Add thread about rounding

Co-authored-by: Claude Opus 4.8 <noreply@anthropic.com>

Author: cb341

pages/threads.md

@@ -9,6 +9,224 @@ math: true
 
 Conversations, thoughts, half-ideas, things I am starting to explore.
 
+## 02.06.2026
+
+exam prep:
+investment accounting ; cashflow, npv, discount factors, present values.
+
+not hard exactly. mostly bookkeeping with time attached.
+a table of cashflows over the years, discount factors, barwerte.
+
+then the table started lying.
+not maliciously. numerically. formula:
+
+$$
+\text{Barwert} = CF \cdot \text{Abzinsungsfaktor}
+$$
+
+simple enough.
+but the displayed values were rounded.
+and somewhere nearby there was a symbol that looked like equality with something on top of it.
+not quite $=$
+not quite $\approx$
+more like $\widehat{=}$
+
+which apparently means "corresponds to".
+that annoyed me into a tangent.
+
+because i realized i do not actually know what i want approximate equality to mean in general.
+
+approximate because why?
+
+because the exact value was rounded for display?
+because a sensor measured it imprecisely?
+because a statistical model estimated it?
+because a numerical method only converged far enough?
+because floating point rounded it?
+because the formula itself is only a simplified model of reality?
+
+all of those collapse into same lazy glyph: $\approx$.
+
+but they are not same failure mode.
+rounding is representational.
+the exact value may exist. i just chose not to print all of it.
+measurement is a knowledge graph.
+the value may exist. i just do not know it exactly.
+numerical error is procedural.
+the mathematical object may be exact. the method only approached it.
+model error is deeper.
+the equation only ever approximated reality.
+
+> "all models are wrong, but some are useful."
+
+i like that line.
+because it accepts lie and asks whether lie earns its keep.
+
+Newton's gravity is nice example.
+
+$$
+F = G \frac{m_1 m_2}{r^2}
+$$
+
+force between two masses.
+clean. compact. almost suspiciously pretty.
+
+but this is only one layer.
+Newton describes gravity as force.
+Einstein replaces that picture with spacetime curvature.
+
+schematically:
+
+$$
+G_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu}
+$$
+
+not exactly same kind of formula.
+not "same thing, more decimals".
+different category entirely.
+
+left side: geometry of spacetime.
+right side: matter and energy.
+
+Newton works extremely well in weak gravitational fields and low velocities.
+falling objects, planets, satellites, engineering calculations.
+
+but push far enough and it breaks.
+Mercury's perihelion.
+black holes.
+strong gravity.
+high precision.
+spacetime itself.
+
+general relativity does not round Newton more carefully.
+it changes what gravity is.
+
+Newton did not round gravity.
+Newton modeled it.
+
+that is different from $1/3 \approx 0.3333$.
+
+there exact value exists.
+display was just shortened.
+
+so i started wanting symbol for that.
+not generic approximate equality.
+display precision equality.
+
+something like $a \dot= b$.
+
+i have not seen it in any engineering textbooks so far.
+but i like it.
+dot feels like decimal point.
+equality sign with decimal damage.
+
+meaning:
+b is displayed finite precision representative of a.
+
+so $1/3 \dot= 0.3333$ would not mean "one third is vaguely close to 0.3333".
+it would mean:
+exact value is one third.
+printed value is rounded.
+
+a [math.stackexchange thread](https://math.stackexchange.com/questions/3986879/correct-usage-of-equiv-and-doteq) makes the point from the other side:
+
+> $\equiv$ and $\doteq$ are used in many different ways in many different areas of mathematics, so there's no single established definition on what they mean. do you know if the use of these symbols is "standard" in the field you are working in? ie, are you getting them from a particular textbook or paper?
+> -- Atticus Stonestrom
+
+
+
+the answerer's first instinct is to ask which textbook or field you took them from. which is precisely my situation. but "i haven't seen it" does not imply "it is free to be specified". my $\dot=$ is just $\doteq$, and that glyph is already overloaded ; in some communities $\doteq$ even means "defined as". so stamping another meaning onto it would not reduce ambiguity. it would add to it.
+
+but then again.
+maybe that distinction often does not matter.
+
+if number is good enough for decision, knowing whether inexactness came from rounding, measurement, or computation may add no useful information.
+for investment table, maybe 0.909 is just 0.909.
+good enough to compare alternatives.
+nobody needs philosophical breakdown of discount factor.
+
+that is probably why $\approx$ survives.
+it is vague, but efficient.
+it says "do not treat this as exact" and leaves reason to context.
+
+maybe people do care why something is approximate.
+maybe they just store reason somewhere else.
+method section, table caption, standard, discipline, implicit context.
+
+fair.
+
+but still.
+i want symbol to carry some of it.
+
+maybe symbols are just compression.
+tiny program.
+reader sees token.
+reader expands token using shared context.
+
+$=$ expands cleanly.
+$\approx$ expands broadly.
+$\dot=$ would expand narrowly, but only if i define it first.
+
+without shared context, it is just another weird glyph.
+
+this reminds me of programming.
+`0`, `null`, and `undefined` can all look like "nothing" from far away.
+but they are different nothings.
+`zero` is value.
+`null` is intentional absence.
+`undefined` is not provided.
+collapse them and you get bugs.
+
+same here.
+collapse rounding, measurement, numerical error, and model error into $\approx$, and you lose information.
+
+sometimes that loss is fine.
+sometimes it is whole point.
+
+$\approx$ says:
+close enough.
+
+$\dot=$ could say:
+equal, after display precision did damage.
+
+or shorter:
+$\approx$ hides error.
+$\dot=$ names compression.
+
+somewhere along way i also learned $ \Box $.
+
+little square at end of proof.
+not part of argument.
+more like punctuation.
+done.
+proof complete.
+
+i like that.
+it is honest about closure.
+proof had target, reached it, marks end.
+
+approximation notation feels like inverse.
+$\approx$ does not close anything.
+it opens metadata.
+approximate why?
+
+rounded?
+measured?
+estimated?
+modeled?
+computed?
+stored in floating point?
+truncated?
+
+maybe that is why $\Box$ feels satisfying.
+it marks certainty.
+and maybe that is why $\approx$ feels slippery.
+it marks uncertainty, but refuses to say what kind.
+
+$$
+\Box
+$$
+
 ## 19.05.2026
 
 started playing chess myself again after watching god-tier play of hikaru, levi, magnus ; thought i understood chess by watching. tried blitz and brutally realized i didn't.