Free Essay

God's Equation

In: Other Topics

Submitted By mohit476
Words 525
Pages 3
God’s Equation
Being one the most astounding formulas in mathematics, Euler’s identity is popularly called God’s equation. Some people also go as far as calling it the mathematical equivalent of Da Vinci's Mona Lisa or Michaelangelo's David.
Named after Leonhard Euler, the formula establishes the deep relationship between trigonometric functions and complex exponential functions.

According to the formula, for any real number x,

In the above formula, e is the base of the natural logarithm, i the imaginary unit. Cos and Sin are trigonometric functions (the arguments are to be taken in radians and not degrees.). The formula applies even if x is a complex number.

Particularly with x = π, or half a turn around the circle,

e^iπ = cos π + i sin π

Since cos π = -1 and sin π = 0,

It can be deduced that e^iπ = -1 + i0 which brings us to the identity

The identity successfully links five fundamental mathematical constants: 1. The number 0(the additive identity). 2. The number 1(the multiplicative identity). 3. The number pi (3.14159265…). 4. The number e (base of all natural logarithms, which occurs widely in mathematics and scientific analysis). 5. The number i (the imaginary unit of the complex numbers)

The formula describes two equivalent ways to move in a circle.
One of its major applications is that in the complex number theory.

The interpretation of the function eix can be that it traces out the unit circle in the complex number plane while x ranges through the real numbers. x in this case refers to the angle that any line that connects the origin with any point on the circle makes with the positive real axis (being measured in radians counter clockwise).

Points in the complex plane are represented by complex numbers that are written in cartesian coordinates. Euler’s formula provides a means of conversion between polar coordinates and cartesian coordinates. The number of terms is reduced from two to one by polar form, which simplifies the mathematics for multiplication or powers of complex numbers. Therefore any complex number, z = x + iy can be written as

In the above, x = Re {z} the real part y = Im{z} the imaginary part the magnitude of z ø = arg z = atan2(y , x) in this is the argument of z or in other words is the angle (clockwise and in radians) between x-axis and the vector z.

Using the Euler’s formula on the derived formula, the logarithm of a complex number can be defined. For this we need to take the definition of the logarithm (as an inverse operator of exponentiation):

and

both being valid for any complex numbers a and b.

Therefore it can be written that

for any number z ≠ 0. On taking the logarithm on both sides, it can be seen that

This in fact can also be used to define the complex logarithm. The logarithm of a complex number is therefore a multi valued function as φ is multi valued.

Finally, the other exponential law

which can be seen to hold for all integers k, together with Euler's formula, implies severaltrigonometric identities as well as de Moivre's formula.

Similar Documents

Premium Essay

2 Step Equations

...|Corps |Aileen Liu |CMA: | |Member: | | | | |Objective 18: SWBAT solve word problems with two-step equations |Knowledge: | | | |Certain words in a word problem can clue us in to the mathematical symbols that| | |A cab ride costs $5 for the first mile and $4 for each additional mile.|relate the values that appear in the word problem. | | |Carlo’s cab ride cost $13. | | | |How many miles was the ride? |Skills: | | |A. 2 |Identify the givens | | |B. 3 |Circle all the values | | |C. 4 ...

Words: 2405 - Pages: 10

Premium Essay

Appendix C

...navigate some of the issues facing bakery owners. Application Practice Answer the following questions. Use Equation Editor when writing mathematical expressions or equations. First, save this file to your hard drive by selecting Save As from the File menu. Click the white space below each question to maintain proper formatting. 1. You have recently found a location for your bakery and have begun implementing the first phases of your business plan. Your budget consists of an $80,000 loan from your family and a $38,250 small business loan. These loans must be repaid in full within 10 years. a) What integer would represent your total budget? The integer is 118,250 b) Twenty-five percent of your budget will be used to rent business space and pay for utilities. Write an algebraic expression that indicates how much money will be spent on business space and utilities. Do not solve. .25(118,250)=u c) How much money will rent and utilities cost? Explain how you arrived at this answer. I solved this problem by converting 25% into a decimal and multiplying it by the total loan amount. d) Suppose an investor has increased your budget by $22,250. The investor does not need to be repaid. Rather, he becomes part owner of your business. Will the investor contribute enough money to meet the cost of rent and utilities? Support your answer, and write an equation or inequality that illustrates your answer. No it does not meet the cost of rent and utilities, it is......

Words: 908 - Pages: 4

Free Essay

Qlt 1 Task 1

...Mike Henkle QLT1 Task 5 Western Governors University A. Create a story problem using one of the above real-world scenarios as a basis, including realistic numeric values, by doing the following: 1. Describe the real-world problem. I was looking into phone plans and stumbled upon T-Mobile, and I decided that I needed a cell-phone and took a look at the plans. T-mobile had one plan that was 50 dollars a month and is unlimited talk, text and web, T-Mobile also has a plan for 30 dollars a month for 1,500 talk and text minutes. After you go through your allotted 1,500 talk and text time was up, the cost skyrockets up to 10 cents a minute. 10 cents a minute comes out to 6.00 per hour, I thought in my head. I decided that instead of jumping into a decision about phone plans, that I should first go home and do the math. I wanted to figure out which plan was going to be the most cost effective, and which plan would suite my needs the best. 2. Explain all needs (e.g., financial, non-financial, situational) of the hypothetical consumer. The needs include many different factors: • The first and most important factor how much do you use your phone? The breaking point on the problem today is 200 talk or text above the 1500 minute plan and I will be paying less up front but more on the back end which is no good. If I chose the 1500 minute plan, I will not want to be going over the 1500 minutes as then the cost would go up to 10......

Words: 1163 - Pages: 5

Premium Essay

Wgu Qlt1 Task 5

...due for the week Option A: f * h = t a. 9 * 40 = $360 Option B: 150 + f(h-20) = t, for h >=20 b. 150 + 10(40-20) = $350 150 = t, for h < 20 c. 150 = total II. I set the hours to "h", the hourly fee to "f" and the total to "t". a. Option A: This option only has an hourly fee for each hour. The fee is $9 and the amount of hours is 40 so I multiplied the hours by the fee. b. Option B: This option has a flat rate of $150 for the first 20 hours and then an hourly rate of $10 thereafter. For the first equation that refers to over 20 hours, I subtracted 20 hours from the total time, multiplied it by $10, and then added it to $150. For the second equation that refers to any amount less than 20 hours, the total is always going to be $150. III. 9h = 150 9 9 h = 16.67 Set the equations equal to each other for under 20 hours. Divide by 9 to get h = 16.67 – the daycare centers are equal in price at 16.67 hours. This solution point is at (16.67, 150). 9h = 150+10(h-20) 9h = 150+10h-200 -10h - 10h -1h = 150 - 200 -1h...

Words: 531 - Pages: 3

Free Essay

Mathematics Performance

...EFFICIENCY LEVEL IN SOLVING POLYNOMIAL EQUATIONS AND THEIR PERFORMANCE IN MATHEMATICS OF GRADE 9 STUDENTS A Thesis Presented to the Faculty of the Teacher Education Program Ramon Magsaysay Memorial Colleges General Santos City In Partial Fulfillment of the Requirement for the Degree Bachelor of Secondary Education Major in Mathematics Armando V. Delino Jr. October 2015 TABLE OF CONTENTS Contents Page Title Page i Table of contents ii CHAPTER I THE PROBLEM AND ITS SETTING 1 Introduction 1 Theoretical Framework Conceptual Framework Statement of the problem Hypothesis Significance of the study Scope of the study Definition of terms CHAPTER II REVIEW OF RELATED LITERATURE CHAPTER III METHODOLOGY Research Design Research Locale Sampling Technique Research Instrument Statistical Treatment CHAPTER 1 PROBLEM AND ITS SETTING Introduction In Mathematics, a polynomial is an expression consisting of variables (or indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents. Polynomials appear in a wide variety of areas of Mathematics and Science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated problems in the Sciences; they are used to define polynomial functions, which appear......

Words: 2716 - Pages: 11

Premium Essay

Businesses

...2 2.1 Introduction to Equations 2.2 Linear Equations 2.3 Introduction to Problem Solving 2.4 Formulas 2.5 Linear Inequalities Linear Equations and Inequalities Education is not the filling of a pail, but the lighting of a fire. — WILLIAM BUTLER YEATS M athematics is a unique subject that is essential for describing, or modeling, events in the real world. For example, ultraviolet light from the sun is responsible for both tanning and burning exposed skin. Mathematics lets us use numbers to describe the intensity of ultraviolet light. The table shows the maximum ultraviolet intensity measured in milliwatts per square meter for various latitudes and dates. Latitude 0 10 20 30 40 50 Mar. 21 325 311 249 179 99 57 June 21 254 275 292 248 199 143 Sept. 21 325 280 256 182 127 75 Dec. 21 272 220 143 80 34 13 ISBN 1-256-49082-2 Source: J. Williams, The USA Today Weather Almanac. If a student from Chicago, located at a latitude of 42°, spends spring break in Hawaii with a latitude of 20°, the sun’s ultraviolet rays in Hawaii will be approximately 249 2.5 99 times as intense as they are in Chicago. Equations can be used to describe, or model, the intensity of the sun at various latitudes. In this chapter we will focus on linear equations and the related concept of linear inequalities. 89 Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by...

Words: 32187 - Pages: 129

Free Essay

Qlt1 Task 5

...QLT1 Task 5 A. Create a story problem using one of the above real-world scenarios as a basis, including realistic numeric values, by doing the following: 1. Describe the real-world problem. I was looking into phone plans and stumbled upon T-Mobile, and I decided that I needed a cell-phone and took a look at the plans. T-mobile had one plan that was 50 dollars a month and is unlimited talk, text and web, T-Mobile also has a plan for 30 dollars a month for 1,500 talk and text minutes. After you go through your allotted 1,500 talk and text time was up, the cost skyrockets up to 10 cents a minute. 10 cents a minute comes out to 6.00 per hour, I thought in my head. I decided that instead of jumping into a decision about phone plans, that I should first go home and do the math. I wanted to figure out which plan was going to be the most cost effective, and which plan would suite my needs the best. 2. Explain all needs (e.g., financial, non-financial, situational) of the hypothetical consumer. The needs include many different factors: • The first and most important factor how much do you use your phone? The breaking point on the problem today is 200 talk or text above the 1500 minute plan and I will be paying less up front but more on the back end which is no good. If I chose the 1500 minute plan, I will not want to be going over the 1500 minutes as then the cost would go up to 10 cents a minute or 6.00 per hour. • Is this replacing a work or office phone? ......

Words: 1158 - Pages: 5

Free Essay

Discussion

...Week 2 Discussion Algebra review. Please respond to one of the following questions: •Imagine you are tutoring a classmate in the four (4) algebraic processes. Outline each procedure in the process of solving algebraic equations and then try to use your outline as a guide to teach a follow student. •The author of your textbook indicates, “If you add percents, you often obtain incorrect results.” Explain in your own words what kinds of errors contribute to inaccurate percent results. •Determine what makes solving an equation with two (2) variables different than one (1) variable. Week 4 Discussion Simple and compound interest. Please respond to one of the following questions: •Explain methods for calculating credit card interest and your reason for going with a particular method. •What are the factors associated with math that you need to know about in order to obtain a mortgage loan. List at least five (5) factors and provide a rationale for those you selected. •A bottle of water cost one dollar in Chicago. However, the same bottle of water cost five dollars in the Mojave Desert. Consider the definitions of cost and value and explain the difference as to why the bottle of water in the desert might be more valuable. Week 5 Discussion Set operations and Venn diagrams. Please respond to one of the following questions: •Create a story problem that demonstrates how a Venn diagram could be used to illustrate combined operation with sets. •Give two (2) reasons......

Words: 410 - Pages: 2

Premium Essay

Tree Value

...Overview and examples from Finite Mathematics Using Microsoft Excel® Revathi Narasimhan Saint Peter's College An electronic supplement to Finite Mathematics and Its Applications, 6th Ed. , by Goldstein, Schneider, and Siegel, Prentice Hall, 1997 Introduction In any introductory mathematics course designed for non-mathematics majors, it is important for the student to understand and apply mathematical ideas in a variety of contexts. With the increased use of advanced software in all fields, it is also important for the student to effectively interact with the new technology. Our goal is to integrate these two objectives in a supplement for the text Finite Mathematics and Its Applications, by Goldstein, Schneider, and Siegel. The package consists of interactive tutorials and projects in an Excel workbook format. The software platform used is the Microsoft Excel 5.0 spreadsheet. It was chosen for the following reasons: • • • suited to applications encountered in a finite math course widespread use outside of academia ease of creating reports with a professional look Use of Excel 5.0 was put into effect in the author's sections of the Finite Mathematics II course in the Spring 1996 semester. It was expanded to cover the Finite Mathematics I course for the Fall semester of 1996. Using a combination of specially designed projects and tutorials, students are able to analyze data, draw conclusions, and present their analysis in a professional format. The mathematical...

Words: 3176 - Pages: 13

Premium Essay

A Treasure Hunt at Castle Rock Using Pythagorean Quadratic

...treasure is 2x+6 paces from Castle rock, whereas, Vanessa map says in order to find the treasure, go to Castle Rock, walk x paces to the north and then walk 2x+4 paces to the east. In order to discover the location of the treasure, we need to factor down the three quadratic expressions by putting the measurements into the Pythagorean Theorem. The first thing we need to do is to write an equation by inserting the binomials into the Pythagorean Theorem, which also states that every right triangle with legs of length have the relationship of a^2+b^2=c^2 x^2+ (2x+4)^2=(2x+6)^2 The binomials into the Pythagorean Theorem. x^2(2x+4) (2x+4)=(2x+6) (2x+6) The equation squared. x^2 4x+8x+8x+16=4x^2+12x+12x+36 Equation FOILED or distributed. x^2+4x^2=5x^2 First two terms added. 5x^2 + 16x+16x=4x^2+24x+36 Like terms combined This is a quadratic equation so to solve this we need to first set it equal to zero that means everything must be moved to one side and then we need to factor. Move everything on the right to the left. Subtract 4x^2 from both sides, subtract 24 from both sides and 36 from both sides. 5x^2+16x+16x=4x^2+24x+36 -4x^2...

Words: 720 - Pages: 3

Premium Essay

I'Ve Got a Secret (Need Help)

...Module 4 Project David Hollingsworth MAT 143 11 November 2015 I’VE GOT A SECRET! Learning outcome: Upon completion student will be able to: * Given linear and exponential data, interpret the rate of change within the given context. * Represent linear and exponential models as equations, tables, graphs and verbal descriptions. Scoring/Grading rubric: Each table is worth 10 points and each question is worth 8 points. Introduction: Everyone has had some experience with gossip. In this lab, you will explore how well rumors (or secrets) spread when this information is passed on to other people. Scenario A: At noon, you get some great news but you need to keep it a secret. It’s just too good to keep to yourself; so 5 minutes after you get the news you call 2 friends and tell them, but swear them to secrecy. They understand how important it is to keep a secret, so they don’t tell anybody. After 5 more minutes you call 2 more friends, and again they keep your secret. You are just so excited that after 5 more minutes you make your third set of calls and tell two more friends. Fill in the table below to show how many people are told and how many people know the secret after each 5 minute interval. (10 points) Time | 5 minute interval | Number of people told | Number of people who know | 12:00 | 0 | | 1 | 12:05 | 1 | | | 12:10 | 2 | | | 12:15 | 3 | | | 12:20 | 4 | | | 12:25 | 5 | | | 12:30 | 6 | | | 12:35 |......

Words: 702 - Pages: 3

Free Essay

Accounty

...corporations and individuals to spend less time on the transactions and more time on major decisions and improvements. Modern accounting is not just for businesses. Individuals can input all of their financial information and make important decisions. People can set up spreadsheets to track checking and savings accounts. They can see where their money is actually going. Also, people can set up different spreadsheets to track financial information to even help them decide on what to do with their money. Such as, invest, save, or retire. Modern accounting also reduces costs. Traditional accounting methods use many resources, such as paper and pencil. There are programs out now so people do not have to write all of the equations down. Some programs automatically do the equation. From a company stand point employees are able to finish more in a short amount of time. This means cost reduction when it comes to employee...

Words: 265 - Pages: 2

Free Essay

Mr Drimple

...Systems of Linear Equations in Three Variables Answer the following questions to complete this lab. Show all of your work for each question to get full credit. 1. Solve the following system of equations: a. x –2y+z=6 b. 2x+y –3z= –3 c. x –3y+3z=10 Add equation b and c: 2X+Y-3Z=-3 +X +3Y+3Z=10= 3X-2Y=7 Add equation a and c, attempting to cancel out variable c. (multiply equation a by -3. -3(x-2Y+Z=6): -3X+6Y-3Z=-18 + X-3Y+3Z=10 = -2X+3Y=-8 Add the new equations together to isolate one variable. Multiple each equation by the necessary coefficient to cancel out a variable. 3(3X-2Y=7) =9X-6Y=21 and 2(-2X+3Y=-8)=-4X+6Y=-16 add together 5X=5. X=1 Back substitute X=1 into one of the two variable equations. -2(1)+3Y=-8, 3Y=-6. Y=-2 Back substitute Y=-2 and X=1 into one of the original three variable equations. 2X+Y-3Z=-3 , 2(1)+(-2)-3Z=-3, -3Z=-3 Z=1 Now check you variables by plugging into one of the original equations and making sure statement is true. X-2Y+Z=6. (1)-2(-2)+(1)=6 True. 2. The opening night of a theater sold a total of 1,000 tickets. The front orchestra area cost $80 a seat, the back orchestra area cost $60 a seat, and the balcony area cost $50 a seat. Total revenue from ticket sales for the night was $62,800. The combined number of tickets sold for the front and back orchestra seats was equal to the number of......

Words: 557 - Pages: 3

Premium Essay

Ten Reasons Why People Life Need to Get Better.

...2 2.1 Introduction to Equations 2.2 Linear Equations 2.3 Introduction to Problem Solving 2.4 Formulas 2.5 Linear Inequalities Linear Equations and Inequalities Education is not the filling of a pail, but the lighting of a fire. — WILLIAM BUTLER YEATS M athematics is a unique subject that is essential for describing, or modeling, events in the real world. For example, ultraviolet light from the sun is responsible for both tanning and burning exposed skin. Mathematics lets us use numbers to describe the intensity of ultraviolet light. The table shows the maximum ultraviolet intensity measured in milliwatts per square meter for various latitudes and dates. Latitude 0 10 20 30 40 50 Mar. 21 325 311 249 179 99 57 June 21 254 275 292 248 199 143 Sept. 21 325 280 256 182 127 75 Dec. 21 272 220 143 80 34 13 ISBN 1-256-49082-2 Source: J. Williams, The USA Today Weather Almanac. If a student from Chicago, located at a latitude of 42°, spends spring break in Hawaii with a latitude of 20°, the sun’s ultraviolet rays in Hawaii will be approximately 249 2.5 99 times as intense as they are in Chicago. Equations can be used to describe, or model, the intensity of the sun at various latitudes. In this chapter we will focus on linear equations and the related concept of linear inequalities. 89 Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by...

Words: 32187 - Pages: 129

Free Essay

Pythagorean Quadratic

...treasure is buried in the desert 2x + 6 paces from Castle Rock. Vanessa has the other half of the map. Her half indicates that to find the treasure, one must get to Castle Rock, walk x paces to the north, and then walk 2x + 4 paces to the east. If they share their information, then they can find x and save a lot of digging. What is x? We need to look at the equation so we can know how far Ahmed will have to walk, which is 2x+6 paces from Castle Rock. Even though Vanessa’s half of the map does not indicate in which direction the 2x + 4 paces should go, it can be assumed that her’s and Ahmed’s paces should end up in the same place. When sketched on scratch paper, a right triangle is formed with 2x + 6 being the length of the hypotenuse, and x and 2x + 4 being the legs of the triangle. When a right triangle is involved, the Pythagorean Theorem helps solve for x. The Pyhagorean Theorem states that in every right triangle with legs of length a and b and hypotenuse c, these length have the relationship of a2 + b2 =c2. Let a = x, and b = 2x + 6, so that c = 2x + 4 Then, putting these measurements into the Theorem, the equation becomes: x2 + (2x + 6)2 = (2x + 4)2 The binomials into the Pythagorean Theorem. x2 + 4x2 + 16x + 16 = 4x2 + 24x + 36 The binomials squared. Notice there...

Words: 699 - Pages: 3