Overview
The Imperial mathematics system exists in two forms. One is "New" or "Merchant" Mathematics, the form commonly used by engineers and merchants around the Empire, consisting of a representational number system and a rich shorthand of symbols - some borrowed from as far away as Vaarta - to express certain concepts. They have a quite developed geometry and a functional algebra, and they're familiar with ideas like absolute value, imaginary numbers, logarithms and trigonometry. Calculus is in its early stages, understood by only a few mathematical theorists, and the mathematics they do have is sharply divided between pure theory and applied concepts.

The problem is that, before any new idea may be taught, it must first be expressed in a paper using Old Arithmetic and approved by the Department of High Mathematics, part of the Ministry of Shining Wisdom. The Old Arithmetic is laid down in the Classic of Ten Thousand Symbols, supposedly dictated by the Divine Minister for the Nine Arts. It manages to be singularly ill-designed for modern purposes: it does not include symbols for such elementary things as negative numbers, powers, roots, grouping symbols, or fractions larger than the unit. The prescribed measure of degree is far too large for most purposes, the method of naming and describing figures is absolutely Byzantine, and all constructs and proofs must be done with only a compass and a marked straight edge. Algebra is nearly impossible, and the number system is not representational.

The result of this is logical - mathematicians don't bother to produce these papers, or they make one their life's work. Mathematics in the Empire has been moving sluggishly, although the recent rebellion of the Anhuine states has resulted in a surge in scholarship, since papers can be written in a form that everyone will understand.

Old Arithmetic
Old Arithmetic, also known as High Mathematics, is an ancient system believed to have been created by the Immortals. Because of its sacred nature, it has remained in use despite the fact that the notation is cumbersome and that it is ill-suited to more than simple arithmetic. It is in Base Four, since four is a sacred number in Imperial philosophy.

In the "Old Math", numbers are identified by both the multiplier column heading (Units, Fours, Sixteens, Sixty Fours etc.) and the value (1, 2 or 3). If the value is zero, the number is ignored.

The multipliers are called:

Units	Yen
Fours	Hanol
Sixteens	Shu
Sixty-Fours	Lasol
Two-Hundred-Fifty-Sixes	Hif
One-Thousand-Twenty-Fours	Suuf

etc....

The values are:

1. Ti
2. Lop
3. Pen

When a number is named, the multiplier is given first, followed by the value, thus one is written as "yen-ti" (Units*1). Two is written as "yen-lop" (Units*2), three as "yen-pen" (Units*3), four as "hanol-ti" (Fours*1) and so on.

When larger numbers are written out, the figures are written with the highest value columns first, rather than in a consistent order. Thus the decimal number 22, which in Base 4 notation would be given as 112 (1*16, 1*4, 2*1) would in Imperial High Math notation be given as Yenlop Shuti Hanolti, or 2*1, 1*16, 1*4. The value of 2 in the Units column is highest, and so in this case the Units come first.

Each column multiplier (Yen, Hanol, Lop, Lasol etc.) has three different written characters for values of Ti, Lop and Pen, but there is no obvious relationship between them. Yenti and Hanolti, for example, do not have a clear "ti" element in common. In addition, some numbers have their own special characters that are not part of the naming system.

In order to perform sums with this system, High Maths users rely on pre-written tables that have been laboriously worked out by predecessors hundreds of years earlier. Architects and engineers use works like the Book of Grand Containment and the Book of Heaven on Earth to guide them with pre-written formulae for structural features.

Merchant Maths
Also called "New" Maths, this alternative notation system has in fact been around for centuries, almost as long as High Maths. It originated as a method of accounting used by merchants, a shorthand version of High Maths. Snobbery against profiteering classes in the Empire meant that it was ignored as "improper" by officialdom but even so found its way into the coinage system and the Official Calendar. Imperial Mathematicians has long used the "merchant" system, but because they were prevented from writing down their discoveries officially, concepts tended to get lost and require rediscovering. Now that many nations have broken away from the Empire they feel little need to stick to the old ways, and in recent decades the use of New Math notation has allowed advances in knowledge.

The earlier forms used simplified versions of the Yen, Hanol, Shu etc. symbols but the values are represented by 1-3 dots above the symbol. Thus the symbol for "three" was the Yen (one) symbol with three marks above it, the symbol for "eight" was the Hanol (four) symbol with two dots above it, and so on. The systems also introduced the concept of zero (no dots), known as "Gaal".

At some point in history, the system changed from Base 4 to Base 5 (possibly for ease of counting on one hand). The value of 4 became known as "Shan". At the same time, values came to be written out in column order, and the column names were dropped. The system became streamlined to just five simple symbols for Gaal, Ti, Lop, Pen and Shan (0,1,2,3 and 4), all of which could be drawn in a single brush or pen stroke. This made arithmetic easy to accomplish without cumbersome tables, essential when calculating the value of cargo quickly. The concept of negative numbers (or "deficit" numbers, as they are known) was also invented.

To compare the two systems, in High Maths, the decimal number 5 is represented as Hanol-Ti Yenti(Fours*1 + Units*1), whereas in Merchant Maths it is 10 (Ti Gaal). The decimal number 27, which in High Maths is the mind bending Yenpen Hanolop Shuti (Units*3 + Fours*2 + Sixteens*1), in Merchant Maths it is Ti Gaal Lop (102, or 25+2).

Table of Comparative Values 0-32

Base 10	High Math	Base 4	Mercantile	Base 5
0	n/a	0	Gaal	0
1	Yenti	1	Ti	1
2	Yenlop	2	Lop	2
3	Yenpen	3	Pen	3
4	Hanolti	10	Shan	4
5	Hanolti Yenti	11	Ti Gaal	10
6	Yenlop Hanolti	12	Ti Ti	11
7	Yenpen Hanolti	13	Ti Lop	12
8	Hanolop	20	Ti Pen	13
9	Hanolop Yenti	21	Ti Shan	14
10	Hanolop Yenlop	22	Lop Gaal	20
11	Yenpen Hanolop	23	Lop Ti	21
12	Hanolpen	30	Lop Lop	22
13	Hanolpen Yenti	31	Lop Pen	23
14	Hanolpen Yenlop	32	Lop Shan	24
15	Hanolpen Yenpen	33	Pen Gaal	30
16	Shuti	100	Pen Ti	31
17	Shuti Yenti	101	Pen Lop	32
18	Yenlop Shuti	102	Pen Pen	33
19	Yenpen Shuti	103	Pen Shan	34
20	Shuti Hanolti	110	Shan Gaal	40
21	Shuti Hanolti Yenti	111	Shan Ti	41
22	Yenlop Shuti Hanolti	112	Shan Lop	42
23	Yenpen Shuti Hanolti	113	Shan Pen	43
24	Hanolop Shuti	120	Shan Shan	44
25	Hanolop Shuti Yenti	121	Ti Gaal Gaal	100
26	Hanolop Yenlop Shuti	122	Ti Gaal Ti	101
27	Yenpen Hanolop Shuti	123	Ti Gaal Lop	102
28	Hanolpen Shuti	130	Ti Gaal Pen	103
29	Hanolpen Shuti Yenti	131	Ti Gaal Shan	104
30	Hanolpen Yenlop Shuti	132	Ti Ti Gaal	110
31	Hanolpen Yenpen Shuti	133	Ti Ti Ti	111
32	Shulop	200	Ti Ti Lop	112