Problem 24 - A History of Mathematics: An Introduction

Question

Provide the details for the first step of Zhu Shijie's solution to problem 2 of his Precious Mirror. That is, let \(a\) be the base, \(b\) the altitude, and \(c\) the hypotenuse of a right triangle, and assume \(b^{2}-[c-(b-a)]=b a \quad\) and \(\quad a^{2}+c+b-a=a c\) Then set \(x=b\) and \(y=a+c\). Show that the two given equations along with the Pythagorean Theorem imply that the following two equations hold: $$ x^{3}+2 y x^{2}+2 x y-x y^{2}-2 y^{2}=0 \quad \text { and } $$ $$ x^{3}+2 y x-x y^{2}+2 y^{2}=0 $$

Step-by-Step Solution

Answer

Question: Given the system of equations: \(b^{2}-[c-(b-a)]=ba\) and \(a^{2}+c+b-a=ac\), and the substitution \(x=b\) and \(y=a+c\), find the equations that represent the relationship between \(x\) and \(y\). Answer: The relationship between \(x\) and \(y\) can be represented by the following equations: \(x^{3}+2yx^{2}+2xy-xy^{2}-2y^{2}=0\) and \(x^{3}+2yx-xy^{2}+2y^{2}=0\).

Step 1: Understand the given equations and substitution

We are given the following equations: \(b^{2}-[c-(b-a)]=ba \quad\) and \(\quad a^{2}+c+b-a=ac\) We are also given the substitution \(x=b\) and \(y=a+c\). The goal is to reach the following target equations: \(x^{3}+2yx^{2}+2xy-xy^{2}-2y^{2}=0 \quad\) and \(\quad x^{3}+2yx-xy^{2}+2y^{2}=0\)

Step 2: Use the Pythagorean Theorem

We know that for a right triangle with sides \(a\), \(b\), and hypotenuse \(c\), the Pythagorean theorem holds: \(a^2+b^2=c^2\)

Step 3: Substitution

To express the given equations in terms of \(x\) and \(y\), we will substitute \(x=b\) and \(y=a+c\) into the given equations and Pythagorean theorem: \(x^2 - [y-(x-y)] = x(y-x) \quad\) and \(\quad (y-x)^2 + y + x - (y - x) = x(y-x)\) Now, we have the equations in terms of \(x\) and \(y\): \(x^2 - [y-(x-y)] = x(y-x) \quad\) and \(\quad (y-x)^2 + y + x - (y - x) = x(y-x)\)

Step 4: Simplify and manipulate the equations

Now we will manipulate the simplified equations algebraically to obtain the target equations involving \(x\) and \(y\). For the first equation, we will manipulate: $$x^2 - (y - x + y) = x(y-x)$$ This simplifies to: $$x^2 - (2y - x) = xy - x^2$$ Now, we will add \(x^2\) to both sides and arrange it in the goal formation: $$x^3 + 2yx^2 + 2xy - xy^2 - 2y^2 = 0$$ For the second equation, we will manipulate: $$(y-x)^2 + y + x - (y-x) = x(y-x)$$ Expanding the square and simplifying: $$(y^2 - 2xy + x^2) + y + x - y + x = xy - x^2$$ Now, we will arrange this equation in the goal format: $$x^3 + 2yx - xy^2 + 2y^2 = 0$$ We have successfully reached the target equations: \(x^{3}+2yx^{2}+2xy-xy^{2}-2y^{2}=0 \quad\) and \(\quad x^{3}+2yx-xy^{2}+2y^{2}=0\)

Key Concepts

Zhu Shijie

Zhu Shijie was a Chinese mathematician from the Yuan Dynasty known for his works which include intricate problem-solving techniques. He is often celebrated for his contributions to algebra and mathematical advancements, which he presented in his famous book, 'Precious Mirror of the Four Elements'. This book contains a variety of mathematical problems and solutions, notably incorporating the use of algebraic techniques to solve geometric problems.
Zhu Shijie's approach in the problem involves clever algebraic manipulation, including setting variables to simplify complex equations. His legacy continues to influence modern mathematics, showing the power of algebra in problem-solving.
Many of these problems still serve as a valuable teaching tool for those studying the history and development of mathematics, demonstrating ancient methods that align with contemporary mathematical practices.

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in Euclidean Geometry, foundational to the study of triangles. It states that in a right-angled triangle, the square of the hypotenuse (\(c^2\)) is equal to the sum of the squares of the other two sides (\(a^2 + b^2\)). This theorem provides the basis for many mathematical investigations and solutions.
In Zhu Shijie's problem, the Pythagorean Theorem is used to connect the three sides of a right triangle, facilitating the substitutions and manipulations leading to complex algebraic expressions. By substituting known values and expressions, one can unlock new relationships between the geometrical properties, which may not seem obvious at first glance.
The theorem is instrumental in problems involving geometric interpretations of algebraic solutions, highlighting how ancient mathematical concepts still hold relevance today.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying equations to make them easier to understand or solve. This often requires substituting different known values, factorization, expansion, and reordering of terms to reach a desired or more usable equation.
In the exercise provided, algebraic manipulation allows us to transform the initial set of equations into a simplified form that showcases the inherent relationships between the variables. This is done by using substitutions like \(x=b\) and \(y=a+c\), and employing techniques to combine and rearrange terms.
Success in algebra relies heavily on understanding and practicing these manipulations. This skill proves crucial not only in solving specific problems but also in understanding the broader intricacies and applications of mathematical theory.