the Frrraction Website for Exact Numeric Math

arith setup

Some paths into Frrraction's User Guide.

Updated with Frrraction Version 0.96.08

the FrrrGuide

frrrUserGuide Intro

Leopold Kronecker (1823–1891) contributed to number theory, algebra and logic. Suspicious of the continuum, he once said "Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk" — or, in my translation, "God made the integers, Satan made the residues, and all the rest is the work of man."

Speaking of, take a look at my thorough take on residues. Those "integers on steroids" are the newest part of this website. Most of my efforts for the last several years have focussed very specifically on modular division. Frrraction's Residue mode is very useful in that study, but my Frrraction program itself is not needed: With the information in residues, anybody using a computer language of their choice can readily write their own collection of software utilities and a way to make them readily accessible to support testing their own ideas about modular arithmetic.

As for the Frrraction program, I recently learned this new reason to continually postpone offering it for sale on the Apple App Store: XKCD has a great cartoon depicting two kinds of tools: (1) Those that solve problems, and (2) those that create problems. The good kind consists of "Tools that don't need a manual". The create-problems kind contains, in worsening order: ♦"Tools that have a manual", ♦"Tools that need a manual but don't have one", and ♦"Tools whose manual starts with 'How to read this manual'". With apologies, Frrraction falls into that worst category. Here goes:

For using the Frrraction app, we recommend a fast skim over this introductory page followed by a breeze through Frrraction's Grade School. If you've done that already then Frrraction 102 in High School is an intermediate place to resume.

Calculators are about numbers, and Frrraction is very good with numbers: Integers, mixed fractions, pure fractions, fixed point decimals, floating point decimals, and residues, all on a screen tailored to be adaptable to their various styles.

Moreover Frrraction is very programmable. Its programming language makes it into a numeric-Applications Programming Interface (n-API). The programming facility lets you create app files that utilize many more functions than any dedicated grid of keys could offer. Its functions include arrays for text strings and numbers; also control functions for iterative computation, if-then-else decisions, looping, subroutines, displaying results--along with all the double-precision math functions in the Unix library math.h commonly found on computers for mathematical and scientific work (roots, exponentials, logarithms, trigonometric, and Bessel functions).

Frrraction's main focus is numbers based directly upon integers: Mixed Fractions, Pure Fractions, and Modular Residues. The fraction facility has been stable for many years, does a great job with what you expect, and expands into some interesting corners such as optimal approximations of decimal numbers by fractions and "continued fractions". The residue facility provides the basic arithmetic operations + - and * along with a couple forms of division; n/d≡a:q provides all solutions q of the congruence d*q≡n (mod m); my favorite is n/d≡iu:q which delivers a well defined unique quotient — if n and d satisfy a certain necessary and sufficient condition for divisibility.

As mentioned above, modular division is what has occupied the bulk of my personal interest for the last several years, and Frrraction has been a superb tool as I try to expand the definition of division to allow more residues to be denominators. Callable residue instructions include several square roots, residues to integer powers, residue-factorials, imaginary residues, and more.

One of Frrraction's functions callable from an applet is an interesting exact integer-square-root function named cfSqrt--yes, you heard right: For any integer, not just 1, 4, 9, 25 etc., the algorithm can produce the exact square root; The Frrraction applet's implementation can show it to you for the first few billion integers. For example, the exact square root of 2 is the continued fraction [1;2,2,2,...], the ellipsis indicating infinitely many copies of "2,". If you truncate it after the eleventh 2 you get the approximate pure fraction 19601/13860 which to nine digits is the approximate decimal 1.41421356. Another example: cfSqrt(2147483)=[1465;2,3,28,5,1,10,16] is exact as far as Frrraction can take it. Using just the first five quotients the square is only off by 0.00019; incorporating the sixth and seventh quotients gets the square off by only 0.000000025. With all infinitely many quotients it would be exact: there would be infinitely many 0's or 9's after the decimal point..

As for Frrraction being very programmable: If you get intrigued by a numeric idea, you write applets in Frrraction's programming language to help you explore the idea. Some starters are pre-installed in the frrrFile collection, but you can modify those or create many of your own.

If this is your first encounter with Frrraction's Website, we recommend reading:
Your Very First Minute with Frrraction

Here is a complete guide to Frrraction. It is formatted to be viewed from within Frrraction's own WEB function on an iThing, but is readable from any browser at URL

By the way, Frrraction is no longer totally about exact arithmetic on fractions. It now also does a swell job with decimal numbers, presenting up to three 'x op y' pairs which make for far more convenient practical calculation. Here's how.

In any case, remember to check the complete guide to Frrraction. It is formatted to be viewed from within Frrraction's own WEB function on an iThing, but is readable from any browser. Apologies if, in my enthusiasm, I repeat myself.

Frrraction is a product of GRS Enterprises, NLC,
a Michigan company since 1978
Most recent update of this guide: June 27, 2022, 4:30 EDT.
Copyright © GRS Enterprises, NLC, 2010-2022
Not void, even where prohibited. Your mileage may vary.