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 …
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 …
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 …
Ratios and proportions are symmetric. A:B = B:A and A:B :: C:D = 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 …
Legend: x-axis coordinate, x; length interval, Δx; unit of length, x̂; t-axis coordinate, t; time interval, Δt; unit of time, t̂. For scalars: Rates Speed/Pace Symbols Space Time Ordinary Time speed Δx(t)/t̂ measure length/distance/ displacement given period/unit Alternate Space speed x̂/Δt(x) given length/distance/unit measure interval/period /dischronment Inverse Space pace Δt(x)/x̂ given length/distance/unit measure interval/period /dischronment …
This post reflects a previous one here. Color (or colour) is both subjective and objective. Objectively, the rays of color light from a glass prism are different wavelengths (or frequencies) of light. The colors we see are those that reflect from objects; the others are absorbed. Colors are additive. Primary colors are red, green, blue; …
This post relates to the previous one here. Speed is the time rate of distance traversed. Pace is the space rate of elapsed time. The time speed of a body is the distance traversed per unit of independent time without regard to direction, Δx/t̂. The instantaneous speed is the time speed at a point in …
The reciprocal difference quotient is The reciprocal derivative of f(x), symbolized by a reversed prime, is the limit of the reciprocal difference quotient as h approaches zero: An alternate limit form of the reciprocal derivative definition of f(a) is The reciprocal derivative of a linear function, f(x) = ax + b, is The reciprocal derivative …
A rate is the quotient of two quantities with different, but related, units. A unit rate is a rate with a unit quantity, usually the denominator. A vector rate is a rate with a vector quantity, usually the numerator. Rates with the same units may be added, subtracted, and averaged. Addition Rates with the same …
This post expands on Harmonic Algebra 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 …
