Numerals
So far, you've seen how the Musa vowels, consonants and punctuation can be used to write the spoken languages of the world. But we also use the same letters to write the formal language of numbers and numerical expressions. This seems odd to us, because we're used to using separate symbols for numbers and arithmetic. But it was actually very common in the past; the Greeks, Hebrews and Romans all used letters to represent numbers, as did many other cultures. For example, the Roman numeral XVI (our 16) is written with Roman letters!
But why don't we just keep using our current system (which is called the Hindu-Arabic numeral system)? Well, for one thing, several Hindu-Arabic numerals look just like Musa letters: 1 2 4 6 7 9 look like . For another, by using the same letters as we do to write words, we can write numbers with the same keyboards and fonts. And the Musa system enables us to write other number systems, too, like hexadecimal or Janus numbers. Finally, just like the letters, Musa numbers are often better; for example, take a look at the Musa equivalent of scientific notation.
The Musa system uses only vowels to represent digits. Since no two vowels can occur next to each other in Musa text, it's always clear that a sequence of vowels is a number, not a word. Numbers are normally written with low vowels, but high vowels are used to indicate ordinals, powers, and bases, just as we now use superscript numbers like x².
Digits
Here are the Musa digits from 0 to 9:
| | | | | | | | | | |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
As you can see, is the same as English 0, and is the same as English 1, but with clown shoes. The digits each have the same number of points as their value:
And these shape families are used for all the digits: multiples of four use square shapes, multiples of three use triangular shapes, and primes use curved shapes.
These digits can be used exactly like our current Hindu-Arabic digits. Here are some examples:
| | | |
|---|---|---|
| 78 | 365 | 1492 |
Repeater
When a digit repeats, we're not always sure it's correct - maybe you "stuttered" with your finger as you were typing, or there's an echo in transmission. That's why we say double-oh seven and not oh-oh seven, or read the digits in pairs, like route sixty-six. It's an even bigger problem when a number includes a long strings of the same digit, like 1230000000456.
In Musa, we usually replace the second consecutive instance of a digit with a repeater, to show that we really mean it. When reading it aloud, instead of the number we say "ditto" (from the Greek word for twin, διδυμος and the Italian word for said). When a digit repeats three or more times, only the second, fourth, etc. use the repeater.
| | | |
|---|---|---|
| 66 | 2008 | 2011 |
The repeater could also be used before the first digit of a number to indicate that some digits are missing, or that they are to be copied from some other source. For example, you might start a telephone number with a repeater to indicate that the number has the same country code or area code as your own number.
A sequence of two repeaters is how we write the ellipsis … in Musa. It's used with no space at the end of a fractional expansion to indicate that there are more digits than shown (but see the section on recurrers below for the case when an expansion repeats), and it's used with a preceding space in formulas to indicate that the pattern repeats or continues.
| | |
|---|---|
| 3.14159… | [1 + 2 + 3 + … |
Negative Digits
Musa also features negative digits, including a digit for negative zero! The best way to understand them is to think about how much easier it is to say 5 (minutes) to 12 (o'clock) instead of 11:55. Now imagine that you could say that a price of $7.99 was actually 1¢ to $8. In fact, the Romans used to write IX instead of VIIII because it's clearer - that first I is negative! Musa has a full set of negative digits:
| | | | | | | | | | |
|---|---|---|---|---|---|---|---|---|---|
| -0 | -1 | -2 | -3 | -4 | -5 | -6 | -7 | -8 | -9 |
| ⓪ | ① | ② | ③ | ④ | ⑤ | ⑥ | ⑦ | ⑧ | ⑨ |
The negative digits are just sideways versions of the corresponding positive digits, and the negative digits are just closed versions of the corresponding positive digits, so they're easy to recognize. When writing them in the Roman alphabet, we surround them with a circle.
We mentioned above that multiples of four use square shapes, multiples of three use triangular shapes, and primes use curved shapes, and the same principle is extended to negative digits:
| Multiples of 4: | |
|---|---|
| Multiples of 3: | |
| Primes: | |
| Units: | |
Once you realize how negative digits can be used, they're really quite useful and easy. For example, the number 28 can also be written 3② . The number 91 can also be written 1①1 . These are examples of balanced numbers that are potentially written with a mix of positive and negative digits. The Janus number system is an example of balanced numbers.
It's more common to see numbers in which only positive digits are used to spell positive numbers, and only negative digits are used to spell negative numbers - these are called signed numbers. They're used, for example, in columns of figures - bank statements, annual reports, etc. - to show which numbers are credits and which are debits (where now parentheses or red ink are used).
In order for you to read Musa numbers aloud, we invented names for the negative digits based on Greek roots for the corresponding positive numbers. These digit names are intended to be used in all languages, albeit with slight variations in pronunciation.
| | ⓪ | -0 | uden | | ① | -1 | mono |
|---|---|---|---|---|---|---|---|
| | ② | -2 | bis | | ③ | -3 | tris |
| | ④ | -4 | tetra | | ⑤ | -5 | penta |
| | ⑥ | -6 | hexa | | ⑦ | -7 | hepta |
| | ⑧ | -8 | octa | | ⑨ | -9 | ennea |
Using this system, we'd pronounce the number as "two bis ditto", and transcribe it as 2②② - it equals 178.
Negative Zero
You're probably wondering why we have a negative zero, since it's equal to positive zero. It actually has a surprising variety of uses. We usually use it in signed numbers but not in balanced numbers. On its own, this symbol is also used to indicate an impossible number, for example the result of division by zero, or 0⁰. In that role, it's similar to the computer term NaN (not a number). We never use it to spell the number zero.
One interesting use is in situations where you have to distinguish between approaching zero from below - negative zero - or approaching it from above. For example, the function f(x) = 1/x has the value +∞ at +0, and -∞ at -0. To see an interesting example of this in depth, take a look at this page on continued fractions.
Another potential use would be to define a new number, let's call it uden after the digit we use to spell it: . Uden is the quotient of 1÷0 (or equivalently, any other numerator); it's the result of division by zero. When used in any other operation, the result is uden:
-
1 + =
2 × =
² =
and so forth...
The complex numbers plus uden don't form an algebraically closed field, a commutative algebra over the reals, or a Euclidean vector space without special treatment. But that's true when division by zero is simply undefined, as well. Uden is only a cosmetic trick.
Other Bases
These negative digits are also used in various situations in which you need more numerals.
To represent binary numbers in hexadecimal notation (base 16), computer programmers are used to seeing the missing digits as A-F. In Musa, we spell F (15) as -1, since it's 16-1: 15 and -1 have the same representation as nybbles in two's-complement notation. Likewise, each of the digits from -2 to -6 is used for a hex digit.
| | | | | | |
|---|---|---|---|---|---|
| F | E | D | C | B | A |
| 15 | 14 | 13 | 12 | 11 | 10 |
To write dozenal or duodecimal numbers (base 12), we just use the last two above: for 10 and for 11.
For vigesimal (base 20), we use an (almost) balanced notation in which represents -10 (instead of -0). In this system, decimal 10 would be written as : a 1 in the twenties place, and a -10 in the units place - there is no digit for 10. Decimal 19 would be written as : a 1 in the twenties place, and a -1 in the units place.
Tetravigesimal (base 24) has a deep connection with the Heegner stances of integers, and is useful in primality proving, the factorization of large semiprimes, public key encryption, and other frontiers of number theory. To write it in Musa, we use the digits from -6 to +6 (the Janus digits) in a balanced notation with a falling accent above to subtract 12, and a rising accent above to add 12: represents the digit +7, represents the digit +11, represents the digit -7, and represents the digit -11. represents the digit +12, and represents the digit -12. If the digits are written high, the accent is below them.
For bases smaller than 10 - binary (base 2), senary (base 6), and octal (base 8), we just use the normal Musa digits. For balanced ternary, we use .
Underlining
Since many Musa digits are rotated versions of other digits, it's important to indicate the orientation in cases where it's not obvious from context, for example on dice. Otherwise, you might confuse with or . To distinguish them, we use underlining: we'd write (not to be confused with ).
For example, these two Musa dice in the foreground are marked 0 to 5 so that two of them provide a distribution of 0-10, much more useful than 2-12. The two in the background are marked 0 to -5 so that one of each pair gives you a distribution centered on 0. You may also see Musa dice marked 0-1-1-2-2-3, two of which provide a good approximation of the normal distribution from 0-6, where each unit represents a standard deviation.
But sometimes, you want the ambiguity. For example, the photo below shows six Musa dice, differing only in color. They're sitting in little trays that indicate the orientation, and that permits them to be turned to display any of the Musa digits from +9 to -9, except -8 (for which we reverse -4) and the zeroes (for which we remove the die). That makes them ideal in situations where you're using dice to keep track of numbers, for example a score, a stock, or a status.
Color Codes
Musa has a standard color code for all these digits, which I introduce here so I can use it as we continue. (The hex numbers at right are RGB triplets.)
Recap of Digits
To recap, here are all the digits in the Musa script :
Signs
But the 21 digits above are not enough to spell numbers - we also need a few signs.
Decimal Point
The English decimal point, for which Continental languages use a comma, is represented in Musa by a Break, a tall vertical line:
| | £ | € |
|---|---|---|
| 3.14 | £5.88 | 4,89€ |
By the way, units always precede numbers in Musa. Units spelled with letters need a space before a following number.
You should also use a Break even when there's no fractional part. It indicates that the number is a integer - not a real number, an ordinal or a code of some kind. A trailing Break also makes it clear that a single digit is a number, not a letter.
| |
|---|
| 98. |
Ordinals
The cardinal numbers above represent quantities, like three or three hundred sixty five. Ordinal numbers, in contrast, represent order, like third or three hundred sixty fifth. To write them in Musa, just write the cardinal number high, followed by a Break (which helps show the height).
However, in Musa zero is always considered the first number, so a Musa ordinal is usually one less than the corresponding English ordinal: 0th means first. For example, the ground floor of a building is the zeroth floor (as in many countries). Because 0 is the first ordinal number in Musa, we avoid the mismatch in the current system between ordinals and cardinals, where the 21st century starts when the dates finally start with 20.
Negative ordinals start counting from the last position, so negative zero refers to the last one in a series, and negative one to the one before it.
| | zeroth (first) |
|---|---|
| | oneth (second) |
| | twoth (third) |
| | next-to-last |
| | last |
Negative Sign
The Musa negative sign is a level accent - it looks just like the Roman minus sign −. As in our current notation, it's used as a prefix to indicate a negative number and for subtraction.
| | |
|---|---|
| -27 | -9.86 |
But you can also use negative digits to write negative numbers. For example, to write -27, you could just write .
Reciprocal Sign
A vertical accent - the long mark - is the Musa sign for reciprocals and division. It's like the slash we now use in fractions like ½. Unit fractions like 1/2 1/3 1/12 are written using just this accent as a prefix; you don't need the 1.
| |
|---|
| 1/5 |
Real Numbers
When we think of numbers, we usually think first of integers - natural numbers like 0, 1, 2, plus their negative complements. But in real life, we more often encounter real numbers: measurements like 2 kilos, 90kmh, 1½ hours, etc. Integers and real numbers are written with the same digits, but they're more different than they seem.
Integers are grounded on the right - the rightmost digit is always the units place. Real numbers are grounded on the left: the leftmost digit is the most significant, and there is a potentially infinite tail of digits trailing off to the right. Since we can't use the count of digits to establish the magnitude, we need another method.
Magnitudes
For most real numbers, we don't use the Break as a decimal point. Instead, Musa uses a variant of scientific notation called magnitude notation. It's reminiscent of the exponential notation used in many computer languages, where the letter e represents the phrase "times ten to the"; in other words 3.45e2 represents 3.45×10².
In scientific notation, the number with the decimal point to the left of the ×10 is called the significand or mantissa, and it's always less than 10 and greater than (or equal to) 1. The integer superscript to the right of the ×10 is called the exponent. In Musa magnitude notation, we reverse the order, writing the exponent first and the mantissa second. In between, we use a Break, but the exponent is written high, so there's no confusion with decimal numbers. For example, 398.6 would be written as below, since it equals 3.986 x 10² :
| |
|---|
| 2°3986 |
| 3.986×10² |
| 398.6 |
Note that we don't use a decimal point in the mantissa. That's because we don't think of this notation as multiplying a decimal number. Instead, we think of the exponent as indicating the magnitude of the number: the 2 magnitude means that this is a number in the hundreds. That's the same thing we do in speech when we read it aloud as three hundred ninety-eight point six: the word hundred indicates the magnitude.
When the magnitude is negative - in other words, the number is smaller than 1 - it's best to use a negative digit in the magnitude. However, if you don't want to use negative digits, you could use a high reciprocal sign in front of the magnitude, since a negative sign would indicate a negative number.
| | | |
|---|---|---|
| | | |
| /2°3986 | -2°3986 | -/2°3986 |
| .03986 | -398.6 | -.03986 |
When using this notation with Hindu-Arabic numerals, we write ° to transcribe the Break. When reading aloud, we use the ordinal form of the magnitude. The first example below would be read "third (magnitude) one two three" :
| | 3°123 | 1.23×10³ | 1230 |
|---|---|---|---|
| | 2°123 | 1.23×10² | 123 |
| | 1°123 | 1.23×10¹ | 12.3 |
| | 0°123 | 1.23×10⁰ | 1.23 |
| | 1°123 | 1.23×10⁻¹ | .123 |
| | 2°123 | 1.23×10⁻² | .0123 |
Recurrer
When a second magnitude sign appears, that indicates that all the following digits recur infinitely. Instead of writing 16.666..., we would write 1°1°6.
| | |
|---|---|
| 2°2°6 | ②°2°6 |
| 266.666… | .0266666… |
Powers
We also use the Break to raise a number to a power. In this case, the power is written to the right of the Break, and high.
| | | | |
|---|---|---|---|
| 9² | 9½ = √9 | 9⁻² | |
| 81 | 3 | 1/81 | |
If 3 squared is 9, then 3 is the square root of 9, the number which, when squared, reverts to the original. But we don't use a dedicated symbol for roots like English √ - instead, we just put a reciprocal sign in front of the power, since the square root is mathematically equivalent to the one-half power. A negative power indicates a reciprocal.
Percentages
When you want to indicate a percentage, just use the magnitude sign with nothing to its left, followed by low digits. If the percentage is less than 10%, insert one or more 0s in front of it. By convention, 100% is written using the magnitude sign followed by negative zero.
| | | | | |
|---|---|---|---|---|
| 43% | 4.3% | 40% | 0% | 100% |
43% of dentists is a percentage, but an interest rate of 4.3% per annum is just a rate expressed 100 times bigger than it is: it could be written an interest rate of 0.043 per annum. Likewise, a 210% improvement is better expressed as a 2.1x improvement. In the latter two cases, better to use magnitude notation. But when you're talking about various portions of a whole, use percentage notation.
Numeric Bases
We use high digits before a number without a Break to specify which numeric base is being used, if it's not clear from the context. Simple bases are represented by their highest digit, one less than the base itself. If the numeric base is balanced (with both positive and negative digits), we write the smallest digit (the largest negative digit). In the chart below, represents any digit.
| | base 2 (binary) |
|---|---|
| | balanced base 3 (ternary) |
| | base 6 (senary) |
| | base 8 (octal) |
| | base 10 (decimal) |
| | base 12 (dozenal) |
| | balanced base 12 (Janus) |
| | base 16 (hexadecimal) |
| | balanced* base 20 (vigesimal) |
| | balanced base 24 (tetravigesimal) |
The bases for ternary and hex are ambiguous … until you look at the number. If it has any digits other than , it's hex.
Of these, the three that are most likely to be confused - to appear in similar contexts - are decimal, hexadecimal, and Janus. Of course, if the number includes a negative digit, it's not decimal, while if it includes 7, 8, or 9, it's not Janus. But we can also use another trick to distinguish them: when it might be ambiguous, we write hex numbers underlined and Janus numbers in italic:
| | | |
|---|---|---|
| decimal 24 | hex 24 (decimal 36) |
Janus 24 (decimal 28) |
Uncertainty
Magnitude notation, or any other notation for real measurements, doesn't represent exact numbers; for example, the number 3.14 represents a value between 3.135 and 3.145 - all we know is that 3.14 is the closest we can get with only three significant figures. If we knew the actual value was between 3.1395 and 3.1405, we would write 3.140. In other words, the normal uncertainty of a real number is plus-or-minus half the place value of the last digit.
But sometimes we want to represent the uncertainty explicitly. For example, if we knew the value was between 3.137 and 3.143, we would write it in English as 3.140±.003, or even more concisely as 3.140(3): the digits of the uncertainty are added and subtracted from the final digits to find the limits of the range of uncertainty. The Musa notation is like the second approach: we write the uncertainty appended to the number as a suffix using high digits:
| |
|---|
| 3.14159±16 |
| 3.14143 > ? > 3.14175 |
Identification Codes
The numerals above - digits and signs - are used to spell numbers, with which we measure and count things. But in the modern world, we also use numerals in identification codes, for instance as telephone "numbers", license plate "numbers", reservation "numbers", credit card "numbers", address or apartment "numbers" , and so on. We call these all "numbers", but they aren't really numbers - they don't measure or count anything - and in many cases they also include letters.
For all these uses, Musa has a separate system of ID codes, which consist of an initial break followed by a sequence of high digits drawn from the full set of 20 digits from -9 to +9, including both -0 and +0. These codes also use the long mark to separate numbers partially, as when separating a country code or area code from a phone number. They always end with a dot (space). The repeater may be used as usual, but only as an alias for the digit it replaces - it shouldn't be sorted on its own. The initial break helps show the height, and works like a Roman # number sign.
This system has several desirable properties. Most important is that they sort in a definite order, read from the left. Another is that every code has a definite next code: just increment the final digit. If that's already a , make it a and increment the one before it, and so on. But the system also has the opposite property: between any two codes, you can always insert another! For example, between and , you can insert . And between and , you can insert : the negative digits sort before the positive digits, but they don't sort before the previous digit - they're not negative.
Often, these codes have a definite length, like a phone number. The first one is then …, and the last one is …. It's also common to use only the positive digits, especially when transcribing from the Roman decimal world, or just the Janus digits.
Codes look like numbers - they're sequences of digits - but they're not numbers. Negative digits aren't subtracted. The value of each digit doesn't depend on its position, like it does in a number (like 12, where the 1 is worth more than the 2). and are not equal, and neither can be ignored when leading or trailing. Codes are anchored at left like fractions, not at right like integers.
Here's what one might look like:
Recap of Signs
To recap, here are the signs (with zero standing for all the digits):
| | negative | | reciprocal |
|---|---|---|---|
| | decimal point | | nonsense :) |
| | integer | | ordinal |
| | percentage | | ID code |
| | magnitude | | power |
| | uncertainty | | base |
Numeric Mode
| < Keyboard | Arithmetic Notation > |
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