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Reverse Numbers | Contents |
On a previous page, you learned how to write numbers using the Shwa digits. But aside from some minor differences, these symbols just replace the current decimal symbols like 4 and +, with no change in the "spelling" of numbers or the "grammar" of expressions.
But just as Shwa letters can be used to write different languages, Shwa numerals can also be used to write different "numeric languages". We are so accustomed to our own way of writing numbers, and it is so widespread, that we forget that it isn't the only one. For example, "Roman numerals" like XIV for 14 are a different way of writing numbers.
This page will introduce another "numeric language", called Reverse Notation. This name was first used by J. Halcro Johnston in his 1937 book, The Reverse Notation, on which this page is based. Numbers written in Reverse Notation are called Reverse numbers.
Before we start, you may wonder where we're heading with this. After all, we have already described how to write our current numbers and formulas in Shwa. We don't need any more than that to begin using the Shwa script for decimal and all the other languages of the world.
But Reverse Numbers and the associated units of measurement (presented on the pages that follow) are interesting advances in their own right which the new script makes feasible - for instance, with the needed digits. For some of you, they may seem one step too far; for others, an additional reason to change scripts. Those in the first group need not continue - you've already seen everything you need to start using Shwa.
You all know that our decimal numbers are base ten, presumably because we have ten fingers. What that means is that when we write a number by putting two digits next to each other, the lefthand digit doesn't represent units, it represents tens of units. For example, in the number 24, the digit 2 doesn't represent the number 2; it represents the number 20 because it's in the tens place. Likewise, in the number 365, the digit 3 doesn't represent the number 3; it represents the number 300 because it's in the hundreds place - each place to the left of the rightmost digit represents one higher power of ten.
The first two of the Shwa hexadecimal digits, 
and 
, can be used to
write dozenal or duodecimal numbers: base twelve. In base twelve, each digit to the left of the
units place represents one higher power of twelve, so the first place to the left represents 12s (the dozens
place), the next place represents 144s (the grosses place), and so on: 1728, 20736, etc. To consider the
same examples, the dozenal number 24 represents the decimal number 28: 2 × 12 + 4. The
dozenal number 365 represents the decimal number 509: 3 × 144 + 6 × 12 + 5.
This is not the place to discuss the advantages of dozenal numbers; if you're interested, you can check out the following websites:
A system in which there are equal numbers of positive and negative digits is called balanced. Reverse Notation combines dozenal numbers and negative digits to form a balanced number system. In Reverse Notation, only the digits from -6 to 6 are used, and each place left of the units place represents the next higher power of twelve.
For example, let's consider the number 9. In decimal notation, it's written with a single digit. But in Reverse Notation, it's written as 13, which means 12 - 3, with a 1 in the twelves place and a minus 3 in the ones place. The principle is the same as with Roman numerals, in which 9 is written IX - writing the I before the X indicates that it is negative. But Roman numerals require different symbols for each place, while Reverse numbers, like decimal numbers, use the same numerals for all magnitudes, relying on position to indicate the magnitude.
Since reverse numbers and decimal numbers use the same digits, we need some way to know which is which. In Shwa, when we write reverse numbers, we replace the oval decimal point, decade, decimal and ordinal signs with rectangular ones, which are called "radix", "degree", "dozenal" and "order" :
Like the decimal point, the radix is often omitted after integers if the context is clear. On this page, in decimal, we'll write reverse numbers in green and underline negative digits.
Let's count from 0 to 100 to show you how Reverse numbers work:
The first surprise is at number 6, which is written as 12-6, or 16. Why don't we just write it using the digit 6? We'll explain below.
7 is likewise written as 12-5, or 15, since 7 is closer to 12 than it is to 0. Rather than make a first approximation of the value of 7 by putting a 0 in the twelves place and then adjusting by 7 units, we put a 1 in the twelves place and then adjust downwards by only 5 units.
Number 13 is the smallest number that needs the repeater: without it, 13 would be written 11.
At 18, numbers start being as close to two dozen as to one dozen, so there's a 2 in the twelves place, and likewise for all the numbers up to 29: they're closer to 24 than to any other multiple of twelve, so they have a 2 in the twelves place.
The next surprise is at numbers 72 and 73. You might think we'd write 73 as 61 : after all, it's 6 dozen + 1 unit. But 73 is actually closer to 144 than it is to 0, so we write it as 1 gross - 6 dozen + 1 unit: 161. But then why don't we write 72 as 60 instead of 160? The rules that govern when you use 6 and when you use 6 are as follows:
The end result of these rules is to make sure that every number with 6 is closer to the number below than to the number above, and every number with 6 is closer to the number above than to the number below. The rules above are also designed so that the digits 6 and 6 can never appear in the same place.
Because of that, they can be regarded as two different forms of the same digit, a digit meaning "halfway in between". And just like Brits, Germans and Russians mean "2:30" when they say "half three", the six usually rounds up : the negative six is more common than the positive six.
The complement of a number is simply that number with the sign reversed, that is, multiplied by -1. So the complement of 123 is -123, which we would write in reverse notation as 123 . Likewise, the complement of 456 is 456. In other words, to find the complement of a number in reverse notation, just replace all the positive digits with their negative counterparts and vice versa.
In reverse notation, the easiest way to subtract two numbers is to complement the subtrahend (the number being subtracted) and then to add that to the minuend (the other number). For example, 7-3 is the same as 7+3. This makes subtraction into addition, which is much easier.
The minus sign is much rarer in reverse notation than in decimal. We don't need it to express negative numbers, since any number that starts with a negative digit is negative, no matter what the other digits are. Of course, we can still use it as an operator, for instance to take the complement of a variable, as in -x. We don't need the dozenal sign to write small numbers, either (although we have other uses for it, explained below).
Shwa has a standard color code for all the digits, which I introduce here so I can use it as we continue :
Here is the addition table for Reverse numbers:
Just like the decimal addition table, the results form diagonal stripes across the table. But this similarity masks a significant advantage of reverse notation: because of the mix of positive and negative digits, there's much less "carrying" between columns when adding lots of numbers. Here is a sample sum in both decimal and reverse notation:
| 11 | 11 |
| 13 | 11 |
| 17 | 15 |
| 19 | 25 |
| ___ | ___ |
| 60 | 50 |
In this example of the four teen primes, you have to carry a 2 in the decimal sum, but nothing at all in the reverse sum: the negative digits cancel out the positive ones. The more numbers involved, the easier the Reverse sum is in comparison to the decimal sum.
Here is the Multiplication table for Reverse numbers:
The multiplication table looks daunting, but then patterns begin to emerge. Some are the consequence of choosing twelve as a base: since it divides all the digits except 5, most of the multiples just repeat the same final digits in the same sequence. In addition, you don't need to memorize very many numbers: only the 15 products in the red outline at lower right. All the others are either multiples of 0 or 1 or the complements of other products. For example, if you learn that 4×5 = 24, then you also know that 5×4 = 24, that 4×5 = 24, that 4×5 = 24, and that 4 ×5 = 24. Compare that to the 36 products you had to learn for the decimal multiplication table!
There is one more trick when representing numbers in reverse notation: it turns out that many fractions, when expanded in reverse notation, form a pattern where the same sequence repeats, but alternating between positive and negative values. In this case, we use the dozenal sign as the recurrer: it means "repeat what follows, but change the sign each time".
To illustrate this, let's consider the decimal expansion for the fraction 1/7, a very interesting case. In decimal, it's written 0.142857... and we would write it in Shwa numerals using the dozenal and recurrer signs: 1*°142857. But when we convert it to balanced decimal notation, it becomes 0.143143... - the same three digits repeated in sequence, but with the sign changing every time. So we replace the recurrer with the decimal sign to indicate this, and just write 1°*143. It works the same way in dozenal, too:
And note that 0.143, rounded off to three decimal places, is a much better approximation to 1/7 than 0.142 , six times better. Because each new digit of a balanced number is no more than half of its place value, fractional expansions are actually a series of closer and closer approximations - you never need to revise a previous digit.
You'll remember that Shwa notation uses the decade sign without any exponent to indicate a true percentage (one restricted to values between 0% and 100%). When using reverse notation, we retain the same notation, substituting the degree sign. But note that percentages from 50% up are written starting with negative digits, to be subtracted from 100%:
There is a separate keypad for the reverse numbers, which does not include the digits 7 8 9 but does include the negative digits. It also excludes the negative and imaginary signs.
To shift into Reverse mode, use the Shift key followed by the Reverse Mode key:
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