The natural numbers provide us with the
means to count indefinitely; they thus allow
us to formalise repetition which, in turn, enables us
to add and multiply them, with sensible
results. Turning repetition back on itself extends that to give a
representation of ratios that enable a form of
division. These and related operations are collectively referred to
as arithmetic
.
Much of what's true of the familiar arithmetic of numbers can also be applied in diverse other situations where we have some suitable notion of addition. This gives rise (as for the counting numbers) to repeated addition, which induces a scaling by the counting numbers; this, in turn, induces a specification of what division by whole numbers would mean, for the things we know how to add. When those things do indeed support such division, we thus obtain the ability to scale our addable things by ratios.
These scalings (by counting numbers and, where relevant, their ratios) are automorphisms of the addition that commute with all automorphisms of the addition. Our addition induces a (pointwise) addition on its automorphisms, whose composition can be construed as a multiplication, so that the automorphisms of an addition (or any chosen sub-collection of them, provided it is closed under addition and composition, e.g. the ones that commute with all automorphisms of the addition) can be understood as addable things that we, furthermore, know how to multiply.
When we have addable things that we know how to multiply
together, in a suitable sense, the interaction between addition
and multiplication gives us more of what we're used to from arithmetic;
and an ordering on values refines what that gives
us to make it look even more familiar. Such an ordering can enable us to define
a notion of nearness
for use in
describing convergence and continuity; when the
automorphisms of some addition, that commute with all others, include limits for
all suitably well-behaved sequences of
values, derivatives become a powerful analytic
tool.
Arithmetic is one of the three pillars of basic education,
along with reading and writing. Because these two at least sound as if
they start with the letter R, and arithmetic begins with the
sound of that letter's name (or, equivalently, because some folk drop the
initial vowel when pronouncing it, as 'rithmetic, which really does start with
an R), the three are collectively known (slighly jokingly) as the three
Rs
. One could make this name slightly closer to accurate by referring to
arithmetic as reckoning
, which has become a somewhat more generic term
(in ways compatible with it being viewed as a pillar of education) but has its
roots in germanic/nordic words: the Norwegian regning
is a
modern cognate of reckoning, meaning arithmetic (or the bill – in a
restaurant, when it's time to pay and go, one asks
for regningen
). To complete the fussy triad, one could
use record
in place of write (generalising in a manner well-suited to
modern multi-media authorship) and get an actual three Rs
: Reading,
Recording and Reckoning – all useful skills.
Any context may define addition and/or multiplications as
(usually flat) combiners
on values in some way suitable to its purposes. When both are defined,
multiplication shall typically distribute over addition, v.(a +b) = (v.a)
+(v.b), and multiplicative operators bind more tightly than
additive
ones, so this last may be written v.a +v.b without the parentheses. Addition is
almost always commutative and cancellable; multiplication may (when we leave
zero aside) be both but some contexts make do without either or both of these
properties.
In any context where addition is defined, its bulk action is called sum; applying it to the list ({v}: |n), whose n entries are all one value v, gives us v.n = sum({v}: |n), the result of scaling v by n. In any context where a multiplication is defined, its bulk action is called product and applying it to a list with one entry repeated gives rise to the power function and thus to polynomials.
Neither addition nor multiplication
necessarily has completions for all
pairs of values. The completions of addition
are known as differences and described as subtracting one value from another; a
−b is a value r for which r +b = a, when such a value exists. Since
multiplication need not be abelian, it may need to distinguish completions on
its two sides, so I define ratios in two forms: b under a
, b\a,
right-completes a←b, so b.(b\a) = a, while a over b
, a/b,
left-completes the same pair, so (a/b).b = a.
When we have well-behaved arithmetic along these
lines, it becomes interesting to consider other things that we can scale
by the numbers
of our arithmetic; when we do so, those numbers get to be
described as scalars
and they may be used to scale diverse things,
usually of kinds on which we also have some form of
addition. Linear algebra is the study of what happens
when we use well-behaved basic arithmetic to scale more interesting things that
we know how to add, with a particular focus on the mappings, between those
interesting things and from them to the relatively mundane scalars, that respect
addition and scaling.
Written by Eddy.