You may have noticed that several Shwa letters resemble European ("Hindu-Arabic") numerals. That doesn't present a problem, because Shwa has its own numerals. Here they are :

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

All of these digits are written starting from the top, without lifting the pen. Unlike European numerals, none of them are the rotated or reflected versions of others, and none can be made by adding lines to another. The odd digits (except 1) have mid-height belts, and only digits 0 4 5 9 start from upper right.

Sometimes, for instance in LED displays, these digits are displayed in a blocky font based on a template :

These digits can be used exactly like European digits. Here are some examples:

Shwa also uses a **repeater**, which indicates that the preceding digit should be repeated a second time.
Just like James Bond's license to kill is numbered *double-oh seven* and not *oh-oh seven*, in Shwa we use the
repeater instead of repeating a digit. Unlike English *double-*, the repeater goes __after__ the digit it
repeats. When a digit repeats three or more times, only the second, fourth, etc. are replaced by the repeater. Using a
repeater prevents errors caused by "stuttering" with your finger while inputting or in transmission.

The repeater is also used as a prefix 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.

The repeater is romanized as an ampersand **&**, but when transcribing Shwa numbers into European numerals, just write the two
digits.

The Shwa script also has digits for the numbers 10 through 15, which are used to represent binary numbers in **
hexadecimal notation**. Computer programmers are used to seeing these digits as A-F.

The last eight hex digits all have a line at the top, while the first eight don't.

The same six hexadecimal digits are also used as **negative digits**. The best way to understand them is to think
about how much easier it is to say *5 of 12 (o'clock)* or *5 minutes to 12* instead of *11:55*. Now
imagine that you could say that a price of $7.99 was actually *1¢ of $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 -1 through -6:

These are the Shwa digits 1 through 6 reflected, and with a line added at the top. They're the same as the hex digits introduced above, since 10 = 16-6, 11 = 16-5, and so forth. There are no negative digits for 7 8 9, and in fact it's rare to see negative 4 5 6 in base 10.

We'll transcribe them in European using Greek letters that resemble the digits upside down. When reading them aloud, we use names based on Greek roots for the corresponding positive numbers. These digit names are intended to be used in other languages, too, albeit with variations.

Once you realize how these can be used, they're really quite useful and easy. For example, the number 28 can also be written 3ζ. The number 91 can also be written 1τ1. And the number -43 can also be written ηε - negative numbers just replace all the positive digits with the corresponding negative ones.

There's even a symbol for negative zero, whose use we'll discuss below. The name comes from a Greek pronoun meaning "nothing".

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).

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

Just like European digits have variants - - some digits can also be written with straight diagonals :

But the 18 digits above are not enough to spell every number - we also need nine more **signs**.

The English decimal point, for which Continental languages use a comma, is represented in Shwa by an empty oval, like the European zero :

You should also use the decimal point after every integer. That's because a series of digits doesn't spell a number until you say it does. The decimal point indicates that this is a cardinal number - not an ordinal, a number in magnitude notation, or a code of some kind - and it also indicates that this number is written in base 10 : that the first digit to its left is the Ones place, the next one is Tens, and so forth.

By the way, **units always precede numbers in Shwa.**

The Shwa negative sign looks like a big X. As in European, it's used as a prefix to indicate a negative number.

The negative sign is only used for negative numbers, not to negate arithmetic expressions - we'll show you how to negate an expression on the Arithmetic page.

For most real numbers, we don't use the decimal point. Instead, Shwa uses a special version of scientific notation.
The Shwa notation is reminiscent of the exponential notation used in many computer languages, where the letter *E*
represents the phrase *"times ten to the"*; in other words *1.23E4* represents *1.23x10 ^{4}*.

The Shwa notation is based on the **decade** sign, which is a belted oval, transcribed into European as the *
degree* sign **°**. The number to the __left__ of this sign is the magnitude (exponent) and the number to
the __right__ of this sign is the **mantissa**, which is always less than 10 and greater or equal to 1. For
example, **98.6** would be written **1°986**, since it equals 9.86 x 10¹ :

When the exponent is negative - the number is smaller than 1 - the negative sign is written __inside__ the
magnitude sign (replacing the belt with suspenders), since a negative sign written in front of the magnitude would
indicate a negative number. We call it the **decimal** sign, and transcribe it by the European *asterisk* *****.

When reading aloud, we use the ordinal form of the number to the left of the decade or decimal signs. The first example below would be read "third decade one two three" :

When a second decade sign appears, that indicates that all the following digits recur infinitely. Instead of writing
**16.666...**, we would write **1°1°6**.

When you want to indicate a percentage, just use the decade sign with nothing to its left. If the percentage is less than 10%, insert one or more 0s in front of it. By convention, 100% is written using the decade 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 as a percent: it could be written *an interest rate of 0.043 per annum*. In the latter case, better to use magnitude notation.)

Rational numbers - one integer divided by another - have a special notation based around a sign that looks like a
capital H. As in European, the number to the left is the numerator, while the number to the right is the denominator.
If the numerator is 1, it can be omitted, and we call it a *reciprocal*.

This sign is __not__ used for division of arithmetic expressions - we'll show you how to divide two expressions on
the Arithmetic page.

Finally, Shwa has signs used for complex and imaginary numbers. In European, 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 symbol that resembles a reflected capital N. We also write *negative i* with
the same symbol, reflected again (and in fact you can choose which one you consider positive). These symbols precede
imaginary numbers and link complex numbers, with the real part on the left.

A leading negative sign does __not__ change the sign of the imaginary part.

Except for the recurrer, the negative sign is the only sign that can be used with any other sign, and it's usually only used at the beginning of the number. But there is another use of the negative sign.

Magnitude notation, or any other notation for real numbers, 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 if the uncertainty is bigger than that, we can express it directly. For example, if we knew the value was between
**3.137** and **3.143**, we would write it in European as **3.140±.003**, or even more concisely as **
3.140(3)**: the digit(s) of the uncertainty are added and subtracted from the final digits of the mantissa to find the
limits of the range of uncertainty.

In Shwa, those last two notations are combined: the uncertainty is appended to the mantissa as a suffix using the
negative sign, which could also be called **plus-or-minus** when used this way.

This plus-or-minus sign can also be used with a default uncertainty of ±1.5 when the preceding digit is ±0, ±3 or ±6. This notation means "the last digit is only shown to the closest quarter". It's like saying "let's meet at a quarter to three", when you don't mean exactly 2:45 but sometime closer to 2:45 than to either 2:30 or 3:00. For example, here's how you'd write "one and a half" when you don't mean exactly 1.5:

I mentioned above that the decimal point is used for cardinal numbers. For *ordinal* numbers like like *
first* or *23rd*, we use an oval with a vertical line inside it. The ordinal symbol comes __after__ the number, replacing the decimal point.

However, in Shwa __zero is always considered the first number__, so a Shwa ordinal is usually one less than the corresponding European 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.

Finally, to show that a sequence of digits spells a hexadecimal number, we use a diamond-shaped **hexadecimal sign** instead of the decimal point.

To recap, here are (almost) all of the signs in the Shwa script :

There is a standard keyboard for entering these decimal numbers:

The decimal keyboad is accessed with this key:

There is a different keyboard for entering hexadecimal numbers:

The hexadecimal keyboard is accessed with this key:

The negative digits are on a different keypad, which I'll show you later.

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 *identification codes*, which follow a simple pattern : **
each consists of a single digit, followed by that number of letters**. For example, **4abcd** would be such a
code, while **3abcd** and **4abc** would __not__ be valid codes. Codes can also be joined in sequence, so that
Paris, Texas, USA might have a post code of **3usa2tx3par** (or just **2tx3par** for domestic mail).

This system has several advantages. One is that the codes sort alphabetically in the correct order (which numbers
don't usually do, e.g. *10* normally sorts before *9* alphabetically). Depending on the letters, it's often
possible either to establish a sequence, so that there's always a single correct next code, or to insert a new code
between any two.

Often, the letters spell the first few letters, or the unusual letters, of the item being identified. For example, Shwa has a system of 3-codes for the world's major cities, similar to the IATA airport codes. In this system, is London and is Paris.

In other cases, the letters are meaningless, assigned in alphabetic order or in some other manner (for instance, consonant + vowel in order to make them pronouncable).

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