Question

Find the solution to problem 3 of chapter 8 of the Nine Chapters using the Chinese method: None of the yields of 2 bundles of the best grain, 3 bundles of ordinary grain, and 4 bundles of the worst grain are sufficient to make a whole measure. If we add to the good grain 1 bundle of the ordinary, to the ordinary 1 bundle of the worst, and to the worst 1 bundle of the best, then each yield is exactly one measure. How many measures does 1 bundle of each of the three types of grain contain? Show that the solution according to the Chinese method involves the use of negative numbers.

Short answer

Answer: The yields of 1 bundle of each type of grain are as follows: the best grain has a yield of -2/3 measures, the ordinary grain has a yield of 5/3 measures, and the worst grain has a yield of -2/3 measures. The Chinese method demonstrates the use of negative numbers in solving this problem.

Step-by-step solution

Define the variables and set up the equations

Let x be the yield of 1 bundle of the best grain, y be the yield of 1 bundle of the ordinary grain, and z be the yield of 1 bundle of the worst grain. From the given information, we can set up the following equations: 1. 2x + 3y + 4z < 1 (None of the yields of 2 bundles of the best grain, 3 bundles of ordinary grain, and 4 bundles of the worst grain are sufficient to make a whole measure.) 2. (x + y) = 1 (If we add 1 bundle of the ordinary grain to the good grain, the yield is exactly one measure.) 3. (y + z) = 1 (If we add 1 bundle of the worst grain to the ordinary grain, the yield is exactly one measure.) 4. (z + x) = 1 (If we add 1 bundle of the best grain to the worst grain, the yield is exactly one measure.)

Solve the system of equations using the Chinese method

To solve the system of equations using the Chinese method, we need to express one variable in terms of the other variables: From equation (2), we have: y = 1 - x From equation (3), we have: z = 1 - y Substituting (2) into (3) we get: z = 1 - (1 - x) = x Now, substitute the expressions for y and z into equation (1): 2x + 3(1 - x) + 4x < 1 Solving for x, we get: 2x + 3 - 3x + 4x < 1 3x < -2 x < -\frac{2}{3} Since x < -\frac{2}{3}, it means that x is a negative number. Now, we can use the expressions for y and z to find their values: y = 1 - x = 1 - (-\frac{2}{3}) = 1 + \frac{2}{3} = \frac{5}{3} z = x = -\frac{2}{3}

State the solution

The solution to the problem is that 1 bundle of the best grain has a yield of -\frac{2}{3} measures, 1 bundle of the ordinary grain has a yield of \frac{5}{3} measures, and 1 bundle of the worst grain has a yield of -\frac{2}{3} measures. The use of the Chinese method in solving this problem demonstrates the use of negative numbers, as the yield of the best and worst grain are both negative.

Key concepts

Nine Chapters

The "Nine Chapters on the Mathematical Art" is a foundational text in Chinese mathematics, crafted during the Han Dynasty. Imagine it as an ancient mathematical cookbook, offering recipes for solving various practical problems. These problems are found in agriculture, construction, commerce, and more.
While not widely known in the Western world, the Nine Chapters served as an essential guide for centuries in China. It consists of 246 problems, with methods outlined concisely. Chapters usually provide a problem statement followed by an algorithmic solution, much like today's textbooks.
One distinctive feature is its structured approach that often involves linear problems and the concept of equations. The exercise we're dealing with is a classic example, illustrating the interplay of different types of grains – a very practical scenario back then.

system of equations

In the problem at hand, a system of equations is used to represent the relationships between different bundles of grain. This technique involves creating multiple equations to represent different conditions or relationships.
Let's break down the equations:

• The first equation is an inequality: \(2x + 3y + 4z < 1\). It represents a situation where the total yield from these bundles is not enough to make a whole measure.
• Other equations use the additive relationships and are equalities: \((x + y) = 1\), \((y + z) = 1\), \((z + x) = 1\). These reveal how yields combine to precisely form complete measures.

A system of equations allows us to solve for multiple unknowns by using substitution or elimination methods. In this context, it provides a way to determine the individual yields of each grain type under the given conditions.

negative numbers

Negative numbers, a concept we often take for granted, played an unexpected role in solving this ancient problem. Although initially odd, negative values can represent deficits or losses in real-world scenarios.
In the given solution, a critical step revealed that the best grain yield, represented by \(x\), needed to be less than \(-\frac{2}{3}\), indicating it was a deficit rather than a gain. Such results may contradict our intuition but are perfectly valid mathematically.
The introduction and acceptance of negative numbers in mathematics history were significant. They allowed for more comprehensive problem-solving, including situations where values dip below zero, broadening the scope of mathematical exploration considerably.

ancient mathematical methods

Ancient mathematical methods often blend practical experience with logical reasoning. The Chinese method exemplifies this by solving equations in a manner that aligns with the practical issues faced during its time.
Methods involved logical transformations, like substituting variables and simplifying calculations. The approach for the grain yields showcases this by expressing variables in terms of others, making it easier to solve a complex system of equations.
Moreover, ancient methods were not just about obtaining a solution but understanding the problem deeply and producing solutions with minimal tools. Simplicity and elegance often characterized these methods. They were rooted in solving tangible problems, ensuring mathematics remained connected to real-world applications.