The ancient Greeks defined ten means in terms of the following proportions (see here): Let a > m > b > 0. Then m represents (1) the arithmetic mean of a and b if (2) the geometric mean of a and b if (3) the harmonic mean of a and b if (4) the contraharmonic […]
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
The Euclidean geometry is a category with point positions as the objects and Euclidean transformations as the morphisms. In kinematics there are two Euclidean geometries: that of length and that of duration. They are in turn categories. Length space is a category with point locations as the objects and Euclidean transformations as the morphisms. Duration
The units of quantities are conveniently ignored in the definition of a derivative, but they should not be. A derivative should be defined as a function of two quantities, both with their own units: where r’ is a vector function of two quantites and r is a vector functon of one quantity. The second derivative is
The traditional ratio, x : y, represents both x / y and y / x. In order to represent a ratio as quotients, both forms are required. Let us define a ratio as an ordered pair of quotients: For vectors this means One can either exclude zero or include infinity as follows; A rate of
Abstract It is easily shown that there are two kinds of addition for rates: arithmetic addition and harmonic addition. The kind of addition required depends on whether the variable in common has the same units as the denominator or numerator. This is shown and illustrated with rates of speed and velocity. Several examples are given
This post relates to the previous post Adding and Averaging Rates. A rate is a fraction, though the denominator is often one (a unit rate). In general a rate could be symbolized as Δx/Δy. And so the general addition of rates follows the general addition of fractions: If, as is usual, the denominator is the
This post is the latest in a series on rates. A rate is a variable quantity measured with respect to a quantity determined independently. A rate is expressed as a ratio of the quantity measured and the independent quantity. A rate of change is a difference of quantities measured with respect to a difference of
This post is a slight modification of section 2.0 “The Parallel Sum of Vectors” from W. N. Anderson & G. E. Trapp (1987) “The harmonic and geometric mean of vectors”, Linear and Multilinear Algebra, 22:2, 199-210. We will consider vectors in a real N dimensional inner product space, although some of the results given herein
The difference quotient is the average rate of change of a function between two points: The instantaneous rate of change is the limit of the difference quotient as t1 and t0 approach each other, which is the derivative of f(t) at that point, denoted by f′(t). Derivatives are added by arithmetic addition, i.e., if f(t)
