mathematics

Arithmetic of rates

Euclid wrote that “A ratio is a relation in respect of size between two magnitudes of the same kind.” Magnitudes of different kinds were not considered until Galileo. A ratio between magnitudes a and b is expressed as either a : b or b : a. A quotient is the result of dividing a dividend […]

Two kinds of vector rates

This post builds on the previous one here. Vector rates rates of change are of two kinds. An ordinary rate for the vector change of f relative to a unit of x is defined as: The reciprocal vector rate is the vector reciprocal of an ordinary rate with a vector change of g relative to

Two kinds of rates

Rates of change are of two kinds. An ordinary rate for the change of f relative to a unit of x is defined as: The reciprocal rate is the reciprocal of an ordinary rate with a change of g relative to a unit of x is defined as: An ordinary rate has its independent variable

From ratios to quotients

Ratios and proportions are symmetric. A:B ≡ B:A and A:B :: C:D iff C:D :: A:B. But when ratios are converted to quotients or fractions, they are no longer symmetric. There must be a convention as to which is the denominator and which is the numerator. In an ordinary fraction or quotient or rate the

Reciprocal derivative

The reciprocal difference quotient is or The reciprocal derivative of f(x), symbolized by a reversed prime, is the limit of the reciprocal difference quotient as x1 and x2 approach x: or as h approaches zero: The reciprocal derivative of a linear function, f(x) = ax + b, is The reciprocal derivative of a power function,

Harmonic vector realm

This post expands on Harmonic Arithmetic posted here. A vector space, or better a vector realm, to avoid connecting it with physical space, is a set V on which two operations + and · are defined, called vector addition and scalar multiplication. The operation + (vector addition) must satisfy the following conditions: Closure: If u

Vector inverse and mean

This post is based on research papers by Anderson and Trapp, Berlinet, and the post on Reciprocal arithmetic. The vector inverse x−1 is defined as with positive norm. For a non-zero scalar k, The reciprocal (or harmonic or parallel) sum is symbolized in various ways, but I prefer a “boxplus” to maintain its relation with addition. The

Galileo’s method

Extracts about Galileo from Scientific Method: An historical and philosophical introduction by Barry Gower (Routledge, 1997): Galileo took great pains to ensure that his readers would be persuaded that his conclusions were correct. p. 23 The science of motion was then understood to be a study of the causes of motion, and to be, like

Mathematics and beauty

Extracts from Scientific Method in Ptolemy’s Harmonics by Andrew Barker (Cambridge University Press 2004): Mathematics is not the study of all quantities and all quantitative relations indiscriminately. It is the science of beauty. Its task, at the theoretical level, is to interpret, in terms of ‘rationally’ or mathematically intelligible form, the features, movements or states

Vectors and functions for space and time

A pdf version of this post is here. The time velocity of an n-dimensional vector variable or vector-valued function Δx per unit of an independent scalar variable t equals Similarly, the space lenticity with Δt and Δx, respectively: The rate of a scalar variable Δx per unit of an independent n-dimensional vector variable or vector-valued