Quadratic Reciprocity Taylor Dupuy
... case 3 Suppose n is not a square mod p. We need two facts. 1. (p − 1)! ≡ −1 mod p (which holds generally) 2. (p − 1)! ≡ n(p−1)/2 . (which holds when n is not a square) First, Z/p is Q a field. We write out (p − 1)! and pairing inverses and get (p − 1)! ≡ c∈F× c = −1, Since the only elements left ove ...
... case 3 Suppose n is not a square mod p. We need two facts. 1. (p − 1)! ≡ −1 mod p (which holds generally) 2. (p − 1)! ≡ n(p−1)/2 . (which holds when n is not a square) First, Z/p is Q a field. We write out (p − 1)! and pairing inverses and get (p − 1)! ≡ c∈F× c = −1, Since the only elements left ove ...
Chapter 3
... You may have noticed that when we write polynomials we use as few of symbols as possible. In math we like to write polynomials in their simplest form, which means that they have as few of symbols as possible. Thus, instead of writing plus a negative and using parentheses, we will simply use a subtr ...
... You may have noticed that when we write polynomials we use as few of symbols as possible. In math we like to write polynomials in their simplest form, which means that they have as few of symbols as possible. Thus, instead of writing plus a negative and using parentheses, we will simply use a subtr ...
Year 2008/09 - Bishopsworth
... What are the steps in the sequence? Show how, if the steps are equal ,a calculator can be used to continue a sequence. On the calculator pad, press 1.5 + =. Explain that each time the + is pressed 1.5 is added to the sequence. Show that the steps in a sequence are not always equal, for example 2,5,9 ...
... What are the steps in the sequence? Show how, if the steps are equal ,a calculator can be used to continue a sequence. On the calculator pad, press 1.5 + =. Explain that each time the + is pressed 1.5 is added to the sequence. Show that the steps in a sequence are not always equal, for example 2,5,9 ...
Revision Notes
... DEFINITION: Assuming we have a square matrix A, which is non-singular ( i.e. det(A) does not equal zero ), then there exists an nxn matrix A-1 which is called the inverse of A, such that this property holds: AA-1= A-1A = I where I is the identity matrix. ...
... DEFINITION: Assuming we have a square matrix A, which is non-singular ( i.e. det(A) does not equal zero ), then there exists an nxn matrix A-1 which is called the inverse of A, such that this property holds: AA-1= A-1A = I where I is the identity matrix. ...
Infinity + Infinity
... Now, when students discuss infinity, assuming they know no set theory or any of Georg Cantor’s work, they are discussing the cardinality of N, which is defined as |N| = ℵ0 (”alephnaught”). We must consider three concepts before we can make sense of ∞ + ∞ = ℵ0 + ℵ0 . 1) Cantor-Bernstein-Schröeder Th ...
... Now, when students discuss infinity, assuming they know no set theory or any of Georg Cantor’s work, they are discussing the cardinality of N, which is defined as |N| = ℵ0 (”alephnaught”). We must consider three concepts before we can make sense of ∞ + ∞ = ℵ0 + ℵ0 . 1) Cantor-Bernstein-Schröeder Th ...
Document
... Multiplication Property of Equality For real numbers a, b, and c, where c =/= 0 if a = b, then a • c = b • c ...
... Multiplication Property of Equality For real numbers a, b, and c, where c =/= 0 if a = b, then a • c = b • c ...
TX_G6_PerformanceTask_U1_TE
... to greatest number. Explain why the list changes from problem 3. ________________________________________________________________________________________ ________________________________________________________________________________________ ...
... to greatest number. Explain why the list changes from problem 3. ________________________________________________________________________________________ ________________________________________________________________________________________ ...
SummerLecture15.pdf
... could have a some maximum or some minimum value (or even some constant value). In the case of the plane, we know it begins its flight by increasing altitude, so we know there is some maximum. What if the plane lands at 11:30 AM and takes off again at 11:45 AM? Do we know there is some maximum (or mi ...
... could have a some maximum or some minimum value (or even some constant value). In the case of the plane, we know it begins its flight by increasing altitude, so we know there is some maximum. What if the plane lands at 11:30 AM and takes off again at 11:45 AM? Do we know there is some maximum (or mi ...
DEPARTMENT OF MATHEMATICS
... Determine which of the following maps are homomorphisms. If the map is homomorphism describe its kernel. a. : Z R under addition given by (n) = n. b. : Z R under addition given by = the greatest integer x. c. : Z6 Z2 given (x) = the remainder of x when divided by 2. ...
... Determine which of the following maps are homomorphisms. If the map is homomorphism describe its kernel. a. : Z R under addition given by (n) = n. b. : Z R under addition given by = the greatest integer x. c. : Z6 Z2 given (x) = the remainder of x when divided by 2. ...
standard error of M - University of Guelph
... What is the distribution of sample means? the set of all Ms for all possible random samples for sample size n for a given population. 1) shape: population must be N or n>30 2) central tendency: M = ...
... What is the distribution of sample means? the set of all Ms for all possible random samples for sample size n for a given population. 1) shape: population must be N or n>30 2) central tendency: M = ...
Law of large numbers
In probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials should be close to the expected value, and will tend to become closer as more trials are performed.The LLN is important because it ""guarantees"" stable long-term results for the averages of some random events. For example, while a casino may lose money in a single spin of the roulette wheel, its earnings will tend towards a predictable percentage over a large number of spins. Any winning streak by a player will eventually be overcome by the parameters of the game. It is important to remember that the LLN only applies (as the name indicates) when a large number of observations are considered. There is no principle that a small number of observations will coincide with the expected value or that a streak of one value will immediately be ""balanced"" by the others (see the gambler's fallacy)