]\!\right]$ of Cauchy sequences of rational numbers.
Let $x = \left[\!\left[{\left \langle {x_n} \right \rangle}\right]\!\right], y = \left[\!\left[{\left \langle {y_n} \right \rangle}\right]\!\right]$, where $\left[\!\left[{\left \langle {x_n} \right \rangle}\right]\!\right]$ and $\left[\!\left[{\left \langle {y_n} \right \rangle}\right]\!\right]$ are such equivalence classes.
Then $x + y$ is defined as:
- $\left[\!\left[{\left \langle {x_n} \right \rangle}\right]\!\right] + \left[\!\left[{\left \langle {y_n} \right \rangle}\right]\!\right] = \left[\!\left[{\left \langle {x_n + y_n} \right \rangle}\right]\!\right]$
The operation of addition on the real numbers is well-defined.
Complex Numbers
The addition operation in the domain of complex numbers $\C$ is written $+$.
Let $z = a + i b, w = c + i d$ where $a, b, c, d \in \R, i^2 = -1$.
Then $z + w$ is defined as:
- $\left({a + i b}\right) + \left({c + i d}\right) = \left({a + c}\right) + i \left({b + d}\right)$
Commutativity of Addition
On all the above number sets, we have that addition is commutative:
- Natural Number Addition is Commutative
- Integer Addition is Commutative
- Modulo Addition is Commutative
- Rational Addition is Commutative
- Real Addition is Commutative
- Complex Addition is Commutative
Associativity of Addition
On all the above number sets, we have that addition is associative:
- Natural Number Addition is Associative
- Integer Addition is Associative
- Modulo Addition is Associative
- Rational Addition is Associative
- Real Addition is Associative
- Complex Addition is Associative
Also see
Sources
- Nathan Jacobson: Lectures in Abstract Algebra: I. Basic Concepts (1951): Introduction: $\S 4$
- Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 2$: Example $2.1$
