SafekipediaThe Nine Chapters on the Mathematical Art
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The Nine Chapters on the Mathematical Art is a very old Chinese book about mathematics. It was written by many different scholars over many years, from about the 10th century BCE to the 1st century CE. This book is one of the oldest known math books from China, along with the Suan shu shu and the Zhoubi Suanjing.
What makes this book special is how it teaches math. Instead of starting with rules and then proving things, like ancient Greek mathematicians often did, it focuses on finding general ways to solve problems. Each problem in the book is presented clearly: first the question is asked, then the answer is given, and finally, the steps to reach that answer are explained.
In the 3rd century, a scholar named Liu Hui wrote helpful comments about the book, making it even more useful for learners. Later, during the Tang Dynasty, the book was collected into a group of important math books called the Ten Computational Canons.
History
The Nine Chapters on the Mathematical Art is an important ancient Chinese math book. Its full name appears on bronze measures from 179 CE, but it may have existed earlier under different names. Scholars think Chinese math developed mostly separate from math in the Mediterranean world until this book was finished.
The book includes methods that were not known in Europe until much later, like a special way to solve equations that a famous mathematician named Carl Friedrich Gauss used centuries after. It also gives a proof for the Pythagorean theorem, which explains a special rule about triangles. The ideas from this book helped math in Korea and Japan grow. A mathematician named Liu Hui wrote detailed notes about the book in 263, checking each step to make sure it was correct.
Table of contents
The book The Nine Chapters on the Mathematical Art is organized into nine main parts, each focusing on different types of math problems. It starts with basic farming and trade calculations and moves on to more complex topics like geometry and algebra. Each section usually presents a problem, then shows the solution and explains how to reach that answer.
The contents include methods for measuring land, calculating taxes, solving equations, and working with shapes. This approach helps readers learn general ways to solve many kinds of problems, making it a key piece of early mathematical writing.
| chapter | contents |
|---|---|
| 方田 Fangtian Bounding fields | Areas of fields of various shapes, such as rectangles, triangles, trapezoids, and circles; manipulation of vulgar fractions. Liu Hui's commentary includes a method for calculation of π and the approximate value of 3.14159. |
| 粟米 Sumi Millet and rice | Exchange of commodities at different rates; unit pricing; the Rule of Three for solving proportions, using fractions. |
| 衰分 Cuifen Proportional distribution | Distribution of commodities and money at proportional rates; deriving arithmetic and geometric sums. |
| 少廣 Shaoguang Reducing dimensions | Finding the diameter or side of a shape given its volume or area. Division by mixed numbers; extraction of square and cube roots; diameter of sphere, perimeter and diameter of circle. |
| 商功 Shanggong Figuring for construction | Volumes of solids of various shapes. |
| 均輸 Junshu Equitable taxation | More advanced word problems on proportion, involving work, distances, and rates. |
| 盈不足 Yingbuzu Excess and deficit | Linear problems (in two unknowns) solved using the principle known later in the West as the rule of false position. |
| 方程 Fangcheng The two-sided reference (i.e. Equations) | Problems of agricultural yields and the sale of animals that lead to systems of linear equations, solved by a principle indistinguishable from the modern form of Gaussian elimination. |
| 勾股 Gougu Base and altitude | Problems involving the principle known in the West as the Pythagorean theorem. |
Major contributions
The Nine Chapters on the Mathematical Art made important contributions to early mathematics. It explored fractions and how to add, subtract, multiply, and divide them. It also introduced the idea of positive and negative numbers, helping to lay the groundwork for the real number system we use today.
The book included practical geometry for agriculture and architecture. It described ways to find the sides of right triangles, a concept known as the Gou Gu Theorem, similar to the Pythagorean Theorem. It also showed methods for solving equations with multiple unknowns, an early form of what we now call Gaussian elimination. These ideas helped advance mathematics in ancient China.
Significance
In ancient China, the number "9" often stood for something grand or powerful. The word "chapter" also had a broader meaning, referring to a section or a full treatise. Because of this, many experts think The Nine Chapters on the Mathematical Art was as important for Eastern math as Euclid's Elements was for Western math.
The book focused on solving practical problems using a step-by-step method: stating the problem, giving a formula, and showing the computation. This problem-solving style is still used today in applied mathematics.
Notable translations
Several translations of The Nine Chapters on the Mathematical Art exist in different languages. In English, Yoshio Mikami wrote an abridged version in 1913, and Florian Cajori also provided a shortened translation in 1919. Lam Lay Yong published another abridged version in 1994.
A complete translation with study was done by Kangshen Shen in 1999. There are also translations in French by Karine Chemla and Shuchun Guo in 2004, in German by Kurt Vogel in 1968, and in Russian by E. I Beriozkina in 1957.
This article is a child-friendly adaptation of the Wikipedia article on The Nine Chapters on the Mathematical Art, available under CC BY-SA 4.0.
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