A Geometric Interpretation of ii Algebraically and numerically: but geometrically the two expressions can be interpreted as two points in two slightly different sets of coordinates lying in two different locations. In the graph below, the natural exponential function is in violet and we introduce and define the imaginary exponential function which is in …
A New Geometry of i Part Two: Complex Slope in 3Di Coordinates by Greg Ehmka Abstract This article is part two of a series. Part one may be reviewed here. 3Di coordinates allows for a clear intuitive geometric interpretation of complex slope which is the sum of real slope and imaginary slope. Real slope occurs …
A New Geometry of i part one: “Elementary Algebraic Functions in 3Di Coordinates” by Greg Ehmka Abstract Imaginary numbers are discussed in relation to human experience and a new coordinate system, “3Di” – an acronym for “Third Dimension Imaginary” is introduced and defined. A three dimensional function of the form: [y + iz = …
The Geometry of Euler’s Formula – part 6 by greg ehmka Euler’s Formula Upgraded for any Base In the eBook excerpt below we are applying a relatively simple intuitive observation to upgrade Euler’s wonderful formula for any base, without formal proof. It is easy enough to show that the upgraded formula is true using a …
The Geometry of Euler’s Formula – part 5 by greg ehmka Euler’s Imaginary Exponential in Multiple Uses In the previous blogs reference was made to two coordinate systems. Those being ‘3Di’ and ‘4Dii’. There are several more and the eBook is organized around these various categories or groups of functions. The underlying principle …
The Geometry of Euler’s Formula – part 3 by greg ehmka Euler’s Formula as a Helix If we take Euler’s formula: and, as before, graph the output on the y and iz axes, but this time, rather than the animation variable θ, have the input variable be the linear x axis, we then have: …
The Geometry of Euler’s Formula – part 2 by greg ehmka Euler’s Formula as an Orbit Operator As we saw in part 1, Euler’s formula: will rotate the point, (x, y) = (0, 1), through an imaginary circle of radius = 1. and will rotate all points (x, ex ): through imaginary circles if …
The Geometry of Euler’s Formula – part 1 by greg ehmka Happy News Mr. Pierce! Benjamin Pierce is often quoted, relative to Euler’s formula, as saying: “Gentlemen, that is surely true, it is absolutely paradoxical; we cannot understand it, and we don’t know what it means. But we have proved it, and therefore we …
The Geometry of Euler’s Formula – part 4 by greg ehmka Euler’s Formula as Points on an ‘Exponential/Natural Log Surface’ In section 12.4 of the eBook we define and explore a ‘rectangular complex region’ (u + iv) with intervals: which, if graphed in 3Di coordinates, appears as: In the previous blogs, the input domain …