x̄ - > Statistics, Applied mathematics and complex analysis

 

Statistics

Descriptive statistics

mean | {21.3, 38.4, 12.7, 41.6} = 28.5

ToeplitzMatrix[{21.3, 38.4, 12.7, 41.6}]

(21.3 | 38.4 | 12.7 | 41.6

38.4 | 21.3 | 38.4 | 12.7

12.7 | 38.4 | 21.3 | 38.4

41.6 | 12.7 | 38.4 | 21.3)

Dimensions - >4 (rows) × 4 (columns)

Properties -> symmetric, Toeplitz

Trace -> 85.2

Determinant - > -358459.

Inverse -> (0.0533794 | 0.0186717 | -0.0633426 | -0.00119043

0.0186717 | 0.059884 | -0.00489768 | -0.0633426

-0.0633426 | -0.00489768 | 0.059884 | 0.0186717

-0.00119043 | -0.0633426 | 0.0186717 | 0.0533794)

Characteristic polynormial - > λ^4 - 85.2 λ^3 - 3754.68 λ^2 + 81200.5 λ - 358459.

Elgen value 

λ_1 = 1/10 (613 + sqrt(261377))

λ_2 = 1/10 (-187 - sqrt(66305))

λ_3 = 1/10 (613 - sqrt(261377))

v_4 = (-1, 1/257 (-16 - sqrt(66305)), 1/257 (16 + sqrt(66305)), 1)

Diagonalization

M = S.J.S^(-1)

where

M = (21.3 | 38.4 | 12.7 | 41.6

38.4 | 21.3 | 38.4 | 12.7

12.7 | 38.4 | 21.3 | 38.4

41.6 | 12.7 | 38.4 | 21.3)

S = (-1 | -1 | 1 | 1

0.939679 | -1.06419 | -1.0318 | 0.969179

-0.939679 | 1.06419 | -1.0318 | 0.969179

1 | 1 | 1 | 1)

J = (-44.4498 | 0 | 0 | 0

0 | 7.04976 | 0 | 0

0 | 0 | 10.175 | 0

0 | 0 | 0 | 112.425)

S^(-1) = (-0.265534 | 0.249517 | -0.249517 | 0.265534

-0.234466 | -0.249517 | 0.249517 | 0.234466

0.242176 | -0.249878 | -0.249878 | 0.242176

0.257824 | 0.249878 | 0.249878 | 0.257824)

Condition number - 16.7347

Statistical inference

The sample size for estimating a binomial parameter

n = ((erf^(-1)(c))/(sqrt(2) M))^2 | 

n | sample size

M | margin of error

c | confidence level

the margin of error | 0.1

confidence level | 0.95

sample size | 96.04

T-interval for a population mean | 

sample mean | 4.15

sample standard deviation | 0.32

sample size | 100

confidence level | 0.95

95 % confidence interval. 4.087 to 4.213

x^_ ± (t_((1 - c)/2) s)/sqrt(n) = 4.15 ± 0.0634949 | 

n | sample size

s | sample standard deviation

x^_ | sample mean

c | confidence level

Regression analysis

fit | data | {{1.3, 2.2}, {2.1, 5.8}, {3.7, 10.2}, {4.2, 11.8}}

model | linear function

Least square best fit - > -1.52256 + 3.19383 x

AIC | BIC | R^2 | adjusted R^2

10.4041 | 8.56296 | 0.989771 | 0.984657

Random Variables

E(-7 + 3 X^4) where 

X distributed Poisson distribution | mean | μ = 7.3

16655.8

probability of A | 0.4

probability of B | 0.6

probability of A intersection B | 0.2

event | probability

P(A conditioned B) | 0.333333

P(B conditioned A) | 0.5

P(A intersection B) = P(A conditioned B) P(B)

P(A intersection B) = P(B conditioned A) P(A)

Applied Mathematics 

Game theory

tic-tac-toe (mathematical game) Assuming it is a mathematical instead of a board game

Two players alternately place pieces (typically X's for the first player and O's for the second) on a 3 × 3 board. The first player to get three matching symbols in a row (vertically, horizontally, or diagonally) is the winner. If all squares are occupied but neither player has three symbols in a row, the game is a tie.

naughts and crosses | three-in-a-row | ticktacktoe | wick wack woe

Payoff matrix

 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9

1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0

2 | 0 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 0

3 | 0 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 1

4 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 1     

5 | 0 | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 1 

6 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 1

7 | 0 | 1 | 1 | 1 | 1 | 1 | 0 | 1 | 1

8 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 0

9 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0

(wherein the first matrix, each player would prefer winning over tying, tying over losing, and losing over crashing; in the second matrix, the benefit of winning is set to 1, the cost of losing is set to -1, and the cost of crashing is assumed to be -10)

fair | finite | futile | perfect information | sequential | two-player | zero-sum

chicken (mathematical game)

Two drivers travel in opposite directions on a collision course toward one another so that if at least one does not swerve, a collision will occur and both may be killed. However, if one driver swerves but the other does not, the swerving driver loses the game and is termed a "chicken, " i.e., coward.

hawk-dove | snowdrift

anticoordination | imperfect information | nonsequential | nonzero-sum | two-player

Fractals

Julia set -0.40+0.65i

Julia set | 

Re(c) | -0.4

Im(c) | 0.65

z_(n + 1) = z_n^2 + c | z_0 = z

(the Julia set is the boundary of the set of z element C for which the orbit of z_n is bounded)

Topological property

Julia set is totally disconnected

c = -0.4 + 0.65 i is not in the Mandelbrot set

packing and covering

estimate | number of baseballs | to fill | interior volume of a Boeing 747

container | Boeing 747

idealized shape | circular cylinder

interior volume | 1753 m^3

object | baseball

idealized shape | sphere

volume | 219 cm^3

packing density | (0.56 to 0.64)

Dynamical systems

logistic map | 

parameter r | 3.56994

initial condition x_0 | 0.1

logistic map | 

parameter r | 3.56994

initial condition x_0 | 0.1

Numerical analysis

solve x cos(x) = 0 using Newton's method to machine precision

x_(n + 1) = x_n - (x_n cos(x_n))/(cos(x_n) - x_n sin(x_n))

x = -1.187250704641182×10^-16

(using the starting point of x_0 = -0.017)

4th order iteration

x_(n + 1) = (x_n^2 (-6 x_n^4 sin^5(x_n) + 1/16 x_n^3 (74 cos(x_n) - 55 cos(3 x_n) + 29 cos(5 x_n)) - 12 sin(x_n) cos^4(x_n) + 14 x_n^2 (sin(3 x_n) - 2 sin(x_n)) cos^2(x_n) + 6 x_n sin^3(2 x_n) csc(x_n)))/(6 (cos(x_n) - x_n sin(x_n))^5)

3 steps to machine precision

solve y'(x) = -2 x^3 y(x)

y(1) = 5 using Euler method from x = 1 to 10

step | x | y | local error | global error

0 | 1 | 5 | 0 | 0

⋮ | ⋮ | ⋮ | ⋮ | ⋮

10 | 10 | 3.62614×10^21 | 0 | -3.62614×10^21

Optimization

Global maximum

max{x (1 - x) e^x} = (sqrt(5) - 2) e^(1/2 (sqrt(5) - 1)) at x = sqrt(5)/2 - 1/2

Complex Analysis

e^z

periodicity - > periodic in z with period 2 i π

1 + z + z^2/2 + z^3/6 + z^4/24 + O(z^5) 

(Taylor series) with Big- O - notation

Indefinite integral - >  integral e^z dz = e^z + constant

Definite integral - > integral_(-∞)^0 e^z dz = 1

lim_(z->-∞) e^z = 0 

Alternative represantation e^z = ζ^z for ζ = e

e^z = 1 + 2/(-1 + coth(z/2)) coth is the hyperbolic cotangent function

(1 + z)^a = ( integral_(-i ∞ + γ)^(i ∞ + γ) (Γ(s) Γ(-a - s))/z^s ds)/((2 π i) Γ(-a)) for (0<γ<-Re(a) and abs(arg(z))<π) 

γ - > 

Γ(s) gamma function

Re(a) Real part of Z

(arg(z) Complex argument of Z

i imaginary unit

x̄ - > Elementary math, Algebra, Calculus, Geometry and Topology

Elementary Math

Arithmetic

125 + 375 = 500

Number line

Fractions 

1/6 + 5/12 + 3/4

Pie chart

percent

convert 1/6 to percent

16.67%

Place values

place values of 6135

Number type arithmetic

1 + (even number * odd number) = odd integer

negative integer/positive integer = negative rational number

rational number + positive even = rational number

Word problems

Rachel has 17 apples. She gives 9 to Sarah. How many apples does Rachel have now?

Rachel has 17 apples.

Rachel gives 9 apples to Sarah.

How many apples does Rachel have?

Algebra 

Equation solving

solve x^2 + 4x + 6 = 0

x^3 - 4x^2 + 6x - 24 = 0  

(x - 4) (x^2 + 6) = 0

(x - 4/3)^3 + 2/3 (x - 4/3) - 560/27 = 0

Polynomials

x^3 + x^2 y + x y^2 + y^3

Rational functions

Properties as a real function

surjective onto R

Domain R (all real numbers)

Range R (all real numbers)

Parity all are even

Definite integrals

integral_(-1)^1 (-1 + x^2)/(1 + x^2) dx = 2 - π≈-1.14159

Simplifications

1/(1+sqrt(2))

simplify | x^5 - 20 x^4 + 163 x^3 - 676 x^2 + 1424 x - 1209

x (x (x ((x - 20) x + 163) - 676) + 1424) - 1209

Matrices

{{0,-1},{1,0}}.{{1,2},{3,4}}+{{2,-1},{-1,2}}

{{0, -1}, {1, 0}} . {{1, 2}, {3, 4}} + {{2, -1}, {-1, 2}}  = {{-1, -5}, {0, 4}}

Trace = 3

Determinant = -4

Inverse is  1/4(-4 | -5

0 | 1)

Characteristic polynomial

λ^2 - 3 λ - 4

Elgen values

λ_1 = 4

λ_2 = -1

Diagonalization

M = S.J.S^(-1)

where

M = (-1 | -5

0 | 4)

S = (1 | -1

0 | 1)

J = (-1 | 0

0 | 4)

S^(-1) = (1 | 1

0 | 1)

Calculus

Integrals  -  integrate sin x dx from x=0 to pi

integral_0^π sin(x) dx = 2

 

integrate x^2 sin^3 x dx

integral x^2 sin^3(x) dx = 1/108 (-81 (x^2 - 2) cos(x) + (9 x^2 - 2) cos(3 x) - 6 x (sin(3 x) - 27 sin(x))) + constant

Derivatives   - a derivative of x^4 sin x 

d/dx(x^4 sin(x)) = x^3 (4 sin(x) + x cos(x))

Sequences - sequence of Fibonacci numbers, 

the sequence in which each term is the sum of the two previous terms with F_0 = 0, F_1 = 1, F_n = F_(n - 1) + F_(n - 2)

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, ...

Limits - lim_(x->0) (sin(x) - x)/x^3 = -1/6 

Sums - sum j^2, j=1 to 100

Products - product (k+2)/k, k=1..25

product_(k=1)^25 (k + 2)/k = 351

Series expansions - Taylor series sin x  

x - x^3/6 + x^5/120 + O(x^7)

(Taylor series)

vector Analysis - grad sin(x^2 y) = (2 x y cos(x^2 y), x^2 cos(x^2 y))

(x: first Cartesian coordinate | y: second Cartesian coordinate)

Application calculus - compute the area between y=|x| and y=x^2-6 

area between | y = abs(x)

y = x^2 - 6area between | y = abs(x)

integral_(-3)^3 (6 - x^2 + abs(x)) dx = 27

Integral transforms - 

Domain and Range - domain of f(x) = x/(x^2-1)

{x element R : x!=-1 and x!=1}

(assuming a function from reals to reals) 

del sin(x^2 y)

Continuity - is y = sin(x - 1.1)/(x - 1.1) + θ(x) continuous?

y = sin(x - 1.1)/(x - 1.1) + θ(x) is not continuous on its domain

(assuming a function from reals to reals)

{x element R : x!=11/10}

Geometry and Topology 

Plane geometry

annulus, inner radius=2, outer radius=5

5, 12, 13 triangle

Solid geometry

Coordinate geometry

line through (1,2) and (2,1)

Distance; from (1, 2) to (2, 1): sqrt(2)≈1.41421d

Midpoint; 

Geometric transformations

rotate 30 degrees

Rotation matrix

Moire patterns

angle between grids | 4° (degrees)

shift between grids | 0

CSC(X) is a cosecant function.

Polyforms

a plane geometric figure made by joining equal squares along common edges

order | count

1 | 1

2 | 1

3 | 2

4 | 5

5 | 12

a plane geometric figure made by joining 6 equal equilateral triangles along common edges

type | count

ignoring orientation (1-sided) | 19

including orientation (2-sided) | 12

  (equilateral triangle)

Topology

packing and covering problems

Curves and surfaces

Tilings

x̄ - > How to Address Food Insecurity and Poverty in Your Community

Sustainability is key to addressing food insecurity and poverty in any community. We must strive to create solutions that are not only effective in the short term but also have a lasting, long-term impact. To do this, we must prioritize initiatives that focus on education, job training, and access to resources and services that can empower those living in poverty and struggling with food insecurity. By understanding the unique needs of each community and implementing strategies to meet those needs, we can begin to see real and lasting change in our communities. 

 The link between food insecurity and poverty

 Food insecurity and poverty are closely intertwined and often go hand-in-hand. People who lack access to enough food for an active, healthy life may find it difficult to participate in their communities or find a job due to their limited resources. Without access to a steady income, it becomes increasingly difficult for people to secure the necessary resources to keep themselves nourished. This can lead to further social issues such as homelessness and health problems. Food insecurity can also lead to an increased risk of mental health issues, due to the stress and anxiety caused by being unable to provide food for oneself and their family. On the other hand, poverty can also lead to food insecurity due to its effect on a person’s ability to purchase food. People living in poverty are more likely to lack access to nutritious food since their limited resources mean they may have to choose between buying fresh produce and paying rent or bills. Thus, food insecurity and poverty are interdependent, reinforcing one another and creating a cycle of deprivation and instability. 

 The scale of the problem 

 Food insecurity and poverty are global issues that affect billions of people across the world. According to the United Nations Sustainable Development Goals, 821 million people are suffering from chronic hunger. That number is made up of both adults and children who lack access to sufficient amounts of nutritious and safe food to live a healthy life. This is a major challenge to the global community and requires collective action to tackle the underlying causes of poverty and food insecurity. The World Bank estimates that over a third of the world’s population lives in extreme poverty and that over 700 million people live in households that are unable to obtain their daily calorie requirements. This shows how food insecurity and poverty are intricately linked, with both issues having devastating consequences for individuals and families around the world. 

 The causes of food insecurity and poverty 

 Food insecurity and poverty are complex issues that can be attributed to a number of different causes. Poverty can be caused by a lack of access to education, jobs, and health care services. It can also be caused by unequal distribution of resources, like land, within a community. This unequal access to resources has been linked to higher rates of malnutrition and stunting in children. In addition, extreme weather events, such as droughts, floods, and storms can have devastating impacts on food production, leading to further economic hardship. Climate change is likely to increase the intensity and frequency of these events, making it more difficult for communities to support themselves. Finally, the globalization of markets has had a profound impact on food production and access. International trade agreements, subsidies, and export taxes can make food more expensive or inaccessible in certain countries. The UN Sustainable Development Goals (SDGs) provide a framework for addressing the challenges posed by food insecurity and poverty. The goals focus on improving access to healthcare services, quality education, job opportunities, infrastructure, and technological innovations to improve food security and reduce poverty. They also emphasize the need for governments and communities to work together to conserve natural resources and build resilient ecosystems. These goals aim to ensure that all people have access to safe and nutritious food, regardless of where they live. 

 The effects of food insecurity and poverty 

 Food insecurity and poverty have devastating effects on communities across the world. Not having access to enough food can lead to malnourishment, health problems, stunted physical and mental growth, and poverty-related stress. Poverty further amplifies these effects by preventing individuals from obtaining adequate nutrition and basic services. It is also associated with poor educational outcomes and a decreased capacity to participate in social, economic, and political activities. The lack of access to basic needs also has broader implications for society as a whole. Food insecurity and poverty contribute to greater social and economic inequality, which can result in greater stress, instability, and conflict in communities. As such, these issues threaten our collective ability to achieve sustainable development goals. In order to ensure that all individuals have access to basic needs and have a chance to lead dignified lives, we must find ways to address food insecurity and poverty in our communities. 

 What can be done to address food insecurity and poverty? 

 The Sustainable Development Goals (SDGs) set out by the United Nations provide a framework for addressing food insecurity and poverty. These goals include ending hunger and poverty, improving nutrition, promoting sustainable agriculture, creating jobs and economic opportunities, strengthening social protection, and ensuring access to financial services and education. At the local level, governments and NGOs can work to reduce food insecurity and poverty through targeted policies and programs. These initiatives could include: providing cash transfers to low-income households; introducing safety nets such as free school meals, job creation schemes, and public works programs; investing in sustainable farming methods to increase crop yields and improve resilience to climate change; and, advocating for more resources to tackle poverty. On an international level, it is important to promote fair trade and eradicate harmful subsidy programs that perpetuate poverty. We must also build capacity in developing countries, particularly in rural areas, to ensure that people have access to education and vocational training opportunities. Ultimately, we need to create an environment where everyone has access to the means of sustenance, which is only possible with a shared commitment to achieving the SDGs. When we create systems of equitable growth and opportunity, we create a better future for everyone.

x̄ - > war and peace Tolstoy Napoleonic period story

The Rostovs and Bolkonskys have a rough and tumble relationship, marked by periods of war, peace, and civil unrest. While the Khersons are peaceful, prosperous, and happy to pass the day in their warm home. The War of 1812 came to be one of the greatest events in Russian and European history. Four families were united on the battlefield. The Rostov and Bolkonsky families have long been enemies, who have little in common except for a common enemy. Same with everyone else. However, during the war all these families managed to reach an agreement and unite against their common enemy Napoleonic Wars was a time of great change, when many countries were unified under one emperor. It brought together four different families of the nobility and helped bring stability to Europe. Napoleonic Wars were a series of wars in Europe between the years 1803 and 1815. The wars took place in many countries and mainly involved France, Great Britain, and their coalitions against a variety of enemies.

The Rostovs and Bolkonskys have had a turbulent history. It is known for its extended periods of combat, peace, and civil unrest. On the other hand, the Khersons are peaceful, prosperous, and happy to pass the day in their warm home.

Rostov and Bolkonsky are intended to be worn together creating a rich, complex color palette. Kherson is a solo act that compliments one, bold statement piece.

The relationship between the Rostovs and Bolkonskys can best be described as tumultuous. The Rostovs are a family of high society while the Bolkonskys live in more rural conditions.

The two families carry a long history of both warm and cold relations with one another.

x̄ - > The Power of Dialogue: How to Use it to Enhance Your Writing

The Power of Dialogue: How to Use it to Enhance Your Writing 

 Introduction 

 In order to improve your writing, it is important to understand the different roles that collaboration and dialogue can play. When used effectively, they can enhance your writing in a number of ways. For example, collaboration can help you to develop new ideas, get feedback on your work, and learn from other writers. Dialogue, on the other hand, can be used to add realism to your writing, create suspense, and advance the plot. In this article, we will explore the power of dialogue and collaboration, and how you can use them to enhance your writing. We will also provide some tips on how to get the most out of these tools. 

The Basics of Writing Dialogue: Writing dialogue is an important part of any story. It allows the reader to experience conversations and conversations between characters and is a powerful way to convey emotions and feelings. In order for dialogue to be effective, you must understand the basics of how to write it. Firstly, it is essential to know the context of the conversation. Knowing the setting and the characters involved can help you decide on the type of language and tone to use in the dialogue. This will help create a more believable and powerful experience for the reader. You should also always strive to capture the true essence of speech. This means avoiding overly long sentences, using contractions and slang appropriately, and including pauses and awkward silences. 

Punctuating Dialogue Correctly: In order to write effective dialogue, you must punctuate it correctly. This means using quotation marks, commas, and other punctuation correctly. Quotation marks should always be used when writing dialogue so that the reader knows when a character is speaking. This will ensure that the dialogue does not blend in too much with the rest of the text. Commas should also be used appropriately. This means placing them before and after a character’s dialogue, as well as when introducing a character’s dialogue. For example, if John said, “I can't believe it,” the correct way to punctuate it would be, “I can’t believe it,” John said. Finally, you should ensure that you are using proper grammar when writing dialogue. This includes avoiding mistakes such as misplaced modifiers, verb tenses, and sentence fragments. 

How to Use Dialogue to Enhance Your Writing: Dialogue can be used to great effect to enhance your writing. It can be used to provide exposition and reveal backstories, introduce new characters, and show relationships between characters. It can also be used to develop suspense and mystery, as well as to introduce humor and illustrate certain scenes. For example, if you were writing a suspenseful scene, you could use dialogue as a tool to create tension. This can be done by introducing a character and then having them talk about something unsettling. Alternatively, if you wanted to create humor, dialogue can be used to showcase the characters’ quirks and personalities.

The Power of Subtext in Dialogue: Another way to enhance your dialogue is to use subtext. The subtext is the hidden meaning behind spoken words and can have a powerful impact on the reader. By using subtext you can add depth and complexity to conversations, as well as reveal hidden aspects of characters. For example, if you wanted to illustrate a character’s bitterness, you could have them say something simple like, “It’s been a long day”. By adding a degree of subtext to this line, you can quickly convey the character’s exhaustion and frustration. Wrapping Things Up – Some Final Tips on Writing Dialogue: Writing dialogue is a skill that takes time to develop, but with practice, you can create powerful and realistic conversations. In order to get the most out of dialogue, you should always strive to create believable characters, use subtext to add depth, vary the pace of conversations, and punctuate correctly. Finally, when it comes to collaboration and dialogue, don’t be afraid to lean on others. Whether you’re looking for feedback on your work or just need a fresh perspective, collaboration and dialogue can be incredibly powerful tools to help you create something truly special.

x

x̄ - > Demoivre's Theorem

Demoivre's Theorem: A Historical Overview 
 

Introduction 

 In mathematics, Demoivre's theorem is a statement about integer roots of complex numbers. It is named after Abraham de Moivre, who proved it in 1707. The theorem is as follows: If z is any complex number such that z^n = 1 for some positive integer n, then z is an nth root of unity. This theorem has a number of applications in number theory, algebra, and complex analysis. It is also the basis for a number of results in these fields, including the arithmetic-geometric mean and the fundamental theorem of algebra. 

 William Demoivre: A Life in Mathematics 

 Abraham De Moivre was born in France in 1667. He studied at the Royal Academy in Paris and later moved to England, where he worked as an astronomer, actuary, and mathematician. He is best known for his work on probability, but he also made significant contributions to the fields of geometry, trigonometry, and algebra. The most significant result of De Moivre's life was the discovery of Demoivre's theorem. Before he arrived at this statement, he had to experiment with a variety of mathematical concepts and ideas. He was particularly interested in complex numbers and their use in trigonometry and calculus. He studied the polar form of complex numbers, deriving a number of important results, including the theorem of addition, subtraction, multiplication, and division for complex numbers. He was also the first to consider complex exponentials, which would later become a part of the core of the complex analysis. Early Work on Complex Numbers Before De Moivre arrived at his theorem, he had to answer the fundamental problem of how to make sense of negative numbers. For centuries, physicists and mathematicians alike had posed the same question. It was De Moivre's insight and ingenuity which allowed him to finally give a satisfactory answer. His approach was to treat negative numbers as points on a line or circle and to use an analogous trigonometric representation. This meant that a negative number could be represented as a point on the 360-degree circle, with the position on the circle determined by the magnitude of the number. This allowed De Moivre to represent the operations on negative numbers as rotations and translations on the circle. This was a breakthrough in mathematics, as it allowed complex numbers and their operations to be viewed geometrically. It was this insight that formed the basis for his subsequent work on complex numbers and which eventually led to the statement of his theorem. 

 

Making Sense of Negative Numbers 

 Once De Moivre had gained an understanding of how to treat negative numbers, he was able to begin to formulate a statement that would eventually become known as Demoivre's Theorem. This statement included the concept of an “nth root” of a complex number and the idea of a “cycle” of such roots. De Moivre's theorem stated that if “z” is any complex number such that z^n = 1 for some positive integer “n”, then z is an “nth” root of unity. This theorem was remarkable at the time, as it gave a description of how the root of any given complex number could be determined. The theorem also provided an algebraic solution to the problem of finding the roots of a polynomial with a negative coefficient. The Significance of Euler's Formula De Moivre's theorem was also significant for its implications for Leonhard Euler's famous formula, e^iθ = cosθ + i sinθ. This equation states that for any real angle θ, the complex number e^iθ is equal to cos θ + i sin θ. Therefore, by combining Euler's formula and De Moivre's theorem, it is possible to express the roots of a complex number in terms of its angle. This discovery was revolutionary at the time, as it allowed complex numbers and complex exponentials to be formulated in terms of trigonometric functions. This opened up a new range of possibilities, as it allowed complex numbers to be manipulated more easily. 

 Demoivre's Theorem and the Binomial Theorem 

 De Moivre's theorem is closely related to the binomial theorem, as the roots of a polynomial equation can be expressed in terms of the binomial coefficients. This can be done using the formula (a+b)^n = a^n + binomial coefficients. The theorem can also be used to prove the arithmetic-geometric mean, which states that the mean of two numbers is equal to their geometric mean multiplied by their harmonic mean. This result is closely related to the binomial theorem, as it is possible to express the geometric mean and harmonic mean in terms of binomial coefficients. A final thought Demoivre's theorem has been of immense benefit to mathematics, as it provided a way to treat complex numbers with greater ease and accuracy. This theorem is closely related to other fundamental mathematical results, such as the fundamental theorem of algebra, the binomial theorem, and the arithmetic-geometric mean. Therefore, Demoivre's theorem is arguably one of the most significant tragedies of mathematics ever formulated. 

 Example

(√ 3 + i)^5

To use DeMoivre's Theorem, we need to convert 

 z = − √ 3 + i = x + i y into Polar Form, i.e., r ( cos θ + i sin θ ), where, r > 0, &, θ ∈ ( − π, π ]. z = x + i y = r ( cos θ + i sin θ ) = r c i s θ ⇒ x = r cos θ , y = r sin θ , r = √ x 2 + y 2 ∴ r = √ ( − √ 3 ) 2 + 1 2 = 2 Hence, from x = r cos θ , cos θ = − √ 3 2 , &, similarly, sin θ = 1 2 clearly , θ = 5 π 6 Thus, − √ 3 + i = 2 c i s ( 5 π 6 ) . Now, by DeMoivre's Theorem, ( r c i s θ ) n = r n c i s ( n θ ) In our case, n = 5, r = 2, θ = 5 π 6 ∴ ( − √ 3 + i ) 5 = ( 2 c i s 5 π 6 ) 5 = ( 2 5 ) ( c i s ( 5 ⋅ 5 π 6 ) ) = 32 ( c i s 25 π 6 ) = 32 ( c i s ( 4 π + π 6 ) ) = 32 ( cos ( 4 π + π 6 ) + i sin ( 4 π + π 6 ) ) = 32 ( cos ( π 6 ) + i sin ( π 6 ) ) = 32 ( √ 3 2 + 1 2 i ) = 16 ( √ 3 + i ).