Numerals

Digits

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:

Repeater

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.

Hexadecimals

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.

Negative Digits

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.

Negative Zero

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).

Recap of Digits

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 :

Signs

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

Decimal Point

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.

Negative Sign

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.

Magnitudes

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" :

Recurrer

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.

Percentages

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.)

Fractions

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.

Complex and Imaginary Numbers

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.

Uncertainty

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:

Ordinals

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.

Hexadecimal Sign

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

Recap of Signs

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

Number Keypad

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.

Numeric 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", 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).


< Cursor Keys Arithmetic Notation >


© 2002-2017 Shwa shwa@shwa.org 14may17