lemÌc. Lemizh grammar and dictionary

Qonvert v1.9.0 (♯13) – Help

Overview

This app lets you convert numbers from the set ℚ of rational numbers between different bases (including nonstandard numeral systems) and between different representations such as positional notation and fractions (including improper, mixed, continued and Egyptian ones). The numbers can have basically arbitrary size and precision.

Long-tap a key on the keyboard to access the second character. You can vanish the keyboard by tapping ◀ on the navigation bar or by swiping down; tap the input line for it to re-emerge.

Short-tap a result to copy it to the input line, and long-tap to copy it to the clipboard.

Choosing input and output systems

Choose an input base via the slider or the shortcut buttons in the lower part of the screen, and an output base via the corresponding controls in the upper part. You can choose the displayed shortcut buttons, including ones to switch to negative bases, in the settings under “Show base buttons”. Long-tapping a button or moving a slider all the way to the right brings up a dialog for choosing a base between ±2 and ±280 or φ (the golden ratio).

The coloured buttons on the right let you switch between various kinds of numeral systems (standard, balanced, bijective, Greek, and Roman), on which more below. Not every option can be combined with every base; for invalid combinations, the buttons are greyed and the app uses the standard numeral system.

You can also prefix a number with one of five tokens (@ # $ % &) to set its base and numeral system on the fly. The meaning of these tokens can be changed in the settings; by default, they represent bases 2, 8, 10, 12 and 16 in the standard system, respectively.

Standard positional notation

If you choose to convert from standard DECimal (base 10) to standard HEXadecimal (base 16), an input of (say) 345 is interpreted as 3×102+4×101+5×100 and converted to 1×162+5×161+9×160, which is displayed as 159.

Digits from 0 to 9 work as expected. Those with values higher than 9 are represented by letters and other characters, from A = 10 to Z = 35, Þ = 36, Ч = 37, ʔ = 38, ʖ = 39, etc. (Swipe left to see the Cheat sheet for further digits and how to type them.)

The digits are case insensitive; and spaces are ignored.

Non-integers

Digits after a radix point . are of course interpreted as the fractional part of the number: 1.A6 in DOZenal (base 12) is 1×120+10×12−1+6×12−2 = 1.875 decimal. If there is a repeating part, mark its start with a dot above ˙ (or with a colon : if you are using the Android keyboard instead of Qonvert’s): For example, 0.1˙486 means 0.1486486486…, and 21˙3.4 means 213.43434….

The result of your conversion has always either a finite number of digits after the radix point or a repeating part. Sometimes, though, the representation can be very long. To keep the app from freezing, the search for an exact representation stops after 300 digits by default. You can change this value in the settings.

If your screen is too small to show all the output formats (positional, fraction, mixed, continued and Egyptian fractions), you can swipe the output area up and down.

Fractions

Type a numerator and a denominator separated with a slash /, as in 6/8 or 1.5/2. Fractions are reduced or expanded as necessary, so both of these inputs will result in 3/4.

For mixed fractions, separate integer and fractional part with an underscore _: for example 1_9/8, or something more complicated such as 1.˙3 _ 1/3. Fractions with an absolute value over 1 will be shown both as improper (17/8 or 5/3 in our examples) and as mixed fractions (2_1/8 or 1_2/3).

Base tokens apply to the individual parts of fractions, so you can type $10_@10/&13 or %251B/14 (where 14 is interpreted according to the currently set base and numeral system). The same is true of the other types of compound numbers described below.

Continued fractions

Continued fractions start with an integer part, followed by a semicolon ; and a comma-separated list of denominators. Square brackets are used for output but are optional for input.

2;3,4, for example, means 2 + 1/(3 + 1/4), which evaluates to 2.˙307692, 30/13, and 2_4/13.

Input of non-integer numbers, including fractions, is okay: 1.5; 2/3, 1_1/5 is simplified to [2; 6], which only contains integers. All continued fractions are displayed with positive denominators, so -3;-1,-1,-18 becomes [−4; 2, 18].

Egyptian fractions

Egyptian fractions are sums of unit fractions. Input works as with continued fractions, only with curly braces: {2;3,4 is interpreted as 2 + 1/3 + 1/4, evaluating to 2.58˙3, 31/12, 2_7/12, and [2; 1, 1, 2, 2].

For every rational number there are infinitely many Egyptian fraction representations. In the settings you can choose between various calculation methods for the main screen. The results of the other methods show up when you tap ⛶ in the title bar – they are identical for some numbers, while for others they differ dramatically.

Calculation of an Egyptian fraction may be cut off if it gets too long; the limit depends on the “Maximum digits in fractional part” in the settings.

Degrees, minutes and seconds

For example, the input 1° 23' 45" equals 1+23/60+45/3600 = 67/48 = 1.3958˙3 in standard decimal. The three parts of this format have to be in that order, but need not all be present; and they can be non-integers as well: thus, 1° 45.6" and 1/2' (half a minute) are valid inputs.

The DMS switch at the top right determines whether numbers are output in normal positional notation or as degrees, minutes and seconds.

Negative numbers

You can type negative numbers either in ordinary notation with a leading minus -, or in complement notation with two leading points ..: these points basically mean “the highest digit in this base repeated all the way to the left”. Complement notation works with positive bases in the standard system. It is known from computing, where −1 is represented as hexadecimal FFFFFFFF, −2 as FFFFFFFE, and so on (with as many F’s as there is space). So, typing ..FFD or simply ..D in hexadecimal, or ..997 or simply ..7 in decimal, all mean −3.

Output of negative numbers is either in ordinary or complement notation, depending on the complement switch at the top right.

Non-standard numeral systems

Balanced

Balanced numeral systems (the best-known being balanced ternary, i.e. base 3) have digits with negative values, notably Z = −1, Y = −2, down to J = −17 and below. (Swipe left to see the Cheat sheet for further digits as defined for balanced systems.) They work with odd bases up to 259. Thus, in balanced ternary, 10Z means 1×32+0×31−1×30 = 8. Negative numbers don’t need a minus sign in this notation: Z01 means −1×32+0×31+1×30 = −8

Bijective

In a bijective system of a base b, the digits go from 1 to b instead of 0 to b−1. There are two variants: “bijective 1” uses the digits 1 to 9 and from A upwards as in standard notation and works up to base 279; “bijective A” uses A = 1, B = 2, … Z = 26, Þ = 27, etc., and only works up to base 270. So, the first natural numbers in decimal bijective 1 are 1, 2, 3, 4, 5, 6, 7, 8, 9, A (=10), 11, 12, …, 19, 1A, 21, …, and the first natural numbers in hexavigesimal (base 26) bijective A are A, B, C, D, …, X, Y, Z, AA, AB, AC, …. Zero is technically an empty string; it is denoted with “/”.

Numbers with a radix point don’t work well with bijective systems, so non-integers are only displayed as fractions.

Greek numerals

Greek (Milesian) numerals are always base 10. The Greek letters Α Β Γ Δ Ε Ϛ Ζ Η Θ have values 1 to 9, Ι Κ Λ Μ Ν Ξ Ο Π Ϟ are 10 to 90, Ρ Σ Τ Υ Φ Χ Ψ Ω Ϡ are 100 to 900, and ͵Α ͵Β ͵Γ etc. are 1000 to 9000. They are normally ordered from largest to smallest: 1231 is ͵ΑΣΛΑ. In the output section they are followed by a keraia (ʹ) to better distinguish them from Roman numerals (see below), but the keraia is optional for input.

Myriads (10 000s) are marked with a dot, following Diophantus: Θ.ΦΛ means 9 myriads 530 = 90530. Repeated dots are used for still larger numbers (Π . . ʹ for 80 0000 0000), following nobody at all.

○ means zero. Non-integers are only displayed as fractions.

When Greek input is set, Arabic (i.e. standard decimal) input is also accepted so you need less switching between input systems. Furthermore, Greek numerals are always accepted regardless of input system as they – consisting of Greek letters – can’t be confused with anything else.

When an output system other than Greek is selected, positive integers are also displayed as Greek numerals in the second output line.

Roman numerals

Roman numerals are also base 10. They can contain the letters I V X L C D M. Symbols for larger values are ↁ = 5000, ↂ = 10 000, ↇ = 50 000, and ↈ = 100 000. (If you are using the Android keyboard instead of Qonvert’s, you can compose them as I)) ((I)) I))) (((I))).)

One traditional way for expressing still larger numbers is to enclose them in a frame to multiply with 100 000, or in nested frames for even higher powers. Qonvert uses pipes | for this purpose: |VI|((I))MM = VIↂMM = 6 12000. You can omit the leading pipes, and replace pipes with exclamation marks: ||V|M|II = V!M!II = 5 01000 00002. (The Romans certainly wouldn’t have recognised this notation, but there doesn’t seem to be any valid way of expressing huge numbers in Roman.)

The integer part may be followed by S (semis) for a half and one or more dots for a twelfth each. N for nulla or nihil means zero.

When Roman input is set, Arabic (standard decimal) input is also accepted; and when an output system other than Roman is selected, positive integers are also displayed as Roman numerals in the third output line.

Special bases

Phinary

Phinary, base φ = (1+√5̄)/2, is available in the standard and balanced numeral systems, the difference being that the digit Z = −1 can be used in the latter. On the sliders, phinary is located on the leftmost tick.

Although the golden ratio φ is irrational, integers expressed in base φ always have finite representations (albeit typically with digits after the radix point), and non-integer rational numbers have repeating representations. However, the reverse is not true: if you type an arbitrary sequence of digits, you will likely get an error message along the lines of “101ᵩ (5/2+√5/2) isn’t rational”.

The output will only contain the digits 0 and 1 (Z for negative numbers in balanced), and never two or more 1’s (or Z’s) in a row. The parts of fractions, including continued and Egyptian ones, aren’t necessarily integers, but they are always numbers without a radix point. (Let’s coin a cheesy term: they are always phintegers.) Egyptian fractions can be calculated with the greedy or the phinary remainder method. Phinary output currently does not support fractions in DMS format.

Negative bases

Negative bases work together with all standard and non-standard systems – including all non-integer notations – except for Greek and Roman numerals and for base φ. Since odd powers of negative numbers are negative, and even powers are positive, 345 in standard nega-decimal is interpreted as 3×102−4×101+5×100, which equals 265, and 3450 as −3×103+4×102−5×101, which equals −2650: thus, we don’t need a minus sign in negative bases.

Note that “bijective” systems with negative bases aren’t actually bijective: for example, in nega-decimal, you can always add or remove a leading 1A without changing a number’s value.

Complex bases

Although Qonvert can’t handle complex bases, there are some hacks:

Unicode characters

Inputs that are integers from hexadecimal 20 to 10FFFF (decimal 32 to 1114111) are not only converted to their representation in the output base and to Greek and Roman numerals, but also to the corresponding Unicode character. (If the output area is too cluttered for your taste, you can hide any or all of these in the settings.)

Conversely, type double quotes " followed by any character to find its Unicode encoding. This only works for single characters, not composed ones such as certain letter-diacritic combinations, people with skin tones, or flags.

The Web has excellent sources to find Unicode characters by name; but if you want to know the Unicode representation of the tortoise in balanced undecimal, Qonvert is your only source. (The answer is 1YX 215.)

And if anybody asks “why”, the answer is “because”.