Question

Given the product of two numbers and their ratio, to find the roots: Let \(A, E\), be the two roots, \(A E=B, A: E=\) \(S: R\). Show that \(R: S=B: A^{2}\) and \(S: R=B: E^{2}\). Viète's example has \(B=20, R=1, S=5\). Show in this case that \(A=10\) and \(E=2\). (Jordanus has the same problem but with different numbers.)

Short answer

Answer: When B=20, R=1, and S=5, the values of A and E are 10 and 2, respectively.

Step-by-step solution

Write the given relations

We are given the following relations: 1. \(A E = B\) 2. \(\frac{A}{E} = \frac{S}{R}\)

Find the relationship between \(R: S\) and \(B: A^{2}\), and \(S: R\) and \(B: E^{2}\)

From the second relation, we can express \(A\) and \(E\) in terms of \(R\) and \(S\): \(A = \frac{S * E}{R}\) Now, substitute the value of A obtained from the second relation into the first relation: \(E * \frac{S * E}{R} = B\) Rearrange to find the relationship between \(R:S\) and \(B:A^2\): \(\frac{R}{S} = \frac{B}{E^2}\) Similarly, for the relationship between \(S:R\) and \(B:E^2\), \(E = \frac{R * A}{S}\) Now, substitute the value of E obtained from the second relation into the first relation: \(A * \frac{R * A}{S} = B\) Rearrange to find the relationship between \(S:R\) and \(B:A^2\): \(\frac{S}{R} = \frac{B}{A^2}\)

Use the example given to find the values of \(A\) and \(E\)

We are given \(B = 20\), \(R = 1\), and \(S = 5\). We use these values to find \(A\) and \(E\) using the relationships obtained above. \(\frac{S}{R} = \frac{B}{A^2}\) \(\frac{5}{1} = \frac{20}{A^2}\) => \(A^2 = 4\) => \(A = 10\) Now, use the second relationship to find the value of \(E\): \(\frac{R}{S} = \frac{B}{E^2}\) \(\frac{1}{5} = \frac{20}{E^2}\) => \(E^2 = 100\) => \(E = 2\) Therefore, the values of \(A\) and \(E\) are 10 and 2, respectively.

Key concepts

Algebraic Equations

Algebraic equations are foundational to understanding mathematical relationships and they play a crucial role in problem solving across various fields. In essence, an algebraic equation is a statement of equality between two expressions involving constants and variables. For instance, the equation used in our problem, \( A \times E = B \) expresses the product of two unknown numbers — A and E — equating to a known number B.

When tackling problems involving algebraic equations, one effective approach is to manipulate these equations to express one variable in terms of others, which then allows for substitution into other equations to find a solution. In our example, we rearranged the given equation to isolate the variable E, leading to a new expression that could be used in further calculations. Such skills in manipulating algebraic equations are not only essential for solving textbook problems but also for critical thinking and analysis in real-world applications.

Ratios and Proportions

Ratios and proportions are important concepts in mathematics and are especially useful when comparing quantities. A ratio is a comparison between two numbers showing how many times the first number contains the second. In our exercise, we look at the ratio \( A:E = S:R \) which compares two unknowns, A and E, in relation to two knowns, S and R. Proportions, on the other hand, state that two ratios are equivalent.

Understanding how to work with ratios and proportions allows for the solving of problems where direct measurement is not possible, or when working with different units of measurement. In our problem-solving process, we used proportional relationships to express one variable in terms of another. This is an example of how the concepts of ratios and proportions can simplify complex problems by setting up relations that can be manipulated algebraically to extract useful information.

Mathematical Roots

Mathematical roots are solutions to the equation where a number or expression is raised to a power and set equal to another number. Basically, finding a root is the inverse operation of exponentiation. In our exercise, we had to calculate the square roots of numbers as part of finding the solution. The square root, which is the most common type of root, answers the question: 'what number, when multiplied by itself, will give me the original number?'

For instance, with \( A^2 = 4 \) from the problem, we consider what number squared equals 4, and we rightly find that \( A = 10 \) (note that we consider the positive root in the context of this problem). Understanding roots is essential for solving quadratic equations and forms the basis for more advanced mathematics, including calculus. Recognizing and calculating roots is not only crucial for academic purposes but also for practical applications, such as in engineering and sciences, where they are used to model and solve real-life problems.