**Mathematics in the Empire**
**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:

- Ti
- Lop
- 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 |