The number `e' by Tanvi Jain

PROGRAM : SUMMER SCHOOL FOR WOMEN IN MATHEMATICS AND STATISTICS ORGANIZERS : Siva Athreya and Anita Naolekar DATE : 13 May 2019 to 24 May 2019 VENUE : Ramanujan Lecture Hall, ICTS Bangalore The summer school is intended for women students studying in first year B.A/B.Sc./B.E./B.Tech. or equivalent degree and having Mathematics as one of the major subjects/courses, during the academic year 2018-2019. Over a period of two weeks, the school shall aim at helping young students gain broader exposure to problem solving skills in mathematics and statistics at the undergraduate level. Specific topics that will be covered are: Mathematical Thinking Advanced Calculus Linear Algebra Probability and Statistics There will be problem solving sessions across topics. Each topic will have a brief introductory lecture session on a focus area followed by problem solving sessions on it. The participants are expected to work together to solve the problem sheets that will be given out in these sessions. Each day will also feature two popular science lectures by distinguished researchers. The topics will range from across all sciences and the lecture will be made accessible to all the participants. Other daily events will include post dinner Problem solving sessions and discussions with popular science speakers of the day. In addition to the above there will be several short video talks that will be shown from aspiring to established women researchers in the areas of mathematical sciences. Further there will be opportunity to interact with many women scientists who will visit the program for short periods. Eligibility: All participants should be enrolled in a first year program (as stated above) B.A./B.Sc./B.E./B.Tech. or equivalent degree and having Mathematics as one of the major subjects/courses, during the academic year 2018-19. *Passport details are not mandatory unless the nationality is other than Indian. Support: All selected participants will be provided sleeper class train fare from home station to ICTS and back, local boarding (for non-Bengaluru participants) plus food and can be eligible for performance based stipend for books. For selected participants who will have 3 or more nights of train travel, a partial airfare travel support will be considered. APPLICATION DEADLINE : 22 March 2019 CONTACT US :[email protected] PROGRAM LINK : https://www.icts.res.in/program/swms2019 Table of Contents (powered by https://videoken.com) 0:00:00 The number ' e' 0:02:11 What is e? Is this just a number 0:02:35 The first hint of 0:05:26 How will you solve this? We will need to solve for the equation: 0:07:09 A question on compound interest 0:08:16 If the amount is compounded twice, each time at the interest rate of 50%. 0:12:44 Observations: 0:15:19 Let's see... 0:20:03 Logarithm and John Napier 0:21:18 So the problem of multiplication and division can be reduced to that of addition and subtraction. 0:23:50 Squaring a hyperbola 0:25:13 Coming to the early seventeenth century 0:26:53 Area of the region bounded by y = x", x = and x = a. 0:36:07 Similarly, the area of the region bounded by the curves y = x-", 0:37:14 A look at the hyperbola 0:41:24 The appearance of logarithm 0:43:36 The series for and the work of Newton 0:47:42 which implies ao = 0. Comparing the coefficient of y. 0:49:49 Derivative of an exponential function 0:51:39 Derivative of bx 0:52:35 The "natural "exponent Derivative of any exponential function is a constant multiple of the function itself. 0:56:07 The notation for the natural exponent 0:56:57 Is an integer? 0:58:04 rational or irrational 0:59:20 Since is not an integer, q 2 2. Hence q + 1 2 3. 1:00:19 Euler's proof of the irrationality of 1:01:13 The exponential function on complex numbers 1:01:30 Occurrences of 1:01:53 e'" Paradox