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Shwa Numerals | Contents |
You may have noticed that several Shwa letters resemble Western numerals. That doesn't present a problem, because Shwa has its own numerals. Here they are :
All of these digits are written starting from the top, without lifting the pen. Unlike English numerals, none of them are the rotated or reflected versions of others, and none can be made by adding lines to another. You might also notice that the odd digits (except 1) add a mid-height "belt" to the previous digit, and the digits divisible by three start at upper right, while the others start at upper left.
Sometimes, for instance in LED or LCD displays, these digits are displayed in a blocky font based on a template :
These digits can be used exactly like English 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 romanized as an ampersand &, but when transcribing Shwa into English, 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. Don't confuse -3 with 9! The digit -5 (B) is usually written with two nice curves:
We'll transcribe them in English by underlining the positive digit: for example, 3 means -3. When reading them aloud, we give them Greek names :
Once you realize how these 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. And the number -43 can also be written 43 - negative numbers just replace all the positive digits with the corresponding negative ones.
To recap, here are all the digits in the Shwa script :
Just like English digits have variants - 
- some digits can also be written with straight diagonals :
But the 17 digits above are not enough to spell every number - we also need eight 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 English zero :
In theory, you should also use the decimal point after every integer. That's because, in theory, 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 .
In practice, the decimal point is often omitted after an integer, especially when we're just talking about numbers and not counting actual things (like cows or days until vacation).
By the way, units always precede numbers in Shwa.
Negative Sign
The Shwa negative sign looks like a big X. As in English, 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 English 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 English 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 the repeater alone.
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 English, 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 English, 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 digits. 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 English 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.
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. 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). The ordinal symbol comes after the number, replacing the decimal point.
To recap, here are all the signs in the Shwa script :
On the Keyboard page, I showed you a double keypad with the Shwa letter shapes on both halves. But you don't need both sets of legends, since they're the same. Instead, the right side of the double keypad is marked with the numerals and signs. Of course, when entering letters, the keys on the right are used as if they had the same legends as the corresponding keys on the left. For example, the key at lower left is the vertical line. But when typing numbers, shift into decimal mode and use the righthand set of keys.
To shift into decimal mode, use the Shift key followed by the Decimal mode 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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