Ratio Algebra
Let us define an algebra of ratios. A ratio consists of two numeric expressions separated by a colon, and for clarity enclosed in parentheses, i.e., (a : b) with a, b ∈ ℝ. The expression on the left is the antecedent, and the expression on the right is the consequent. (0 : 0) is excluded.
Operations within each expression are ordinary algebraic operations. If either a or b is a vector, then there are the same properties with scalar multiplication. If both a and b are vectors, then there are the same properties with the dot or inner product.
An independent or free quantity is one that is determined independently of the experiment. A dependent quantity is one whose value depends on the value of at least one independent quantity.
If the antecedent is an independent quantity, the ratio (a : b) is represented as (a \ b) and is called a left ratio. If the consequent is an independent quantity, the ratio (a : b) is represented as (a / b) and is called a right ratio.
Ratios are asymmetric: in general, (a : b) ≠ (b : a).
Scalar multiplication is noncommutative.
Left scalar multiplication: k (a : b) := (ka : b).
Right scalar multiplication: (a : b) k := (a : kb).
Ratio multiplication is commutative and is defined as the compound ratio:
(a : b) (c : d) := (ac : bd).
Scalar addition is noncommutative.
Left scalar addition: k + (a : b) := ((a + kb) : b).
Right scalar addition: (a : b) + k := (a : (ka + b)).
There are two forms of ratio addition: right addition and left addition, which are represented by a forward and backward slash respectively:
Right addition: (a / b) + (c / d) := (ad + bc / bd)
Left addition: (a \ b) + (c \ d) := (ac \ ad + ba)
Right addition is also known as arithmetic addition; left addition is also known as harmonic addition.
Ratio operations are associative and distributive:
(a / b) ((c / d) + (e / f)) = (a / b) (c / d) + (a / b) (e / f)
(a \ b) ((c \ d) + (e \ f)) = (a \ b) (c \ d) + (a \ b) (e \ f)
The multiplicative ratio unit is (1 : 1). The multiplicative ratio inverse of (a : b) is (b : a). Thus ratio multiplication is a group over the reals.
The right ratio addition unit is (0 : 1). The left ratio addition unit is (1 : 0). The ratio addition inverse of (a : b) is (−a : b) for both left and right addition. Thus ratio addition is a group over the reals.
Theorem 1 : The definition of ratio multiplication leads immediately to:
(ac : bc) = (a : b), with c ≠ 0.
Corollaries: for any c ≠ 0, (1 : 1) = (c : c); (0 : 1) = (0 : c); and (1 : 0) = (c : 0).
It is a convention that the value of expressions are kept small (i.e., close to one or negative one if less than zero) by use of Theorem 1. The consequent is also kept non-negative.
A proportion is an equality of two ratios with a constant (or factor) of proportionality:
(a : b) :: (c : d) iff (a : b) = k (c : d) for some non-negative k ∈ ℝ.
The antecedent of the left ratio and the consequent of the right ratio are called the extremes. The consequent of the left ratio and the antecedent of the right ratio are called the means.
Theorem 2 : the product of the extremes equals the product of the means, or if (a : b) :: (c : d), then ad = bc.
Corollary: three quantities a, b, c with the same kind of units are in continuous proportion if the following holds:
(a : b) :: (b : c).
By theorem 2, b² = ac or b = √(ac) and b is called the geometric mean.
Theorem 3 Inversion (invertendo) :
if (a : b) :: (c : d), then (b : a) :: (d : c).
Theorem 4 Permutation (alternendo) :
if (a : b) :: (c : d), then (a : c) :: (b : d).
Theorem 5 Composition (componendo) :
if (a : b) :: (c : d), then ((a + b) : b) = ((c + d) : d).
Theorem 6 Divison (dividendo) :
if (a : b) :: (c : d), then ((a − b) : b) = ((c − d) : d).
Theorem 7 Mixing (componendo and dividendo) :
if (a : b) :: (c : d), then ((a + b) : (a − b)) = ((c + d) : (c − d)).
Theorem 8 Isomorphism
(a / b) and (b \ a) are isomorphic.
Demonstration: the mapping (a / b) → (b \ a) and its inverse mapping (b \ a) → (a / b) are bijective homomorphisms.
Ratio Calculus
The ratio derivative is (dx : dy).
The right ratio derivative (dx / dy), is the derivative of a function x(y) at a point y, which is defined as
The left ratio derivative (dx \ dy), is the derivative of a function y(x) at a point x, which is defined as
Each ratio derivative has its ratio antiderivative. The right ratio antiderivative of x′(y) is X(y) if X′(y) = x(y). The left ratio antiderivative of y′(x) is Y(x) if Y′(x) = y(x).
Each ratio antiderivative has its ratio integral. The right ratio integral of X(y) is ∫ x(y) dy = X(y) + C. The left ratio integral of Y(x) is ∫ y(x) dx = Y(x) + C.