i’m currently organizing my research.
https://min.togetter.com/id/yuuki26

this is called a generating function n was discovered by de Moivre in the 18c, according to wiki.

seems using aₙ = aₙ₋₁ + aₙ₋₂, it follows
𝑓(x) = x/(1 - x - x²).

n some reasoning leads to
aₙ = (ϕⁿ - (-ϕ)⁻ⁿ)/√5.

• Recurrence relation
seems one way to find an exact formula for a sequence like
aₙ = 1, 1, 2, 3, 5, 8, …
is the “inverse of the Taylor expansion.”

its known that
eˣ = 1/0! x⁰ + 1/1! x¹ + 1/2! x² + …,
but the idea is to find 𝑓 such that
𝑓(x) = 1x¹ + 1x² + 2x³ + 3x⁴ + ….

0.999…=1 is obvious if the lhs is viewed as a limit.
there’s no doubt that 0.999… is getting closer to 1.

• Limit
lim[x→∞] f(x) = L
basically means that the point where the value gets closer (as x gets larger) is L.
nothing more, nothing less.
not smtg like “equals L after infinite steps.”
(idk if such an interpretation is possible in higher math.)

• Consider drawing a figure using only a function that draws a point (x,y) on a plane, say, drawPoint(x,y).

considering drawing a line leads to the equation of a line.

considering drawing a circle leads to trig functions.

however, it was discovered that this can be interpreted as the point (x,y) being moved to (e,f) by the matrix ((a,b),(c,d)). (by Cayley in the 19th century, according to wikipedia.)

so matrix multiplication represents the composition of maps f∘g, and so on.

My current understanding of mathematics.

• Matrices are linear maps.
matrices were historically just a notation for simultaneous equations with coefficients and variables separated.

e.g. writing

ax + by = e
cx + dy = f

as

(a b)(x) = (e)
(c d)(y) = (f).

deleted my ts complex number class.
(archive) https://web.archive.org/web/*/https://raw.githubusercontent.com/yuuki15/Complex.js/main/%2A

i forgot my ROG Phone 5s Pro (2021–2023)