Mathematics
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This book is an introduction to the standard methods of proving mathematical theorems. It has been approved by the American Institute of Mathematics' Open Textbook Initiative . Also see the Mathematical Association of America Math DL review , and the Amazon reviews .
Mathematics can sometimes make smart people dumb. Let me explain what I mean by this. I don't mean that it is dumb not to be good at mathematics. After all, mathematics is a highly abstract and challenging discipline requiring many years (decades even) of study, and there are plenty of very smart people who have little understanding of it, and little ability to use it. What I mean is that mathematics quite often bamboozles people into accepting very silly arguments — arguments that are so silly that if you stated them without draping them in mathematical negligee, you would instantly become an object of ridicule to all those people who flunked out at basic algebra back in high school.
How Mathematics Can Make Smart People Dumb - Ben O'Neill
Euler's identity
The exponential function e z can be defined as the limit of (1 + z / N ) N , as N approaches infinity, and thus e i π is the limit of (1 + i π / N ) N . In this animation N takes various increasing values from 1 to 100. The computation of (1 + i π / N ) N is displayed as the combined effect of N repeated multiplications in the complex plane , with the final point being the actual value of (1 + i π / N ) N .
Baron Augustin-Louis Cauchy ( French pronunciation: [ogysˈtɛ̃ lwi koˈʃi] ) (21 August 1789 – 23 May 1857) was a French mathematician who was an early pioneer of analysis . He started the project of formulating and proving the theorems of infinitesimal calculus in a rigorous manner, rejecting the heuristic principle of the generality of algebra exploited by earlier authors. He defined continuity in terms of infinitesimals and gave several important theorems in complex analysis and initiated the study of permutation groups in abstract algebra . A profound mathematician, Cauchy exercised a great influence over his contemporaries and successors. His writings cover the entire range of mathematics and mathematical physics .
Augustin-Louis Cauchy
Infinite Series
In elementary algebra , the binomial theorem describes the algebraic expansion of powers of a binomial . According to the theorem, it is possible to expand the power ( x + y ) n into a sum involving terms of the form ax b y c , where the exponents b and c are nonnegative integers with b + c = n , and the coefficient a of each term is a specific positive integer depending on n and b . When an exponent is zero, the corresponding power is usually omitted from the term. For example,
Binomial theorem
This article provides an introduction. For a more detailed and technical article, see Ramsey's theorem . Ramsey theory , named after the British mathematician and philosopher Frank P.
Ramsey theory
Set theory
Set theory is the branch of mathematics that studies sets , which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. The language of set theory can be used in the definitions of nearly all mathematical objects .
Grelling–Nelson paradox
The Grelling–Nelson paradox is a semantic self-referential paradox formulated in 1908 by Kurt Grelling and Leonard Nelson and sometimes mistakenly attributed to the German philosopher and mathematician Hermann Weyl . [ 1 ] It is thus occasionally called Weyl's paradox as well as Grelling's paradox . It is closely analogous to several other well-known paradoxes, in particular the Barber paradox and Russell's paradox . [ edit ] The paradox Suppose one interprets the adjectives "autological" and "heterological" as follows:"I think about math constantly and I see and look for math in everything around me." Zachary Abel is a second year Ph.D student in the MIT Mathematics department, but he also creates sculptures out of everyday objects. "By transforming often-overlooked household items into elaborate, mathematical sculptures, I hope to share this sense of excitement, curiosity, and beauty that a mathematical outlook has instilled in me." Zachary Abel, "Impenetraball", 2011. The "Impenetraball" protects its hollow interior with a dense, chainmail-like mesh made from 132 binder clips (and pliers). With his intricate formations of otherwise mundane objects, Abel hopes to expose the "hidden geometric beauty" in our everyday lives.
