JavaScript is off, so the live parts of this page are not running. Everything written below is complete on its own and says what each computation does and what it produced — the empty rectangles are where it would have run in front of you. Nothing here needs an account, and nothing is hidden behind the scripts; they compute, they do not reveal.

Moving visuals on this page — measured at 0.26% brightness change per frame and zero flashes per second; your reduce motion setting stops them. What moves, and how it was measured

Mathematical plates that draw themselves — prime counts, orbits and lattices rendered live rather than shipped as pictures. It runs continuously while on screen, and it pauses on its own whenever it scrolls out of view.

We measured it rather than guessing: the largest frame-to-frame change in average brightness is 0.26%, and it produces zero flashes per second against the threshold of three set by WCAG 2.3.1. That is well under the limit, and it is not a promise about any individual: if moving images are a problem for you, you should not have to assess ours.

Everything animated on this site honours your system’s reduce motion setting — turn it on in your operating system or browser and these pages stop moving, with nothing else lost. All the mathematics is in the text and the still figures.

The error from adding 0.1 twenty times in floating point, which should be zero and is not.

Exact arithmetic calculator Machines

The calculator that never rounds

Most programming languages use binary floating point, which means 0.1 + 0.2 silently gives 0.30000000000000004 instead of 0.3. This one never does that: every operation is exact fraction arithmetic (BigInt numerator/denominator), so the result is always exactly correct — shown as a fraction, an exact decimal (with repeating digits called out, not truncated), and a continued fraction expansion. Supports + - * / and parentheses. Type an expression below and press Compute exactly.

Math: a fraction is a pair of BigInts kept in lowest terms after every operation (GCD reduction, not left to accumulate), so nothing here ever loses precision the way Number arithmetic does — the continued-fraction expansion underneath is the same construction that gives 355/113 as a famously good rational approximation of π. Dev: a small hand-written parser and evaluator, no eval() anywhere — the same self-hosted-only, zero-dependency rule as the rest of this site.

decimal digits to show continued-fraction terms

exact-calc — output

raw output, click to copy

Next tool →Collatz trajectory explorer

Back to the tools · The Notebook, if you would rather read than compute.

Name	Type	Set by	Purpose	Duration
`ksec_theme`	localStorage	This site	Remembers your light/dark mode choice	Until you clear it
`ksec_cookie_ack_v1`	localStorage	This site	Remembers you've seen this notice	Until you clear it
`ksec_cookie_choice`	localStorage	This site	Remembers whether you accepted or rejected, so we do not ask again	Until you clear it
`ksec_build_notice_v1`	localStorage	This site	Remembers that you collapsed the build notice in the corner	Until you clear it
`ksec_notice_rev_v1`	localStorage	This site	The site revision you last loaded, so a page left open across a deploy can tell you it is stale	Until you clear it
`ksec_catalog_seen`	localStorage	This site	How many entries the shared catalogue had the last time you looked at it, so the nav can tell you when it has grown and stay quiet when it has not. One number. It says nothing about you — only about how much of a public list you have already seen	Until you clear it
`ksec_findings_seen`	localStorage	This site	The measurements of Life objects this browser has already sent to the shared catalogue, so it does not send the same one twice. A short list of numbers such as `p6:d0-0:n14` — period, displacement, population. It describes what the background drew, not you, and it never leaves your browser: what goes to the catalogue is the pattern itself, and the server measures it there	Until you clear it
`ksec_notebook`	localStorage	This site	The mascot’s research notebook on /mascot: the claims it tested in this browser, the seed, and the verdict, at most 40 entries. Never sent anywhere.	Until you clear it (there is a button on the page)
`ksec_last_visit`	localStorage	This site	When you were last here, so the mascot can tell you the site was rebuilt while you were away — and stay quiet when it was not. A timestamp, nothing else, and it never leaves your browser	Until you clear it
`ksec_tool_uses_…`	localStorage	This site	A per-tool daily-use counter for the free tools, one key per tool (a fair-use nudge, not an enforced limit, never sent anywhere) — set only if you actually use one of those tools	Resets daily; until you clear it
`__cf_bm` and similar	HTTP cookie	Cloudflare, Inc. (our security & CDN provider)	Distinguishes legitimate traffic from automated abuse — the same security gate described in Methodology	Session, typically under 30 minutes