VEDIC MATHEMATICS -1 (VAC) PROJECT .pptx

Vedic maths project

1. VEDIC MATHEMATICS (VAC) PROJECT SUBMITTED BY Harshal Mehra 1110/22 Bcom (H) Utkarsh Singh 1010/22 Bcom (H) Vishal 1062/22 Bcom (H) Kartik Hans 1052/22 Bcom (H)

2. ABOUT BAUDHAYANA Baudhayana, an ancient Indian mathematician and scholar, lived around the 6th century BCE. He is renowned for his significant contributions to mathematics, particularly in geometry and algebra. He formulated geometric principles and solved mathematical problems, demonstrating advanced knowledge of geometry and algebraic techniques. One of Baudhayana's most notable achievements is the approximation of the square root of 2, known as the Baudhayana theorem. This theorem is found in the Baudhayana Sulbasutra, a collection of ancient Indian mathematical texts dealing with geometry and construction of altars for Vedic rituals. Baudhayana's contributions to mathematics laid a foundation for further developments in the field and helped shape mathematical thought in ancient India.

3. DISCOVERIES OF BAUDHAYANA

4. BAUDHAYANA THEOREM Baudhayana's theorem is found in the Baudhayana Sulba Sutra, a collection of ancient Indian mathematical texts dating back to around 800 BCE. The Sulba Sutras deal with various geometric principles related to the construction of altars used in Vedic rituals, indicating the practical applications of mathematics in ancient Indian society. Baudhayana's theorem, in essence, provides a method for constructing a right-angled triangle with sides of specific lengths, similar to the Pythagorean theorem. The theorem states that a rope stretched along the diagonal of a rectangle creates areas which the vertical and horizontal sides make together, equal to the sum of the areas formed separately by the horizontal and vertical sides. This principle, when translated into modern mathematical language, can be expressed as: a^2 + b^2 = c^2 Where 'a' and 'b' are the lengths of the two shorter sides of the right-angled triangle, and 'c' is the length of the hypotenuse.

5. The similarities between Baudhayana's theorem and the Pythagorean theorem are notable, given that they were discovered independently by scholars from two different ancient civilizations. This raises intriguing questions about the transmission of mathematical knowledge and the universality of mathematical truths . USE OF THIS THEOREM- Calculating missing side lengths If you know two side lengths of a right triangle, you can use the theorem (a² + b² = c²) to find the length of the missing side (the hypotenuse or a leg). This is crucial in carpentry, construction, engineering, and many other fields where right angles and precise measurements are important. Determining distances By forming right triangles with landmarks or other reference points, the theorem can be used to calculate indirect distances that are difficult or impossible to measure directly. This could be useful in surveying land, laying out foundations, or even tasks like navigating with sightlines. Checking squareness The Pythagorean theorem (and by extension, Baudhayana's theorem) can be used to verify if an angle is truly a 90-degree right angle. This is helpful in ensuring square corners in construction projects, furniture making, or any situation where perfect right angles are critical.

6.   Circling a Square One of Baudhayana’s remarkable achievements was his ability to construct a circle almost equal in area to a square and vice versa. Let’s explore this intriguing concept: Construction : Baudhayana’s procedure involves constructing two squares and their circumscribed circles. Here are the steps he followed: Draw a square (let’s call it  Square A ). Construct another square (let’s call it  Square B ) with the same side length as  Square A . Circumscribe a circle around  Square A . Circumscribe another circle around  Square B . Observe the relationship between the areas of these squares and their circumscribed circles.

7. 2. Observations : Baudhayana realized that the area of the inner circle (circumscribed around  Square A ) was exactly  half  of the area of the larger circle (circumscribed around  Square B ). He knew that the area of a circle is proportional to the  square of its radius . By comparing the areas of the squares and the circles, he deduced that the inner circle’s radius was related to the side length of  Square A . 3. Perimeter Relationship : Baudhayana extended his observation to perimeters: The perimeter of the outer circle (circumscribed around  Square B ) should be  √2 times  the perimeter of the inner circle (circumscribed around  Square A ). This relationship holds true for regular polygons as well.

8. 4. Value of π : Baudhayana is also credited with discovering an approximation for the value of π. His Sulba Sutras mention approximations such as: π ≈  676/225  (approximately 3.004) π ≈  1156/361  (approximately 3.202) π ≈  900/289  (approximately 3.114)  Baudhayana's method for approximating the area of a circle doesn't involve calculating pi directly. It relies on geometric relationships and the Pythagorean theorem. Here's the breakdown: Initial Square: Let's say the side of the original square is "a" units. The area of the square ( A_square ) is simply a x a = a^2 square units. New Square: After dividing each side in half, the sides of the new inscribed square become a/√2 units (due to the Pythagorean theorem applied to the right triangle formed by dividing the side). Area of the New Square: The area of this new square ( A_new_square ) is (a/√2) x (a/√2) = a^2 / 2 square units. 5. Calculations:

9. According to Baudhayana's theorem, this area ( A_new_square ) is approximately equal to the area of the original circle ( A_circle ). Here's the key point: While this method doesn't give the exact value of pi, it establishes a relationship between the areas. We can express this as an approximation: A_circle ≈ A_new_square = a^2 / 2 Note: This is an approximation. The actual value of pi is irrational and cannot be expressed as a simple fraction. However, we can take it a step further. If we know the area of the circle (say, from measurement), we can rearrange the equation to get an approximate value of pi: π ≈ ( A_circle x 2) / a^2 This provides a way to estimate pi based on the measured area of a circle and the side length of the original square.

10. Concept Angles : Baudhayana Number Euclid's geometry is studied all over the world, considering it to be authentic in the subject of geometry. But it should be remembered that before the great Greek geometer Euclid, many geometry in India had discovered important rules of geometry, Baudhayana's name is paramount among those geometry. At that time, geometry or geometry was also called Shulba Sutra in India. Shulba Sutras are the instructions for constructing various geometrical shapes of different angles to use in making or building the 'fire altars'.For fire altar construction cardinal directions east West line was required this line was created by following instructions given in the Shulb Sutra. Took a long stick and erected on a plane ground and tie a string to the stick and the other end of the string tied to a nail. Then draw a circle around the stick this has to be done along before the sunrise once sun rises the shadow of this chick is casted on the opposite direction so the longest shadow will be in the early morning so the shadow protrudes out of the circle that was drawn and the Sun progresses over the day proportionally. The shadow reduces and in the evening again it protrudes towards the other end. So the two spots which crossover the circular path give East and West direction. This East and West line gives zero degree angle.A square can be constructed with a measuring cord this square contains 90 degree angle at each corner. When a rhombus of the size of a (given) square is desired, produce a rectangle with area twice that of the square and join the midpoints of the sides of the rectangle. Thus a rectangle and a rhombus also consist of a 90 degree angle.To construct an isosceles triangle of the size of a (given) square, form a square with twice the area as the given square and join the midpoint of one of the sides to the vertices of the side opposite to it. This isosceles triangle consists of a 45 degree angle.

11. When a square is desired to be converted to one that is pointed on one side (in the form of an isosceles trapezium, having the same area as the initial square) keeping a transverse segment of the size desired for the shorter side (at one end), the remaining (rectangular) part is to be divided along its diagonal and the (triangular) excess part is to be inverted and adjoined on the other side. Thus we get 30 degree and 60 degree angles in isosceles trapezium.

12. Anurupyena Sutra The " Anurupyena Sutra" is a foundational text in the field of mathematics, particularly in ancient Indian mathematics. The term "sutra" refers to a concise and aphoristic style of writing, often used in Indian texts to convey profound or fundamental principles. The " Anurupyena Sutra" specifically pertains to the concept of proportionality or proportionate increase. " Anurupyena " can be translated to mean "in proportion" or "in accordance with." Therefore, the " Anurupyena Sutra" likely contains mathematical principles or methods related to proportional reasoning, such as those used in solving mathematical problems involving ratios, proportions, or scaling. In ancient Indian mathematics, sutras were often used as mnemonic devices to aid in the memorization and transmission of mathematical knowledge. They encapsulated mathematical concepts concisely, allowing scholars to easily remember and apply them in problem-solving contexts.

13. Examples: Example 1:  Since both these numbers are far away from 1000, we take 1000 as our theoretical base and 1000/2 = 500 as our working base We then work-out the multiplication as before and to the answer obtained, we divide the left-hand portion of the result in the same proportion as our theoretical base is to the working base (in this example divide by 2) Example 2:  A case where right hand portion of the answer generates an extra digit, we carry it over and add to the left hand portion after division    

14. Examples: Example 3:       A case where left hand portion of the answer generates a remainder after final division, we add the fractional part to the right hand portion of the answer                                                                                               Example 4:      In all the examples above, the reason we divide only the left hand portion of the answer proportionately is that when we divide a number by different numbers that have a certain ratio amongst themselves, e.g 10 divided by 3,6,9, in all 3 cases we notice that the remainder is always 1. So the quotient reduced in the same ratio in which the divisor increased, but remainder remained constant 1.   We now look at an example where the working base is a multiple of the theoretical base.                                                                                                           e.g 365 * 380

15. Adyamadyenantyamantya In  Adyamadyenantyamantya  (Commonly called as  Adyamadyena ), we divide the first term’s  coeff  of eq with 1st term of factor obtained above and last term of eq with the last term of the same factor. Sanskrit Name (For  Adyamadyenantyamantya ): आद्यमाद्ये   नान्त्यमन्त्येन English Translation (For  Adyamadyenantyamantya ): 1st by 1st and last by last The " Adyamadyenantyamantya  Sutra " is not a well-known or widely recognized term outside of Sanskrit or Hindu philosophical circles. However, it seems to be a compound phrase composed of several Sanskrit words: "Adya," "Madhya," " Antya ," and " Antyamantya .

16. Examples: Suppose we are asked to find out the area of a rectangular card board whose length and breadth are respectively 6ft . 4 inches and 5 ft. 8 inches. Generally we continue the problem like this.