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1) Ancient Indian mathematicians made several contributions to geometry, including calculating the square root of 2 to high precision in the Sulvasutras from 1600 BC and developing geometrical proofs. 2) In 1932, Dutta B.B. developed a geometric construction to prove that the square root of 2 is irrational using strips divided into squares of fractional widths. 3) This construction was one of the earliest geometric proofs that the square root of 2 cannot be expressed as a ratio of two integers, and calculated its value to 5 decimal places, very close to the precise modern value.

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Transformation numerical

The document describes scaling a square object by 4 units in the x-direction and 3 units in the y-direction while keeping one point fixed. It provides the step-by-step process to derive the composite matrix for this scaling transformation, which involves translating the object, scaling it, and translating it back. To check if a point remains fixed after the transformation, its homogeneous coordinates are multiplied with the composite matrix. This proves the given point is fixed, while another arbitrary point is not fixed after the transformation.

Combinations

This document provides an overview of isometries and their composition. It outlines objectives for students to classify isometries and find compositions, references relevant Iowa Common Core standards, defines isometries as transformations that preserve distance, describes the four kinds of isometries including reflections, rotations, translations, and glide reflections, and provides examples of reflections over parallel and intersecting lines as well as glide reflections using Sketchpad. Homework assignments are included.

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This document discusses reflections of geometric shapes and why it is important to label transformed shapes. It includes warm-up questions, essential questions about reflections, learning objectives related to rotations, reflections and translations, definitions of translations, reflections and reflection lines, examples of reflections and translations, and a closing question about what was learned about reflections.

"A Metaheuristic Search Technique for Graceful Labels of Graphs" by J. Ernst...

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Computational techniques have been presented that will identify a graceful labeling for a given graph provided one exists, thereby confirming that the graph is indeed graceful. Supported by the necessary theory to guarantee a solution, these routines primarily rely upon constrained iterative methods and are often quite computationally expensive. A number of other methods for graceful labelings have been proposed, including those employing deterministic backtracking and tabu search, among others. Here, a genetic algorithm-inspired, metaheuristic search technique to attempt to ascertain graceful
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Transformation numerical

The document describes scaling a square object by 4 units in the x-direction and 3 units in the y-direction while keeping one point fixed. It provides the step-by-step process to derive the composite matrix for this scaling transformation, which involves translating the object, scaling it, and translating it back. To check if a point remains fixed after the transformation, its homogeneous coordinates are multiplied with the composite matrix. This proves the given point is fixed, while another arbitrary point is not fixed after the transformation.

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The document discusses exploring characteristics of trigonometric functions using a graphing calculator. It describes investigating how changing parameters of trig functions affects the graph, and discussing the difference between parameters and variables. It also summarizes a student presentation about investigating relationships between perimeter and area of squares and triangles graphically.

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This document discusses reflections of geometric shapes and why it is important to label transformed shapes. It includes warm-up questions, essential questions about reflections, learning objectives related to rotations, reflections and translations, definitions of translations, reflections and reflection lines, examples of reflections and translations, and a closing question about what was learned about reflections.

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Presented at the 44th annual Southeastern International Conference on Combinatorics, Graph Theory, and Computing at Florida Atlantic University in 2013.

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International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.

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Kk[1]

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Copy of समासविचारः

work done by students in digital library of gltsbm

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Kk[1]

This document is a Sanskrit dictionary of mathematical terms translated to Sanskrit. It includes translations of common geometry terms like circle, angle, area, axis, base, center, diameter, hypotenuse, parallel, perpendicular, radius, side, and triangle. Each entry provides the Sanskrit translation of the term. The document aims to expand the Sanskrit vocabulary for mathematics and geometry terms.

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Climate

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The documents discuss the status of women in India historically and currently. While women had high status in ancient scriptures, in practice their status was inferior to men. Today, women face many issues including gender discrimination, domestic violence, dowry deaths, female foeticide, acid attacks, honor killings, and sexual harassment. Several laws and reforms have tried to protect and empower women, such as banning dowry and sex-selective abortion. However, violence against women remains a serious problem in India.

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Diabetes mellitus

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Copy of समासविचारः

work done by students in digital library of gltsbm

Sridharacharya[1]

Sridharacharya was an Indian mathematician from the 9th century AD who made several important contributions to geometry. He wrote two major mathematical treatises called Patiganita and Trisatika that covered topics like arithmetic, algebra, geometry, and the early concepts of calculus. In these works, Sridharacharya provided early formulas for the volumes of spheres and segments of circles. He also explicitly defined the rules of zero for the first time, such as how it affects addition, subtraction, multiplication and division.

The road not taken

The road not taken

S104

S104

Work and energy by ayushman maheswari

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Mera anubhav meri siksha 7th

Mera anubhav meri siksha 7th

Kk[1]

Kk[1]

Tenses

Tenses

Save the tigers

Save the tigers

Earthday (shikha)

Earthday (shikha)

Copy of समासविचारः

Copy of समासविचारः

Rational numbers

Rational numbers

Climate

Climate

Trigonometry

Trigonometry

Women empowerment (1)

Women empowerment (1)

Child labour

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Bramhagupta dec[1]

Bramhagupta dec[1]

Diabetes mellitus

Diabetes mellitus

Copy of समासविचारः

Copy of समासविचारः

Sridharacharya[1]

Sridharacharya[1]

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- Defining polynomials as expressions involving variables and coefficients using addition, subtraction, multiplication, and exponents.
- Discussing the history of polynomial notation pioneered by Descartes.
- Explaining the different types of polynomials like monomials, binomials, and trinomials.
- Outlining common uses of polynomials in mathematics, science, and other fields.
- Describing how to find the degree of a polynomial and graph polynomial functions.
- Explaining arithmetic operations like addition, subtraction, and division that can be performed on polynomials.

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Chemistry

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Issue of Shares

The document discusses key differences between private and public companies. It states that private companies have restrictions on the number of members and cannot invite the public to subscribe to its shares, while public companies can have an unlimited number of members and can invite public subscription. Additionally, private companies have restrictions on the transfer of shares while public companies do not.

My experience my values 7th

The document discusses the benefits of exercise for mental health. Regular physical activity can help reduce anxiety and depression and improve mood and cognitive functioning. Exercise causes chemical changes in the brain that may help protect against mental illness and improve symptoms.

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The document summarizes key concepts about crystalline and amorphous solids, including:
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Smart class

This document discusses the factors that influence demand, including price, income, and the prices of substitute and complementary goods. It defines quantity demanded as a change in demand due to a change in price, while a change in demand is due to other factors. A demand schedule shows the inverse relationship between price and quantity demanded in a table. A demand curve graphs this relationship, with quantity demanded on the y-axis and price on the x-axis. Movement along the demand curve occurs when quantity demanded changes due to a change in price.

S107

Cancer immunotherapy harnesses the body's immune system to fight cancer by developing antibodies or T cells that target and destroy cancer cells. Researchers are working to develop new immunotherapy approaches that use antibodies or T cells to identify and eliminate cancer cells while leaving normal cells unharmed. Immunotherapy holds promise as a revolutionary new approach for treating many cancer types by giving the body's own immune system tools to fight the disease.

S103

This document summarizes different types of work, energy, and collisions. It defines positive, negative, and zero work done based on the angle between the applied force and displacement. Kinetic energy is defined as the energy of motion, while potential energy is the energy due to position. The equations for kinetic and potential energy are provided. Conservation of energy and conservative versus non-conservative forces are also summarized. Kinetic energy and total linear momentum are stated to be conserved, while inelastic collisions are defined as those where the initial and final kinetic energies are not equal.

S102

This document discusses the structure and function of neurons and the nervous system. It describes the key parts of a neuron including the cell body, dendrites, and axon. It explains how nerve impulses are transmitted across synapses using neurotransmitters. The document then provides an overview of the major structures of the brain including the cerebrum, cerebellum, brainstem, and how the brain acts as the command center to integrate sensory input and control coordination.

S101

The document provides tips for conserving electricity and reducing energy costs, such as turning off lights and appliances when not in use, using more efficient appliances like CFL bulbs and gas water heaters, avoiding unnecessary opening of refrigerators, not leaving devices on standby, doing laundry in batches, and installing solar panels. It also notes that Energy Star certified devices use 20-30% less energy.

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Mansi

Anees Jung's book Lost Spring focuses on stories of children from deprived backgrounds who face difficult circumstances like being kidnapped and forced to work in the carpet industry. Some children are maltreated by alcoholic fathers, married off early, or sexually abused. In this chapter, Jung expresses concern over the exploitation of children forced to do hazardous jobs like bangle making and rag picking due to poverty and traditions. This results in the loss of childhood, education, and opportunities. One story focuses on Mukesh, who lives with his brother doing bangle making in Firozabad but wants a different career, though people believe their fate is predetermined by their previous birth. The theme of the book and awards is to create awareness about child

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From the begning

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English aphotograph

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Volume of sphere[1]

The sphere is a three-dimensional shape with the following properties:
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providing crucial environmental data for scientific, resource management, policy purposes, and
diverse human activities.
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changes, conversion trends, and other related patterns. The spatial dimensions of land use and
cover support policymakers and scientists in making well-informed decisions, as alterations in
these patterns indicate shifts in economic and social conditions. Monitoring such changes with the
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9
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- 1. Presented by: Mrs.Mamta Verulkar Vimaltai Tidke convent & Jr. College, Beltarodi , Nagpur 05/08/15 1India's Contribution to Geometry
- 2. • Mathematicsisthescienceof structure, orderandrelationthat developsgraduallyfrombasic practiceofcounting, measuringand describingtheshapesofobjects. 05/08/15 2India's Contribution to Geometry
- 3. Geometrical Proof • AGeometricalproofinvolveswriting reasoned, logicalexplanationsthatuse definitions, axioms, postulatesand previouslyprovedtheoremstoarrive ataconclusionaboutageometric statement. 05/08/15 3India's Contribution to Geometry
- 4. SquareRootof2 • Theoldestvalueofsquarerootof2is obtainedfromBabyloniantimes (1600B.C.) • Thesexagesimalequivalentvalueof squarerootof2asgiveninthe Sulvasutra is1.24, 51, 10, 37 05/08/15 4India's Contribution to Geometry
- 5. Proof • Construct ABC such that • AB = BC = n • AC = m, B = • Supposemandnareintegers • Let us assume that m:n is the ratio A B C m n 05/08/15 5India's Contribution to Geometry n ∆ ∠ 90 0 n m AB AC = 2= n m
- 6. Draw arcs of lengths m and n with centre A AB = AD AC = AE • BAC and DAE coincide • ABC ADE byS.A.S. test 05/08/15 India's Contribution to Geometry 6 ∠ ∠ ∆ ≅
- 7. EBF = 05/08/15 India's Contribution to Geometry 7 0 90∠ (Rightangle) ∠ BEF= = ishalfofrightangle BEFisrightisosceles triangle. ∆ 0 45
- 8. Hence BE = m– n BF = m – n • DF = m – n • Then FDC isalso right Isoscelestriangle • FC = n – ( m – n ) = 2n - m 05/08/15 India's Contribution to Geometry 8 ⇒ ∆ EBF ∆ CDF≅
- 9. ABC & FDC are similar 05/08/15 India's Contribution to Geometry 9 ∆ ∆ FC AC FD AB = FD FC AB AC = nm mn n m − − = 2
- 10. But triangle ABC is biggerin size than FDC i.e m > 2n-m n > m-n But assumption is is in lowest form. i.e is also in lowest form 05/08/15 India's Contribution to Geometry 10 n m nm mn − −2
- 11. • i.e m and n have common integral factors but it contradicts the assumption m:n is in lowest form. • Therefore cannot be expressed in form (rational form) hence is irrational. 05/08/15 India's Contribution to Geometry 11 2 n m 2
- 12. DuttaB.B. (1932) Considertwosquareswhosesidesareofunit length. Dividethesecondsquareintothree equalstripesI, IIanIII. 05/08/15 12India's Contribution to Geometry III III
- 13. • Sub-dividethelast stripintothree smallsquaresIII1, III2, III3 of sides⅓rd each. 05/08/15 India's Contribution to Geometry 01 III2 III1 III3 III III III1
- 14. • ThenonplacingI, IIandIII1 about thefirstsquareS inthepositionI’, II’ andIII1 ’ anew squareisformed. 05/08/15 India's Contribution to Geometry 01 I II III1
- 15. Nowdivideeachofthe portionsIII2 andIII3 into fourequalstripes. 05/08/15 15India's Contribution to Geometry Arrange four partsof III2,III3 asshown in figure
- 16. • Introducing asmall square at thesouth-east corner, a largesquarewill beformed. 05/08/15 India's Contribution to Geometry 16 ++ )4)(3( 1 3 1 1 Each side of which is equal to
- 17. 05/08/15 India's Contribution to Geometry 17 In order to get theequivalent arearemovetwo thin stripes, say of width x from either side of thesquare.
- 18. 05/08/15 India's Contribution to Geometry 18 2 2 )4)(3( 1 )4)(3( 1 3 1 12 =− ++ xx Neglecting x2 as to small 2 )4)(3( 1 12 17 2 = x )34)(4)(3( 1 2 1 17 12 )4)(3( 1 2 = =x 2 6271.41421568 )34)(4)(3( 1 )4)(3( 1 3 1 1 = = −++=Side of a square after subtracting x
- 19. • In decimal form this value equals 1.41421568627 Today’s value of is 1.41421356237,a difference of 0.000021239 05/08/15 India's Contribution to Geometry 19 2
- 20. References • Book by Saraswati Amma. • Ganit Kaumidi (Mathematics book) • www.cut-the-knot.orgproofs/sq_root.shtml • httphttp://www.math.ubc.ca/~ • http://www.math.ubc.ca/~cass/courses/m309 -0l1a/kong/sulbasutra_geometry.htm • Guided by Professor DR.Anant Vyawhare. 05/08/15 India's Contribution to Geometry 01
- 21. 05/08/15 India's Contribution to Geometry 21

- Mamta R. Verulkar
- Geometry
- By Pythagorus theorem
- BEF is half of right angle BEF is right isosceles triangle.
- CDF
- Side of a square after subtracting x