Problem 35 - A History of Mathematics: An Introduction

Question

The dowry of Francis's wife is 100 aurei more than Francis's own property, and the square of the dowry is 400 more than the square of his property. Find the dowry and the property. (Note the negative answer for Francis's property; Cardano interpreted this as a debt.)

Step-by-Step Solution

Answer

Answer: The dowry of Francis's wife is 48 aurei, and Francis's property is a debt of 52 aurei.

Step 1: Setting up the equations

From the given information, we can set up the following two equations: 1. y = x + 100 (the dowry is 100 aurei more than Francis's property) 2. y^2 = x^2 + 400 (the square of the dowry is 400 more than the square of his property)

Step 2: Solving the first equation for x

From equation 1, we can solve for x by subtracting 100 from both sides: x = y - 100

Step 3: Substituting the value of x from the first equation into the second equation

Now we can substitute the value of x from the first equation into the second equation to eliminate one variable: (y - 100)^2 = y^2 + 400

Step 4: Expanding and simplifying the equation

Expand the left side of the equation and simplify: (y^2 - 200y + 10000) = y^2 + 400

Step 5: Canceling out the y^2 terms and solving for y

As we can see, the y^2 term on both sides will cancel out, leaving us with: -200y + 10000 = 400 Now we can solve for y by adding 200y to both sides and then dividing by 200: 10000 - 400 = 200y 9600 = 200y y = 48

Step 6: Finding the value of x

Now that we've found the value of y, we can use it to find the value of x by substituting it back into the first equation: x = 48 - 100 x = -52 Since Francis's property has a negative value, we can interpret this as a debt of 52 aurei.

Step 7: Conclusion

The dowry of Francis's wife is 48 aurei, and Francis's property is a debt of 52 aurei.

Key Concepts

Quadratic Equations

Quadratic equations are a vital part of algebra and are widely used in various fields such as physics and engineering. They take the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants. When solving quadratic equations, we often look for their roots, which are the values of \( x \) that make the equation true. To find these roots, you can use different methods:

• Factoring: This involves rewriting the equation as a product of its factors. It's most useful when the quadratic can be easily factored.
• Completing the square: This method involves rearranging the equation so that one side is a perfect square trinomial. This is useful for deriving the quadratic formula.
• Quadratic formula: A general formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) that can solve any quadratic equation.

In our exercise, we use algebraic manipulation and substitution to solve for the variables. This involves expressing one variable in terms of the other and inserting that into the quadratic equation, leading us to the solution through simplification.

Negative Numbers and Debt

Negative numbers can be puzzling, but they are crucial in both mathematics and practical applications. In the context of debt, a negative number simply represents something less than zero. For example, if Francis's property value is \(-52\) aurei, it indicates a debt of 52 aurei rather than a lack of physical property. Negative values can appear in many scenarios, such as:

• Financial situations: Debt, losses, or expenses exceeding income.
• Temperature: Values below zero, like sub-freezing conditions.
• Coordinates: Positions below or to the left of a reference point, like the origin in a graph.

In problem-solving, recognizing the context helps us understand and interpret negative answers. Cardano's interpretation of a negative value as a debt illustrates how mathematical solutions need contextual understanding. It shows how algebra connects abstract numbers to real-world concepts.

Problem Solving in Mathematics

Effective problem-solving in mathematics requires a blend of logical reasoning and creative thinking. Typically, it involves carefully following steps to arrive at a solution. Here's a structured approach:

• Understand the problem: Comprehend the given conditions and what the problem is asking for. Identify known variables and relationships.
• Plan a strategy: Decide on the methods that might help solve the problem. This often involves setting up equations or using geometric relationships.
• Carry out the plan: Execute the chosen strategy step by step. This may involve calculations, substitutions, or drawing diagrams.
• Review the solution: After solving, check that your answer makes sense in the context of the problem, and revise if necessary.

In our exercise, we faced multiple unknowns, so setting equations based on the given relationships was key. We then used substitution and simplification to find a viable solution, demonstrating how mathematical theory translates into real-world problem-solving.