## Knowledge in Logic & Mathematics

Does Rational Thinking Depend on the Culture

This project lays an emphasis on the effects of culture and surrounding on Rational Thinking.

A New way, The Area of Trapezium

A New way, The Area of Trapezium by Piyush Goel Lot of mathematicians have proved Pythagoras theorem in their own ways. If you google it you will indeed found hundred of ways. Meanwhile I was also sure that maybe one day I could find something new out of this incredible Pythagoras theorem and Recently I got something which I would like to share with you. To Prove: Deriving the equation of area of trapezium using Arcs Proof: There is a triangle ABC with sides a b and c as shown in the figure. Now, Area of ∆&nbsp; BCEG = Area of ∆&nbsp; BDC +Area of ⌂ DCEF + Area of ∆ EFG c^2=ac/2+ Area of ⌂ DCEF + (c-b) &nbsp;c/2 (2c^2&ndash; ac &ndash;c^2+ bc )/2=Area of ⌂ DCEF (c^2&ndash; ac+ bc )/2=Area of ⌂ DCEF c(c&ndash; a+ b)/2=Area of ⌂ DCEF Area of ⌂ DCEF=BC(DE+CF)/2 Copyrighted&copy;PiyushGoel

How the Factorial Function came into Existence.

&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;A note on the Factorial Function Write down 0,1,2,3,4,5 and &nbsp;put parallel&nbsp;square&nbsp;of each number like 0,1,4,9,16,25, then start to subtract the bigger one to the lower one (1&ndash;0),(4&ndash;1),(9&ndash;4),(16&ndash;9) and (25&ndash;16) to get 1,3,5,7,9 and again subtract the bigger one to the lower one (3&ndash;1),(5&ndash;3),(7&ndash;5) and (9&ndash;7) to get&nbsp;(2,2,2,2). Again we&nbsp;squared each number, at the same time we&nbsp;cubed each number&nbsp;(0,1,8,27,64,125,216) and the &nbsp;same procedure follows, subtract the bigger one to the lower one (1&ndash;0),(8&ndash;1),(27&ndash;8),(64&ndash;27) ,(125&ndash;64) and(216&ndash;125) to get (1,7,19,37,61,91) and again (7&ndash;1),(19&ndash;7),(37&ndash;19),(61&ndash;37) and (91&ndash;61), to get (6,12,18,24,30) same again(12&ndash;6),(18&ndash;12),24&ndash;18) and (30&ndash;24) &nbsp;till the result come out&nbsp;here we get (6,6,6,6). At the same time if we do it again for 4 and 5 (power).When we get&nbsp;2 for 2 ,6 for 3 ,24 for 4 and 120 for 5. The result is the factorial function. https://piyushtheorem.wordpress.com/2017/02/08/a-note-on-the-factorial-function/

Graph Theory

Number Theory

Number theory is a branch of pure mathematics devoted to the study of the natural numbers and the integers. It is the study of the set of positive whole numbers which are usually called the set of natural numbers. As it holds the foundational place in the discipline, Number theory is also called &quot;The Queen of Mathematics&quot;.The number theory helps discover interesting relationships between different sorts of numbers and to prove that these are true . Number Theory is partly experimental and partly theoretical. Experimental part leads to questions and suggests ways to answer them. The theoretical part tries to devise an argument which gives a conclusive answer to the questions. Here are the steps to follow: 1. Accumulate numerical data 2. Examine the data and find the patterns and relationships. 3. Formulate conjectures that explain the patterns and relationships. 4. Test the conjectures by collecting additional data and check whether the new information fits or not 5. Devise an argument that conjectures are correct. All the steps are important in number theory and in mathematics. A scientific theory is an ability to predict the outcome of experiments. In mathematics one requires the step of a proof, that is, a logical sequence of assertions, starting from known facts and ending at the desired statement.

Piyush Theorem

&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;PiyushTheorem PiyushTheorem: In a Right-Angled Triangle with sides in A.P. Series, the distance between the point of intersection of median &amp; altitude at the base is 1/10 Th&nbsp;the sum of other two sides. This Theorem applies in Two Conditions: 1.The&nbsp;Triangle&nbsp;must be&nbsp;Right-Angled. 2.Its&nbsp;Sides&nbsp;are in&nbsp;A.P. Series. 1.Proof with Trigonometry Tan&nbsp;&nbsp;&alpha;&nbsp;&nbsp;&nbsp;=AD/DC AD= DC&nbsp;Tan&nbsp;&nbsp;&alpha;&nbsp;&nbsp;&nbsp;&mdash;&mdash;&mdash;&mdash;&mdash;&ndash;1 Tan&nbsp;&nbsp;&alpha;&nbsp;= AD/DE AD= DE&nbsp;Tan2&nbsp;&alpha;&nbsp;&nbsp;&nbsp;&mdash;&mdash;&mdash;&mdash;&mdash;-2 DC&nbsp;Tan&nbsp;&nbsp;&alpha;&nbsp;= DE&nbsp;Tan 2&nbsp;&alpha; (DE+EC)&nbsp;&nbsp;Tan&nbsp;&nbsp;&alpha;&nbsp;= DE&nbsp;Tan 2&nbsp;&alpha; DE&nbsp;Tan&nbsp;&nbsp;&alpha;&nbsp;&nbsp;+ EC&nbsp;Tan&nbsp;&nbsp;&alpha;&nbsp;= DE&nbsp;Tan 2&nbsp;&alpha; DE&nbsp;Tan&nbsp;&nbsp;&alpha;&nbsp;&nbsp;+ EC&nbsp;Tan&nbsp;&nbsp;&alpha;&nbsp;= 2 DE&nbsp;Tan&nbsp;&nbsp;&alpha;&nbsp;/ (1-&nbsp;Tan2&nbsp;&nbsp;&alpha;&nbsp;&nbsp;&nbsp;) DE&nbsp;Tan&nbsp;&nbsp;&alpha;&nbsp;&nbsp;&nbsp;&ndash; DE&nbsp;Tan3&nbsp;&nbsp;&alpha;&nbsp;+ EC&nbsp;Tan&nbsp;&nbsp;&alpha;&nbsp;&ndash;EC&nbsp;Tan3&nbsp;&nbsp;&alpha;&nbsp;&nbsp;= 2DE&nbsp;Tan&nbsp;&nbsp;&alpha; EC&nbsp;Tan&nbsp;&nbsp;&alpha;&nbsp;&ndash;EC&nbsp;Tan3&nbsp;&nbsp;&alpha;&nbsp;&ndash; DE&nbsp;Tan3&nbsp;&nbsp;&alpha;&nbsp;= 2DE&nbsp;Tan&nbsp;&nbsp;&alpha;&nbsp;&ndash; DE&nbsp;Tan&nbsp;&nbsp;&alpha; Tan&nbsp;&nbsp;&alpha;&nbsp;(EC &ndash; EC&nbsp;Tan2&nbsp;&nbsp;&alpha;&nbsp;&ndash; DE&nbsp;T an2&nbsp;&nbsp;&alpha;&nbsp;)= DE&nbsp;Tan&nbsp;&nbsp;&alpha; DE&nbsp;Tan2&nbsp;&nbsp;&alpha;&nbsp;&nbsp;&ndash; DE = EC&nbsp;Tan2&nbsp;&nbsp;&alpha;&nbsp;&nbsp;&ndash; EC -DE (&nbsp;Tan2&nbsp;&nbsp;&alpha;&nbsp;+ 1) = -EC (1 &ndash;&nbsp;Tan2&nbsp;&nbsp;&alpha;&nbsp;) DE (sin2&nbsp;&alpha;&nbsp;&nbsp;/cos2&nbsp;&alpha;&nbsp;+ 1) = EC (1-&nbsp;sin2&nbsp;&alpha;&nbsp;&nbsp;/cos2&nbsp;&alpha;&nbsp;) DE (sin2&nbsp;&alpha;&nbsp;+&nbsp;cos2&nbsp;&alpha;&nbsp;/cos2&nbsp;&alpha;&nbsp;) = EC (cos2&nbsp;&alpha;&nbsp;&ndash;&nbsp;sin2&nbsp;&alpha;&nbsp;/cos2&nbsp;&alpha;&nbsp;) DE (sin2&nbsp;&alpha;&nbsp;&nbsp;+&nbsp;cos2&nbsp;&alpha;&nbsp;) = EC(cos2&nbsp;&alpha;&nbsp;&nbsp;&ndash;sin2&nbsp;&alpha;&nbsp;) DE (sin2&nbsp;&alpha;&nbsp;&nbsp;+&nbsp;cos2&nbsp;&alpha;&nbsp;) = EC (cos2&nbsp;&alpha;&nbsp;&nbsp;&ndash;sin2&nbsp;&alpha;&nbsp;) &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;where (sin2&nbsp;&alpha;&nbsp;&nbsp;+&nbsp;cos2&nbsp;&alpha;&nbsp;=1) &amp; (cos2&nbsp;&alpha;&nbsp;&nbsp;&ndash;sin2&nbsp;&alpha;&nbsp;=&nbsp;cos2&nbsp;&alpha;&nbsp;&nbsp;) &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;DE= EC&nbsp;cos2&nbsp;&alpha;&nbsp;&nbsp; cos&nbsp;&alpha;&nbsp;&nbsp;&nbsp;=a/a+d &nbsp;&nbsp;&amp;&nbsp;sin&nbsp;&alpha;&nbsp;= (a-d)/ (a +d) cos2&nbsp;&alpha;&nbsp;&nbsp;= a2/ (a +b)&nbsp;2 sin2&nbsp;&alpha;&nbsp;&nbsp;= (a-d)&nbsp;2/ (a+ d)&nbsp;2 DE= EC (cos2&nbsp;&alpha;&nbsp;&nbsp;&nbsp;&nbsp;&ndash;&nbsp;sin2&nbsp;&alpha;&nbsp;) = EC (a2&nbsp;/ (a +b)&nbsp;2&nbsp;&ndash; (a-d)&nbsp;2/ (a +d)&nbsp;2 = EC (a2&nbsp;&ndash; (a-d)&nbsp;2/ (a +d)&nbsp;2 = EC (a &ndash;a +d) (a+ a-d)/ (a+ d)&nbsp;2 = EC (d) (2a -d)/ (a+ d)&nbsp;2 = (a +d)/2(d) (2a -d)/ (a +d)&nbsp;2&nbsp;&mdash;&mdash;&mdash;&mdash;- where EC= (a +d)/2 = (d) (2a -d)/2(a +d) = (d) (8d -d)/2(4d+d) &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&mdash;&mdash;&mdash;&mdash;&mdash;&mdash;where a= 4d (as per the Theorem) = 7d2&nbsp;/2(5d) = 7d /10 = (3d+4d)/10= (AB+AC)/10 2.Proof with Obtuse Triangle Theorem AC2=EC2&nbsp;+AE2&nbsp;+2CE.DE&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;where EC = ( &nbsp;a +d) /2,AE=( a +d)/2 a2&nbsp;= (a +d/2)2&nbsp;+ (a+ d/2)2&nbsp;+ 2(a +d)/2DE = (a +d/2) (a+d+2DE) = (a +d/2) (a+d+2DE) &nbsp;&nbsp;where a=4d 16d2&nbsp;= (5d/2) (5d+2DE) 32d/5 = 5d + 2DE 32d/5 &ndash; 5d = 2DE 32d -25d/5 = 2DE DE =7d/10 = (3d+4d)/10 = (AB+AC)/10 3.Proof with Acute Triangle Theorem AB2= AC2+BC2&nbsp;&ndash; 2BC.DC (a-d) 2= a2&nbsp;+ (a+ d)&nbsp;2&nbsp;-2(a+ d) (DE+EC) &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;where AB= (a-d), AC=a, BC =( a +d) &amp; EC= (a +d)/2 (a-d)&nbsp;2&nbsp;&ndash; (a +d)2&nbsp;= a2&nbsp;&nbsp;-2(a +d)(DE+EC) (a- d &ndash;a-d) (a -d +a +d)&nbsp;&nbsp;= a2&nbsp;-2(a+ d) (2DE+a+d)/2 2(-2d) (2a)&nbsp;=&nbsp;2a2&nbsp;-2(a +d) (2DE+a+d) -8ad&nbsp;&ndash;&nbsp;2a2&nbsp;= -2(a +d) (2DE+a+d) -2a (4d&nbsp;&nbsp;&nbsp;+a) = -2(a +d) (2DE+a+d) a (4d&nbsp;&nbsp;+ a) = (a +d)(2DE+a+d) 4d (4 d&nbsp;+&nbsp;4d) = (4d+d) (2DE+4d+d) 4d (8d)&nbsp;= (5d) (2DE+5d) 32d2/5d = &nbsp;&nbsp;(2DE+5d) 32d/5 = &nbsp;&nbsp;(2DE+5d) 32d/5 &ndash; 5d = &nbsp;&nbsp;2DE (32d &ndash; 25d)/5 = &nbsp;&nbsp;2 DE DE = 7d/10 = (3d+4d)/10 = (AB+AC)/10 4. Proof with Co-ordinates Geometry Equation of BE Y &ndash; 0 =b-0/0-a(X &ndash; a) Y = -b/a(X) + b&mdash;&mdash;&mdash;&mdash;&mdash;&mdash;- (1) M1&nbsp;= -b/a For perpendicular M1M2= -1 So M2=a/b Equation of AC Y &ndash; 0 = a/b(X-0) Y=a/b(X) &mdash;&mdash;&mdash;&mdash;&mdash;&mdash; (2) Put Y value in equation (1) a/b(X) + b/a(X) =b X (a2+b2/a b) = b X = ab2/ (a2&nbsp;+ b2) To get Value of Y, put X value in equation (2) Y = a/b (ab2/ (a2+b2) Y = a2b/ (a2+b2) Here we got co-ordinates of Point C &ndash; ab2/ (a2&nbsp;+ b2), a2b/ (a2+b2) and co-ordinates of point d is (a/2, b/2) because d is midpoint. As per the &ldquo;Theorem&rdquo; a=z-d, b=z, c = z+ d (z +d)&nbsp;2= (z-d)&nbsp;2+z2&nbsp;from here z=4d so a=3d and b=4d Put value of a &amp; b ab2/ (a2&nbsp;+ b2), a2b/ (a2+b2) &amp; (a/2, b/2) ab2/ (a2&nbsp;+ b2) = 48d/25 a2b/ (a2+b2) = 36d/25 a/ 2=3d/2 b/ 2 =4d/2 CD2= (48d/25 -3d/2)2-(36d/25-4d/2)2 = (96d-75d/50)2&nbsp;+ (72d-100d/50)2 = (21d/50)2&nbsp;+ (-28d/50)2 = (441d2/2500) + (784d2/2500) = (1225d2/2500) CD= 35d/50 = 7d/10 = 7d/10 = (3d+4d)/10 = (AB+AE)/10 https://piyushtheorem.wordpress.com/2017/02/08/a-theorem-on-right-angled-triangles/

Annova

Application of annova and regression

JCG Case study LP solution

Linear Programming and Its solution for the JCG case. its excel solution with the sensitivity analysis report .

Mathematics-I for KIIT University

Notes on Mathematics-1 for B.tech

Numerical methods & Probability:The complete reference

Probability is the measure of the likelihood that an event will occur. See glossary of probability and statistics. Probability quantifies as a number between 0 and 1, where, loosely speaking, 0 indicates impossibility and 1 indicates certainty.In numerical analysis, a numerical method is a mathematical tool designed to solve numerical problems. The implementation of a numerical method with an appropriate convergence check in a programming language is called a numerical algorithm.The content provides a self explanatory insight on the subject.primarily for students of 2nd year.

Derivatives

This file will help you gain a holistic view on the derivatives rule in mathematics and it will enhance your overall knowledge.

simple and compounding rate