Vedic Mathematics:
Vedic Mathematics is a system of mathematics derived from ancient Indian texts called the Vedas. It is believed to have been discovered by the Indian mathematician Bharati Krishna Tirthaji in the early 20th century. Vedic Mathematics is based on 16 sutras (aphorisms) and 13 sub-sutras that provide various techniques and shortcuts for solving complex mathematical problems.

Modern Mathematics:
Modern Mathematics refers to the mathematical concepts and techniques that have evolved over time and are widely used today. It is based on rigorous logical reasoning and is taught in schools and universities around the world. Modern Mathematics includes various branches such as algebra, geometry, calculus, statistics, and more.

Differences between Vedic and Modern Mathematics:

1. Approach:
Vedic Mathematics follows a holistic and intuitive approach. It focuses on mental calculations, simplification techniques, and pattern recognition to solve problems quickly. On the other hand, Modern Mathematics emphasizes logical reasoning, formal proofs, and abstract thinking.

2. Methods:
Vedic Mathematics offers a wide range of methods and shortcuts for solving mathematical problems. These techniques are often based on patterns and symmetries, and they can be applied to various branches of mathematics. In contrast, Modern Mathematics relies on algorithms, formulas, and standard procedures to solve problems systematically.

3. Scope:
Vedic Mathematics is known for its versatility and applicability to various areas of mathematics. It covers arithmetic, algebra, geometry, calculus, and more. However, it may not always provide the most efficient or generalizable solutions. Modern Mathematics, on the other hand, offers a comprehensive and rigorous framework that can handle complex mathematical problems in a consistent manner.

4. Teaching:
Vedic Mathematics is often taught as a separate subject or as a supplementary approach to traditional mathematics. It aims to improve mental calculation skills and speed up problem-solving. Modern Mathematics, on the other hand, is the primary focus of formal education systems worldwide. It follows a structured curriculum and emphasizes theoretical understanding and problem-solving techniques.

5. Acceptance and Recognition:
While Vedic Mathematics has gained popularity in some educational circles, it is not universally accepted or recognized as a formal mathematical system. Modern Mathematics, on the other hand, is widely accepted and used by mathematicians, scientists, engineers, and professionals in various fields.

In conclusion, Vedic Mathematics and Modern Mathematics represent different approaches to solving mathematical problems. Vedic Mathematics offers techniques and shortcuts for quick mental calculations, while Modern Mathematics provides a comprehensive and rigorous framework that is widely accepted and recognized in formal education and professional settings.