He then introduced sine and cosine as functions and

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He then introduced sine and cosine as functions and calculated values as decimals. He made tables with all the values of sin and cos, something that is impossible to calculate half angles… Napier Introduced the log (base 10) function and calculated values of log as decimals and found the basic log formulas. Notation: Before 1500, all algebra was written in words. No mathematical symbols existed. The Italians started abbreviating the words. o They used “cosa a ” for the unknown (e.g. quadratto cosa b quad c) o “piu” = more (e.g.: 3 p 5 = 3 + 5) o “multiplicare” = times (e.g.: 3 m c = 3 x ) o “radice” = root (e.g.: rad 3 p 5 m r 3 m c = 3 + 5√3? ) François Vieta (~1600) He started using x for the unknown, +, × He introduced arbitrary coefficients for solving equations Cardano: x 3 + 5 x 2 = 2 x + 4 x 3 = 5 x 2 + 2 x + 4 x 3 + 7 x = x 2 = 9 Vieta: ax 3 + bx 2 + cx + d = 0 He only has to do it once and applies to all equations. For the next century, more mathematics was created than what was created within the last 2000 years. This has provided the tools to do more mathematics. Lecture 21 Friday, October 25, 2019 a “thing” in Italian b “square thing” in It alian
30 ©2019 An Li. All rights reserved. The Great Explosion (1630 1750) The “new” algebra has suddenly been available to mathematicians. Fermat* Number theory Fermat’s Little Theorem : If p is prime, then a p a is divisible by p . (True) He actually proved the theorem. See . He also said 2 2 𝑛 + 1 is always prime. (False) In fact, 4294967297 = 641 × 6700417. Fermat’s Last Theorem : You know x 2 + y 2 = z 2 has many integer solutions (e.g. 3 2 + 4 2 = 5 2 , 5 2 + 12 2 = 13 2 , etc.). “I can prove that x n + y n = z n has no integer solutions for n > 2, I have a beautiful proof but the margin of this book is too small for me to write it. This became the most famous unsolved problem in mathematics. In 1730, Saint Leonard Euler** proved x 3 + y 3 = z 3 has no solution. It was very, very hard. In 1800, Legendre showed that x 4 + y 4 = z 4 has no solution. The whole theorem was first proved in 1994 by Andrew Wiles. Analytical geometry Old problem: Find the tangent to a given curve. The slope of a curve is given by 𝑓(?+ℎ)−𝑓(?) when h = 0. It was “sort of” the derivative. This does not make sense because cannot divide by 0. Descartes* (1640) wrote Géométrie Universale . He invented graphs for curves, as curves can be expressed as equations. This unified geometry and algebra.
©2019 An Li. All rights reserved. 31 Shortly after that, calculus was invented. Lecture 22 Monday, October 28, 2019 Short note on Descartes: Descartes did not believe in negative numbers, so his graphs looked like this: A few years later, John Wallis used negatives to get: Up until 1600, all physics were based on Aristotle from 400 BC. Not everything he said was true. Examples: 1. Something must be pushing for something to move. 2. Action at a distance was impossible, gravity made no sense. Around 1600: Two extremely important changes in physics: 1. Galileo realized velocity and acceleration are vital. However, no formula exists for acceleration… 2. Kepler found that the planets travel on ellipses with the sun at a focus. Equal areas in equal time
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