Introduction to Discrete Mathematics
... Question: If a and b are irrational, can ab be rational?? We (only) know that √2 is irrational, what about √2√2 ? Case 1: √2√2 is rational Then we are done, a=√2, b=√2. Case 2: √2√2 is irrational Then (√2√2)√2 = √22 = 2, a rational number So a=√2√2, b= √2 will do. So in either case there are a,b irr ...
... Question: If a and b are irrational, can ab be rational?? We (only) know that √2 is irrational, what about √2√2 ? Case 1: √2√2 is rational Then we are done, a=√2, b=√2. Case 2: √2√2 is irrational Then (√2√2)√2 = √22 = 2, a rational number So a=√2√2, b= √2 will do. So in either case there are a,b irr ...
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... Lemma 1. Let S be a subset of C that contains a nonzero complex number and α ∈ C. Then α is constructible from S if and only if there exists a finite sequence α1 , . . . , αn ∈ C such that α1 is immediately constructible from S, α2 is immediately constructible from S∪{α1 }, . . . , and α is immediat ...
... Lemma 1. Let S be a subset of C that contains a nonzero complex number and α ∈ C. Then α is constructible from S if and only if there exists a finite sequence α1 , . . . , αn ∈ C such that α1 is immediately constructible from S, α2 is immediately constructible from S∪{α1 }, . . . , and α is immediat ...
1. Complex Numbers and the Complex Exponential
... The set of all complex numbers (conventionally denoted by a fancy letter C: C, or C, or C) is the set of everything that can be written in the form x + yi, where x and y are real numbers. Addition and multiplication are performed as they would be for polynomials in the character “i” with the underst ...
... The set of all complex numbers (conventionally denoted by a fancy letter C: C, or C, or C) is the set of everything that can be written in the form x + yi, where x and y are real numbers. Addition and multiplication are performed as they would be for polynomials in the character “i” with the underst ...
Math
... 1.4 I can use Right triangle trig, law of sines, or law of cosines to solve application problems. 2.1 I can choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, vertical shift and phase shift. 2.2 I can understand that restricting a trigonometric function t ...
... 1.4 I can use Right triangle trig, law of sines, or law of cosines to solve application problems. 2.1 I can choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, vertical shift and phase shift. 2.2 I can understand that restricting a trigonometric function t ...
List of important publications in mathematics
This is a list of important publications in mathematics, organized by field.Some reasons why a particular publication might be regarded as important:Topic creator – A publication that created a new topicBreakthrough – A publication that changed scientific knowledge significantlyInfluence – A publication which has significantly influenced the world or has had a massive impact on the teaching of mathematics. Among published compilations of important publications in mathematics are Landmark writings in Western mathematics 1640–1940 by Ivor Grattan-Guinness and A Source Book in Mathematics by David Eugene Smith.