- Frrraction 001 — Getting started
- YellowKey is a big deal!
- Summary of the yellow functions
- M|P|R — from M to P to R
- Proper, Improper, and Reduced fractions
- the HINT gesture
- M|P|R — from Mixed fractions to Pure
- M|P|R — Residues
- Delete and Clear
- Frrraction 102 — Fractions, actually

YellowKey itself isn't *the* big deal — what it is is the
entry point to all the power that raises Frrraction above being just several
numeric calculators; that's **the big deal**:

Key idea

**tap**to use the**white**keys — digits 0..9 and ops +-x/**y-tap**to use the**yellow**keys — extended functionality

InfoTopic:YellowKey—that yellow key over on the right-hand side in the ops-key strip — and within it, the YellowKey status indicator which is usually a 'W' for 'whitekeys'. Try it: Tap YellowKey and consider the labels on the yellow keys where the white keys had been. Here's the reason for YellowKey: The only way to tap M|P|R or XCH or APRX or any of the
other yellow functions is what we call a y-Tap: first tap YellowKey to reveal all the yellow keys
(and turn the YellowKey status indicator from 'W' into 'Y' (for
'yellowkeys') then tap a function key.So. From now on when this Guide says to use M|P|R or to tap M|P|R or any other yellow function key, it goes without saying that you must
first tap YellowKey. That "y-tap" is the only way to tap
any yellow function key. |

Yellow Label | Yellow Function | ||
---|---|---|---|

M|P|R | Each tap advances MAINview from Mixed Fractions to Pure Fractions to Residues then starts over at Mixed Fractions. | ||

XCH | Exchanges field F1 and field F2 if the active fCell
contains an integer or residue; or exchanges just the active fCell with its mate if they
are a decimal pair. | ||

APRX | Approximates the active fraction by a simpler fraction. If the active fCell has a decimal point, the approximating stack exactly matches all the given decimal digits, accurate within 1 in the last given digit (up to 7 digits). Otherwise the approximation is the best "kitchen fraction" with denominator 2,3,4,6,8,or 16. | ||

DUP | Duplicates the active number (copies it to its mate). If the active fCell is an integer then its whole field gets duplicated; if the active fCell and its mate are a decimal pair then just the single active Fcell gets duplicated. | ||

MAX | Loads the active fCell with the largest integer that 32-bit computers natively represent, alternating between pMAX (largest positive) and nMAX (largest negative). | ||

PROG | Switches from the MAIN to the programming view; brings up the system
keyboard so you can edit the frrrNOTE. The note can contain an applet that
interacts with all the MAIN subviews (hInfo, hShow, etc.) using CF
operations. The PROG view also gives access to the frrrFile system via a
filePicker that plants a file command into a button that performs the created
command when you tap it. The file commands are : ?|n| peek, <|n| get, >|n|
save, and -|n| zap (delete). The Get and Save operations handle the numeric
fCells and the main storage registers (the first hundred of the up-to-2000
hpNumber registers (hp s-registers)), but none of the hpString
registers. | ||

EXEC | Executes the active Composite Function operation closest to the top of frrrNote and may continue from there, depending on the CF {{bracket}} decorations. | ||

WEB | Opens a web view into Frrraction's website http://www.frrraction.com to keep you up-to-date with the App. | ||

B|O|Z | Loads simple numbers into the active fraction. With each touch it cycles from Blank=0+0/0 to One=0+1/1 to Zero=0+0/1 then back to Blank. | ||

GCD | Shows the greatest common divisor of the active numerator and denominator. Also shows the decimal forms of both fields if Frrraction is displaying Mixed fractions. Alternates greenly with the RDC function. | ||

RDC | Cancels all common factors between the active numerator and denominator to reduce the active stack to lowest form. Alternates greenly with the GCD function. | ||

D|F | Converts between fractions and decimals. If the active number field is a fraction (all integers) then it converts the field to a decimal in F.active.i with 0's in F.active.n and F.active/d. If the active fCell is decimal, it converts it into a fraction using the standard inefficient conversion to fill the whole active field. | ||

TIK | Stops|Restarts the key-click sound. (The sound itself is licensed from: http://www.freesound.org/forum/profile.php?mode=register) | ||

CHS | Changes the sign of the integer, decimal, or residue in the currently active fCell. | ||

INV | Calculates the inverse (the reciprocal, 1 over the numeric value) of the fraction in the currently active field, or of the decimal or residue in the currently active fCell. | ||

STO | Stores the active fraction into rational storage Register Rn (if active fCell is integer) or active decimal into decimal storage register Dn (if the active fCell is decimal) with n specified by the next digit-key tapped. | ||

RCL | Recalls rational storage Register Rn into the active field (if active fCell is integer) or decimal storage register Dn into only the active fCell (if the active fCell is decimal) with n specified by the next digit-key tapped. Empty registers contain 0+0/1. |

TryIt (with 3+58/82 in
F1 with F1d active):Y-tap M|P|R. (Remember, to do that you must first tap YellowKey — until then you can't access the M|P|R-key.) |

Explanation:Three M|P|R things happened: - The integer-cells F1i and F2i of the mixed fractions became darker green, suggestively hiding them as they are irrelevant on the Pure Fractions screen;
- The mixed fraction 3+58/82 was replaced by the equivalent pure fraction: 152/41.
- The status indicator W in YellowKey and the M|P|R key itelf turned green, indicating that the M|P|R key can be tapped again without first tapping YellowKey. We'll return to this convenience later in the guide.
F1 was numerically larger than 1,
so its integer part was nonzero and its fractional part 58/82 was a proper
fraction (had numerator smaller than its denominator). The pure form of
the same fraction of course has no integer part so it is improper (has
numerator larger than its denominator). The next section of this guide
addresses how Frrraction handles proper vs. improper issues in more
detail. |

Each of the three parts of a mixed fraction can be independently positive or negative, but there is always an implied addition+sign between the integer part and the stack. Frrraction's preferred sign combinations are: integer and numerator should have the same sign and the denominator should always be positive. It gets to that condition at its first opportunity—certainly after y-tapping M|P|R. The only exception is a pathological case called "the Nemesis". More about that fascinating case later.

TryIt (in MixedFraction mode with 152/41 in active
F1):Make F1d negative (Tap F1d, then y-tap CHS, changing F1 from
152/41 to 152/−41.This makes F1 be a negative number,
but not in Frrraction's preferred form.Give Frrraction a chance to express its preference: Y-tap M|P|R then tap M|P|R again to get back to Mixed Fractions, and notice what you got: −3 + −29/41; then tap M|P|R again (it's still green, so the y of y-tap isn't necessary) to get back to Pure Fractions. Now F1 is the preferred denominator-positive -152/41 instead of the
non-standard 152/-41.Change the sign of F1d again, changing F1 from −152/41 to
−152/−41.Now the numeric value of F1 is back to
the original but its signs are non-standard. If you triple-tap M|P|R again.
Voila! Frrraction got back to its preferred signs as soon as it
could.Starting with any irregular sign pattern, a cycle of M|P|R's always fixes it, as does adding or subtracting 0/1 or 0 + 0/1 (fraction forms of zero). Similarly, multiplying or dividing by any fraction form of one always does it. Using GCD and RDC, to reduce the stack to lowest form, also remedies the signs. |

Key ideas: **Kinds of fractions**

**Pure**is a/b = only a stack**Mixed**is a + b/c = integer + stack

**Proper**is a/b = smaller / larger**Improper**is a/b = larger / smaller or a/a = any / same**Reduce**replaces a⋅c / b⋅c by a / b

**M|P|R**tries for integer + proper stack**RDC**produces reduced stack

M|P|R BackgroundTopic:We saw M|P|R do sign-corrections in the previous section of the guide, but actually, in transforming from Pure through Residue to Mixed, M|P|R did two other things too: It made F1n smaller, to make F1's stack into a
proper fraction, meaning that its numerator became
smaller than its denominator, so the stack became truly a fractional
number—less than one. Then, to preserve the numeric value of the
fraction while reducing F1n, M|P|R made up for the reduction by
simultaneously increasing F1i — a nice balancing act. Thus, 304/82
became 3 + 58/82. The other thing M|P|R did is put the stack into reduced form, which means eliminating all factors shared by the numerator and denominator. That converted 3 + 58/82 into the final, preferred, result: 3 + 29/41. [Editor's note: Internally, Frrraction always begins all fraction-processing by reducing the involved stacks if possible, then doing whatever else the task requires. So in the above case it would have first reduced 304/82 to 152/41, then converted that to 3 + 29/41. By working with smaller integers when possible, it avoids some numeric overflows that could occur with larger numerators and denominators.] |

Key idea:

**y-tap y-tap key**
*describes* the yellow function of that key

The HINT gesture is: YellowKey YellowKey <*any key*>,

*i.e.* tap YellowKey twice then some other key.

TryIt:Tap the Yellow key two times (tapping speed is no issue, this is not intended to be an iPhone-style 'double-tap gesture''). The first tap changes the YellowKey status indicator from a white W into a yellow Y, and the second tap changes that into a yellow H. H stands for Hint. H, like Y, is about
the yellow functions. While the status is
H, tap any of the other keys and Frrraction temporarily covers the
Notes area with a hint about the yellow function of that key.To see this in action, tap the M|P|R key while the H is showing and notice the Hint. Notice that the YellowKey status indicator reverts to the W after the Hint is displayed. The Hint vanishes and the hidden Notes reappear as soon as you do anything else. |

You can abort the HINT operation by tapping YellowKey one more time, changing the status back to W.

By now you have noticed that every time you tap
YellowKey, Frrraction temporarily covers the Notes area with a brief
description of what to expect next. Here's welcome news about that:

The three consecutive taps Y Y Y which start-then-abort the HINT
operation also serve to stop those repeated YellowKey status reminders for
the rest of this Frrraction session. (The next time you run Frrraction from a
cold start, the reminders will be back, so remember Y Y Y.)

TryIt (start with mixed
fraction 3 + 58/82 in F1):Y-tap M|P|R ( Remember,
that means tap YellowKey then tap the M|P|R-key). This switches
the display to PureFraction mode. Notice that Frrraction correctly restored
F1 to a pure fraction again—albeit in the reduced form
F1 = 152/41 rather than the original 304/82 we started with
a few screens ago.Ytap CHS, notice F1. Ytap CHS again,
notice F1 again. All is in order.Return the display to MixedFraction mode by greenly tapping M|P|R twice more, with 3 + 29/41 (the reduced form of 3 + 58/82) in F1. Confirm that all is as it should be.Here's a simpler example and you should predict the results before having Frrraction do it: Put 3 + 10/8 into F1; convert it to a pure fraction—any
surprises from M|P|R? Then convert it back to a mixed fraction—any
surprises from M|P|R M|P|R? |

Like several of the other yellow functions,
M|P|R has 'cyclic action': Using it several times
in a row does a repetitive series of actions, like
Mixed-to-Pure-to-Residue-to-Mixed-to-Pure-to... *etc*. Other such
functions are MAX, GCD, and B|O|Z.

↑ to Frrr101 Menu

TryIt (start in Pure mode with
152/41 in F1):Y-tap M|P|R ( Remember, that means
tap YellowKey then tap the M|P|R-key). Starting in Pure mode,
this switches the display to Residue mode. Notice that Frrraction correctly
keeps F1i and F2i hidden, since modular fields also need only two cells: one
for residue and the other for modulus.Now yTap CHS and notice F1. Ytap CHS again, notice F1
again. Do DUP CHS +(add) and notice — the residue plus its negative
is indeed zero. Do +(add) again to confirm that 0 plus the original residue
is the original residue. If you're rusty on residue arithmetic, then the
sequence 152 CHS 12 CHS 29 CHS 12 might be disconcerting. It's correct, and
Frraction Guide's HighSchool course on residues will make it all
clear.If you're new to residues, introductions are readily available on the web and here's an intermediate treatment on Frrraction's site. If you're OK with residues, do this simple example and predict the results of each step before having Frrraction do it: Starting in mixed fraction mode, put the fraction 3 + 10/8 into F1; convert it to a pure fraction; yTap INV,
then MPR into residue mode. Field1 is now a residue, modulo...modulo
what? — any surprises from M|P|R? yTap DUP then INV; then tap x
(residueMultiply) — note that 13 surely was the reciprocal of 4 modulo 17.
ResidueMultiply again — note that 1 surely was a multiplicative identity element
modulo 17. Let's try that again, but with CHS instead of INV:Staying in residue mode, ytap CHS — reciprocal equals negative!? That's unusual even for residues, but not impossible; confirm it in this case by adding the 4 (mod 17) that's still in Field2 (tap + for residueAdd). Then add again and CHS once more, to restore Field1 to where we started when we first entered residue mode. Convert Frrraction back to mixed fraction; ytap INV one last time — any final surprises? Return to MixedFraction mode with 3+10/8 in F1 and try taking
the INVerse once — then INV again. Repeat INV,INV in Pure mode and
again in Residue mode. |

↑ to Frrr101 Menu

TryIt (start with 22/3 in
F1):Put the cursor back into F1n and tap the Backspace-Delete BDel key (it's the key below YellowKey). See what happened in F1n? Tap several digit keys, then tap the BDel key several times. That's what the BDel key does. Finally, notice the Clear or Clr Button in the F1 numerator&mdask;the odd little white x in a gray
circle. Tap several digit keys and then tap the cell's Clr Button.That's what the two delete keys do: One of them back-deletes a single character at a time, the other clears its whole cell to 0. |

There. You passed Frrraction 101. You now know the App's basic idioms.

↑ to Frrr101 Menu ↑ to Main Menua Michigan company since 1978

Most recent update of this guide: February 7, 2018, 1030 GMT

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