Question

The correct match of the following columns is given by \begin{tabular}{|l|l|} \hline \multicolumn{1}{|c|} { Column I } & \multicolumn{1}{c|} { Column II } \\\ \hline 1. Leibnitz & A. \(\mathrm{e}^{\mathrm{i} \theta}\) \\ 2\. Euler & B. Mathematical logic \\ 3\. Cayley-Hamilton & C. Calculus \\ 4\. George Boole & D. \(\left(\mathrm{e}^{\mathrm{i} \theta}\right)^{\mathrm{n}}=\mathrm{e}^{\mathrm{i}(\mathrm{n} \theta)}\) \\ 5\. De-moivre & E. Theory of Matrices \\ \hline \end{tabular} (a) \(1-\mathrm{D}, 2-\mathrm{A}, 3-\mathrm{E}, 4-\mathrm{B}, 5-\mathrm{A}\) (b) \(1-\mathrm{B}, 2-\mathrm{D}, 3-\mathrm{A}, 4-\mathrm{C}, 5-\mathrm{E}\) (c) \(1-\mathrm{C}, 2-\mathrm{A}, 3-\mathrm{D}, 4-\mathrm{B}, 5-\mathrm{E}\) (d) \(1-\mathrm{C}, 2-\mathrm{A}, 3-\mathrm{E}, 4-\mathrm{B}, 5-\mathrm{D}\)

Short answer

The correct match of the given columns is (d) \(1-\mathrm{C}, 2-\mathrm{A}, 3-\mathrm{E}, 4-\mathrm{B}, 5-\mathrm{D}\).

Step-by-step solution

- Identifying Mathemeticians Contributions

In this step, we need to identify the mathematical contributions of the mathematicians provided in the list. Some general knowledge or research is needed for this step. - Leibnitz is well-known for developing **Calculus**, along with Sir Isaac Newton. - Euler made significant contributions in many areas, but his famous formula is **\(\mathrm{e}^{\mathrm{i} \theta}\)**. - The Cayley-Hamilton theorem is about **Theory of Matrices**. - George Boole is known for his work in **Mathematical logic**. - De Moivre's theorem deals with the formula **\(\left(\mathrm{e}^{\mathrm{i} \theta}\right)^{\mathrm{n}}=\mathrm{e}^{\mathrm{i}(\mathrm{n} \theta)}\)**.

- Match the Mathematicians with their Contributions

Now that we have identified the mathematicians' contributions, let's match them correctly. 1. Leibnitz - C. Calculus 2. Euler - A. \(\mathrm{e}^{\mathrm{i} \theta}\) 3. Cayley-Hamilton - E. Theory of Matrices 4. George Boole - B. Mathematical logic 5. De-moivre - D. \(\left(\mathrm{e}^{\mathrm{i} \theta}\right)^{\mathrm{n}}=\mathrm{e}^{\mathrm{i}(\mathrm{n} \theta)}\)

- Choose the Correct Option

The correct matching is as follows: \(1-\mathrm{C}, 2-\mathrm{A}, 3-\mathrm{E}, 4-\mathrm{B}, 5-\mathrm{D}\) From the given options, we can see that this matches with the option (d). So, the correct match of the given columns is (d) \(1-\mathrm{C}, 2-\mathrm{A}, 3-\mathrm{E}, 4-\mathrm{B}, 5-\mathrm{D}\).

Key concepts

Calculus

Calculus is a branch of mathematics that plays a crucial role in understanding changes and motion. Developed in the 17th century by mathematicians like Leibnitz and Newton, calculus offers tools to study rates of change (derivatives) and accumulation of quantities (integrals).

This field deals with two primary concepts:

• Derivatives: Describes how a function changes as the input changes. It's like finding the slope of a curve at any given point.
• Integrals: Concerned with adding up small quantities to determine the whole, such as areas under curves.

Understanding calculus is essential for advanced studies in mathematics, physics, engineering, and economics. It enables us to model real-world problems, such as predicting the path of a falling object or determining areas and volumes.

Theory of Matrices

The theory of matrices is a vital area in mathematics that deals with arrays of numbers known as matrices. These arrays are pivotal in solving linear equations and transforming geometric data.

• Matrices: Rectangular arrangements of numbers, symbols, or expressions in rows and columns used to represent linear transformations.
• Determinants: A special number calculated from a matrix that helps in finding solutions to matrix equations and understanding properties of the matrix.
• Eigenvalues and Eigenvectors: Important concepts that provide insights into the matrix's properties, especially in transforming spaces, like rotation and scaling.

The Cayley-Hamilton theorem is a key result in matrix theory, stating that every square matrix satisfies its own characteristic equation. This fundamental concept aids in simplifying matrix expressions and solving equations efficiently.

Mathematical Logic

Mathematical logic forms the backbone of mathematics, focusing on formal logical systems and reasoning. It is essential for developing algorithms, proofs, and solving complex problems.

• Symbolic Logic: Involves using symbols to represent logical expressions and structures.
• Propositional Logic: Deals with propositions, which are statements that can be true or false.
• Predicate Logic: Extends propositional logic by considering variables and quantifiers, useful in computer science and linguistics.

George Boole's contributions to mathematical logic laid the foundation for Boolean algebra, which is crucial in computer science and digital circuit design, enabling the conceptualization of binary systems and logic gates.

Euler's Formula

Euler's formula, \[e^{i\theta} = \cos\theta + i\sin\theta\],is a stunning result in mathematics that establishes a deep relationship between trigonometry and exponential functions. It is hailed for its beauty for connecting these abstract concepts seamlessly.

• This formula is particularly useful in engineering, physics, and complex analysis, enhancing the understanding of wave functions, electrical circuits, and signal processing.
• Euler’s formula allows for a simple format of representing sinusoidal functions as exponentials, thereby simplifying many mathematical operations.

Using Euler's formula can simplify complicated trigonometric expressions and provide deeper insights into the behavior of complex numbers.

De Moivre's Theorem

De Moivre's theorem provides a powerful method for raising complex numbers to integer powers efficiently. Stated as:\[(e^{i\theta})^n = e^{in\theta}\],it expresses complex numbers in polar form.

• This theorem connects the powers of complex numbers with trigonometric functions, simplifying calculations drastically.
• It is often used to find roots of complex numbers and play a crucial role in signal processing and quantum mechanics.

By using De Moivre’s theorem, mathematicians and scientists can perform operations on complex numbers more seamlessly, aiding in the integration of different mathematical disciplines.