Complex Plane / Imaginary Axis Probe [S1244–S1248]

Theme: Foundations · 30 sentences.

CPX-001 · Complex Plane / Imaginary Axis Probe [S1244–S1248]

S1244 The imaginary unit is a quarter-turn relation

S1245 Axis I is an imaginary axis

S1246 Axis I is not a spatial dimension

S1247 The complex model has two axes

S1248 The R-axis is a quantity axis, and the I-axis is a walf-axis

S1249 Point Z is represented by the pair {2, 0}

S1250 Therefore Z lies on the R-axis

S1251 Point Y is represented by the pair {0, 2}

S1252 Therefore Y lies on the I-axis

S1253 A point with two non-zero coordinates lies on neither axis alone

S1254 Point Z is represented by the pair {2, 0}

S1255 Multiplying Z by walf yields {0, 2}

S1256 Therefore walf·Z lies on the I-axis, not the R-axis

S1257 Point O is represented by the pair {0, 0}

S1258 Multiplying the origin by walf leaves it at the origin

S1259 walf·Z is represented by the pair {0, 2}

S1260 Applying walf again yields {−2, 0}

S1261 The second-quarter-turn result lies on the R-axis, not the I-axis

S1262 Applying walf a third time yields {0, −2}

S1263 The third-quarter-turn result lies on the I-axis, not the R-axis

S1264 walf³·Z is represented by the pair {0, −2}

S1265 Applying walf a fourth time yields {2, 0}

S1266 The fourth-turn result lies on the R-axis, not the I-axis

S1267 Point Z is represented by the pair {2, 0}

S1268 Therefore four walf applications return to point Z

S1269 The imaginary unit is exactly a quarter-turn

S1270 Four applications of walf return Z to Z

S1271 Therefore four applications of walf function as one full-turn angle quantity

S1272 A full-turn angle quantity is represented by worn

S1273 worn is exactly two pi-quantities