Important Theorems for Algebra II and/or Pre
... illustration/example on exactly one side of a sheet of notebook paper. You may use both the front side and backside of the notebook paper, but only one theorem may be completed per side. Your illustration/example should be brief and to the point, not just one part in the solution of a long problem. ...
... illustration/example on exactly one side of a sheet of notebook paper. You may use both the front side and backside of the notebook paper, but only one theorem may be completed per side. Your illustration/example should be brief and to the point, not just one part in the solution of a long problem. ...
Three Connections to Continued Fractions
... More generally, we can look at so-called generalized continued fractions, i.e. expressions of the form x = b0 + ...
... More generally, we can look at so-called generalized continued fractions, i.e. expressions of the form x = b0 + ...
Idiosynchromatic Poetry
... Ramsey theory is generally concerned with problems of finding structures with some kind of homogeneity in superstructures. Often a structure contains an homogeneous substructure of a certain sort if it is itself large enough. In some contexts the notion of size can not only be interpreted as cardina ...
... Ramsey theory is generally concerned with problems of finding structures with some kind of homogeneity in superstructures. Often a structure contains an homogeneous substructure of a certain sort if it is itself large enough. In some contexts the notion of size can not only be interpreted as cardina ...
Math 201 – Homework 5 – solutions
... A Venn diagram for the three sets A, B, C with the two given regions highlighted will show how to construct lots more examples. But any one example is enough. Problem 9.28 Let a, b ∈ Z. Is it true that if a|b and b|a then a = b? (Disproof). Solution. No, eg a = 4 and b = −4 (not part of this problem ...
... A Venn diagram for the three sets A, B, C with the two given regions highlighted will show how to construct lots more examples. But any one example is enough. Problem 9.28 Let a, b ∈ Z. Is it true that if a|b and b|a then a = b? (Disproof). Solution. No, eg a = 4 and b = −4 (not part of this problem ...
![[Part 2]](http://s1.studyres.com/store/data/008795795_1-c00648edd6f578e3e44ef8aca9f22ea2-300x300.png)
When to Use Indirect Proof
... If 2 were rational then we could write it as 2 = x/y where x and y are integers and y is not 0. By repeatedly cancelling common factors, we can make sure that x and y have no common factors so they are not both even. Then 2 = x2 /y 2 so x2 = 2y 2 so x2 is even. This means x is even, because the squa ...
... If 2 were rational then we could write it as 2 = x/y where x and y are integers and y is not 0. By repeatedly cancelling common factors, we can make sure that x and y have no common factors so they are not both even. Then 2 = x2 /y 2 so x2 = 2y 2 so x2 is even. This means x is even, because the squa ...
THE E.IRREGULAR PRIMES
... is called irregular if it divides the numerator of at least, one of the Bernoulli numbers B, , 8a,..., Bb-g (in the even suffix notation); see e,g. F, pp. 367-3391. carlitz l2l has given the simplest proof of the fact that the number of irregular primes is infinite. Metsänkylä [5] has proved that fo ...
... is called irregular if it divides the numerator of at least, one of the Bernoulli numbers B, , 8a,..., Bb-g (in the even suffix notation); see e,g. F, pp. 367-3391. carlitz l2l has given the simplest proof of the fact that the number of irregular primes is infinite. Metsänkylä [5] has proved that fo ...
Fermat's Last Theorem
In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two. The cases n = 1 and n = 2 were known to have infinitely many solutions. This theorem was first conjectured by Pierre de Fermat in 1637 in the margin of a copy of Arithmetica where he claimed he had a proof that was too large to fit in the margin. The first successful proof was released in 1994 by Andrew Wiles, and formally published in 1995, after 358 years of effort by mathematicians. The theretofore unsolved problem stimulated the development of algebraic number theory in the 19th century and the proof of the modularity theorem in the 20th century. It is among the most notable theorems in the history of mathematics and prior to its proof it was in the Guinness Book of World Records for ""most difficult mathematical problems"".