So far, you've seen how the Shwa 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 sounds 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 Shwa 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 Shwa system enables us to write other number systems, too, like hexadecimal or Janus numbers. Finally, just like the letters, Shwa numbers are often better; for example, take a look at the Shwa equivalent of scientific notation.

The Shwa system uses only vowels to represent digits. Since no two vowels can occur next to each other in Shwa 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*.

Here are the Shwa digits from 0 to 9:

0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
---|

And some hints to help you learn them:

- and are the same as 0 and 1.
- is a
*tally mark*, a single vertical line. - is two tally marks, connected by a simple U-turn.
- is an open triangle.
- is an open square.
- is less obvious, but it's the bottom of our current 5.
- , we close the triangle to show that 3 is doubled to make 6.
- is the top of our current 7.
- , we close the square to show that 4 is doubled to make 8.
- is a reflected N for "nine".

These digits can be used exactly like our current Hindu-Arabic digits. Here are some examples:

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 Shwa, we sometimes write a level accent above (or below) the second digit, 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. display the repeater. The repeater isn't used with the digit , since Shwa doesn't have that letter.

The repeater could also be used on 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.

Shwa 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 easier - that first **I** is negative! Shwa has digits for the negative numbers -0 through -6:

-0 | -1 | -2 | -3 | -4 | -5 | -6 |
---|

The digits are easy to explain: they're just upside down. The was turned sideways to make , while were flattened to make . So the negative digits are easy variants of their positive twins.

Once you realize how these can be used, they're really quite useful and easy. For example, the number 28 can also be written . The number 91 can also be written .

You're probably wondering why we have a negative zero, since it's equal to positive zero. It actually has several uses, which I'll present below in context. On its own, this symbol is also used to indicate an impossible number, for example the result of division by zero. In that role, it's similar to the computer term *NaN* (not a number). We never use it to spell the number zero.

In order for you to read Shwa 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. And there are also official transcriptions using Greek letters that resemble the digits upside down, so that you can use negative digits with Hindu-Arabic numerals, too.

Shwa also uses the negative digits to represent binary numbers in **hexadecimal notation** (base 16). Computer programmers are used to seeing these digits as A-F. The digit for 10 is the same as -6, since 10 = 16 - 6, and so on.

To help you remember, the digit for **A** (10) looks like an A.

To recap, here are all the digits in the Shwa script :

But the 17 digits above are not enough to spell numbers - we also need a few **signs**.

The English decimal point, for which Continental languages use a comma, is represented in Shwa by a Break, a vertical line twice as tall as a 1:

By the way, **units always precede numbers in Shwa**. Units spelled with letters need a space before a following number.

You can also use a decimal point even when there's no fractional part. It indicates that the number is a integer - not a real number or a code of some kind. A trailing Break also makes it clear that a single digit is a number, and shows the orientation when ambiguous. Otherwise, you might confuse and , or and .

But often, knowing that the digits are low, their position is enough to indicate their orientation, for instance, on these dice. They're marked 0-5 so that two of them provide a distribution of 0-10, much more useful than 2-12.

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 Shwa, just write the cardinal number high, followed by a Break (which helps show the height).

However, in Shwa __zero is always considered the first number__, so a Shwa 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 Shwa, 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.

The Shwa negative sign looks like an upside-down T, like our minus sign hanging from a string. As in our current notation, it's used as a prefix to indicate a negative number.

But you can also use negative digits to write negative numbers. For example, to write -43, you could just write .

Unit fractions like *1/2 1/3 1/12* are written using a rightside-up T as a *division* or *reciprocal* sign, transcribed **÷**. It has several other uses that you'll meet below.

For most real numbers, we don't use the Break as a decimal point. Instead, Shwa uses a version 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*, 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 Shwa magnitude notation, we reverse the order, writing the exponent first and the significand 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 :

Note that we don't use a decimal point in the significand. 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

When the magnitude is negative - in other words, the number is smaller than 1 - we use a reciprocal sign in front of the magnitude, since a negative sign would indicate a negative number.

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" :

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**.

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.

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 exponent, since the square root is mathematically equivalent to the one-half power. A negative exponent indicates a reciprocal.

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% 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.

We also use a Break followed by high digits to specify which numeric base is being used, if it's not clear from the context. When used this way, there's always a space before the Break. The base is represented by its highest digit, which is always __one less__ than the base itself. If the numeric base is balanced (with both positive and negative digits), we write both the largest negative and positive digits, in that order.

Finally, Shwa has signs used for complex and imaginary numbers. In our current notation, we write those numbers using the letter *i*, which represents the square root of negative one. But in Shwa, we consider that *i* is a __sign__, not a factor, and we write it with a rising accent . To write *negative i*, we use a falling accent . These symbols precede imaginary numbers and link complex numbers, with the real part on the left. If the digits are high, so is the sign.

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 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 Shwa notation is like the second approach: we write the uncertainty appended to the number as a suffix using high digits:

To recap, here are the signs (with zero standing for all the digits):

All of these numbers and signs can be written using a normal Shwa keyboard, on which every letter requires two consecutive keys. But if you're only entering numbers, it's convenient to use a numeric keypad, where each number uses only one key. The usual keypad for decimal numbers includes a Break and a Shift key for entering high digits, but no other signs, no space, not even negative digits. it looks like this:

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", 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, Shwa has a separate system of ID codes, which consist of a sequence of vowels written high. To show that it's a code, and to show that the vowels are high, the first one is written with a "tail": the bottom is a instead of a space . Another particularity: we never use or in these codes; they're not vowels! Nor do we use the repeater. Here are some codes:

All of the 15 vowels are also digits, so why do I keep calling them *vowels* instead of *digits*? Because they sort in alphabetical order, not numerical order! You may never have thought about it before, but our English numbers also don't sort correctly in alphabetical order: **100** sorts before **99** (because 1 comes before 9). So the vowels in Shwa ID codes sort in canonical order, as on the keyboard:

If they're vowels, why do we use digits to transcribe them (and to read them aloud)? Because most languages don't use all 15 vowels, and in any case, the vowel sounds are too close to be clearly distinguished. So we treat them as if they were an arbitrary sequence of digits, __not__ a number.

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 except the last has a definite next code: just increment the final vowel. 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 . In effect, the space is the first vowel in the order, so there are a total of 16 possible code letters; it's hexadecimal (but with no internal spaces).

And because the first vowel is marked by its tail - which doesn't change its order - you can sequence codes with no space. For example, a post code might be , with a three-letter country code, a two-letter state code, and a four-letter code for the post office.

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