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- https://math.libretexts.org/Courses/SUNY_Schenectady_County_Community_College/Professor_Holz'_Topics_in_Contemporary_Mathematics/03%3A_Number_Bases_and_Modular_Arithmetic/3.05%3A_Cryptography/3.5.03%3A_Public_Key_Cryptography\(\begin{array}{llll} & \textbf{Alice} & & \textbf{Bob} \\ \text{Alice and Bob publically} & g=3, p=17 & \text{Common info} & g=3, p=17 \\ \text{share a generator and prime} & & & \\ \text{modulus.} &...\(\begin{array}{llll} & \textbf{Alice} & & \textbf{Bob} \\ \text{Alice and Bob publically} & g=3, p=17 & \text{Common info} & g=3, p=17 \\ \text{share a generator and prime} & & & \\ \text{modulus.} & & & \\ \text{Each then secretly picks a} & n = 8 & \text{secret number} & n=6 \\ \text{number n of their own.} & & & \\ \text{Each calculates } g^n \bmod p & 3^{8} \bmod 17=16 & & 3^{6} \bmod 17=15 \\ \text{They then exchange these} & A=16 & & B=15 \\ \text{resulting values.} & B=15 & & A=16\\ \te…
- https://math.libretexts.org/Bookshelves/Geometry/Geometry_with_an_Introduction_to_Cosmic_Topology_(Hitchman)/02%3A_The_Complex_Plane/2.01%3A_Basic_NotionsThe set of complex numbers is obtained algebraically by adjoining the number i to the set R of real numbers, where i is defined by the property that i^2=−1. We will take a geometric approach and defin...The set of complex numbers is obtained algebraically by adjoining the number i to the set R of real numbers, where i is defined by the property that i^2=−1. We will take a geometric approach and define a complex number to be an ordered pair (x,y) of real numbers.
- https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Gentle_Introduction_to_the_Art_of_Mathematics_(Fields)/01%3A_Introduction_and_Notation/1.01%3A_Basic_SetsIt has been said that “God invented the integers, all else is the work of Man.” This is a mistranslation. The term “integers” should actually be “whole numbers.” The concepts of zero and negative valu...It has been said that “God invented the integers, all else is the work of Man.” This is a mistranslation. The term “integers” should actually be “whole numbers.” The concepts of zero and negative values seem (to many people) to be unnatural constructs. Indeed, otherwise intelligent people are still known to rail against the concept of a negative quantity – “How can you have negative three apples?” The concept of zero is also somewhat profound.
- https://math.libretexts.org/Bookshelves/Analysis/Tasty_Bits_of_Several_Complex_Variables_(Lebl)/00%3A_Introduction/0.01%3A_Motivation_Single_Variable_and_Cauchy's_FormulaIf \(f\) is holomorphic, the derivative in \(z\) is the standard complex derivative you know and love: \[\frac{\partial f}{\partial z} (z_0) = f'(z_0) = \lim_{\xi \to 0} \frac{f(z_0+\xi)-f(z_0)}{\xi} ...If \(f\) is holomorphic, the derivative in \(z\) is the standard complex derivative you know and love: \[\frac{\partial f}{\partial z} (z_0) = f'(z_0) = \lim_{\xi \to 0} \frac{f(z_0+\xi)-f(z_0)}{\xi} .\] That is because \[\begin{align}\begin{aligned} \frac{\partial f}{\partial z} = \frac{1}{2} \left( \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} \right) + \frac{i}{2} \left( \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y} \right) & = \frac{\partial u}{\partial x} + …
- https://math.libretexts.org/Bookshelves/Applied_Mathematics/Math_in_Society_(Lippman)/16%3A_Cryptography/16.05%3A_Public_Key_Cryptography\(\begin{array}{llll} & \textbf{Alice} & & \textbf{Bob} \\ \text{Alice and Bob publically} & g=3, p=17 & \text{Common info} & g=3, p=17 \\ \text{share a generator and prime} & & & \\ \text{modulus.} &...\(\begin{array}{llll} & \textbf{Alice} & & \textbf{Bob} \\ \text{Alice and Bob publically} & g=3, p=17 & \text{Common info} & g=3, p=17 \\ \text{share a generator and prime} & & & \\ \text{modulus.} & & & \\ \text{Each then secretly picks a} & n = 8 & \text{secret number} & n=6 \\ \text{number n of their own.} & & & \\ \text{Each calculates } g^n \bmod p & 3^{8} \bmod 17=16 & & 3^{6} \bmod 17=15 \\ \text{They then exchange these} & A=16 & & B=15 \\ \text{resulting values.} & B=15 & & A=16\\ \te…
- https://math.libretexts.org/Bookshelves/Precalculus/Elementary_Trigonometry_(Corral)/06%3A_Additional_Topics/6.03%3A_Complex_NumbersThere is no real number \(x\) such that \(x^ 2 = −1\). However, it turns out to be useful to invent such a number, called the imaginary unit and denoted by the letter i.
- https://math.libretexts.org/Courses/Las_Positas_College/Math_for_Liberal_Arts/06%3A_Cryptography/6.05%3A_Public_Key_Cryptography\(\begin{array}{llll} & \textbf{Alice} & & \textbf{Bob} \\ \text{Alice and Bob publically} & g=3, p=17 & \text{Common info} & g=3, p=17 \\ \text{share a generator and prime} & & & \\ \text{modulus.} &...\(\begin{array}{llll} & \textbf{Alice} & & \textbf{Bob} \\ \text{Alice and Bob publically} & g=3, p=17 & \text{Common info} & g=3, p=17 \\ \text{share a generator and prime} & & & \\ \text{modulus.} & & & \\ \text{Each then secretly picks a} & n = 8 & \text{secret number} & n=6 \\ \text{number n of their own.} & & & \\ \text{Each calculates } g^n \bmod p & 3^{8} \bmod 17=16 & & 3^{6} \bmod 17=15 \\ \text{They then exchange these} & A=16 & & B=15 \\ \text{resulting values.} & B=15 & & A=16\\ \te…
- https://math.libretexts.org/Bookshelves/Geometry/Geometry_with_an_Introduction_to_Cosmic_Topology_(Hitchman)/08%3A_Cosmic_Topology/8.01%3A_Section_1-Euclidean geometry is the geometry of our experience in three dimensions. Planes look like infinite tabletops, lines in space are Euclidean straight lines. Any planar slice of 3-space inherits two-dim...Euclidean geometry is the geometry of our experience in three dimensions. Planes look like infinite tabletops, lines in space are Euclidean straight lines. Any planar slice of 3-space inherits two-dimensional Euclidean geometry. The Poincaré disk model of hyperbolic geometry may also be extended to three dimensions. Three-dimensional elliptic geometry is derived from the fact that the 3-sphere consists of all points in 4-dimensional space one unit from the origin.
- https://math.libretexts.org/Courses/Mount_Royal_University/Higher_Arithmetic/3%3A_Modular_Arithmetic/3.2%3A_Modulo_ArithmeticLet \( a, b, c,d, \in \mathbb{Z}\) such that \(a \equiv b (mod\,n) \) and \(c \equiv d (mod\, n). \) Then \((a+c) \equiv (b+d)(mod\, n).\) Let \(a, b, c, d \in\mathbb{Z}\), such that \(a \equiv b (mod...Let \( a, b, c,d, \in \mathbb{Z}\) such that \(a \equiv b (mod\,n) \) and \(c \equiv d (mod\, n). \) Then \((a+c) \equiv (b+d)(mod\, n).\) Let \(a, b, c, d \in\mathbb{Z}\), such that \(a \equiv b (mod\, n) \) and \(c \equiv d (mod \,n). \) Let \( a, b, c,d, \in \mathbb{Z}\) such that \(a \equiv b (mod \, n) \) and \(c \equiv d (mod \,n). \) Then \((ac) \equiv (bd) (mod \,n).\) Let \(a, b, c, d \in \mathbb{Z}\), such that \(a \equiv b (mod\, n) \) and \(c \equiv d (mod \, n). \)
- https://math.libretexts.org/Courses/Cosumnes_River_College/Math_373%3A_Trigonometry_for_Calculus/09%3A_Polar_Coordinates_and_Complex_Numbers/9.03%3A_Complex_NumbersThis section introduces complex numbers, covering their standard form + , where is the imaginary unit. It explains operations with complex numbers, including addition, subtraction, multiplication...This section introduces complex numbers, covering their standard form + , where is the imaginary unit. It explains operations with complex numbers, including addition, subtraction, multiplication, and division. The section also covers how to represent complex numbers graphically in the complex plane and discusses the polar form of complex numbers, including how to convert between rectangular and polar forms. Practical examples and exercises reinforce these concepts.
