The Euler’s Formula
Euler’s formula and Identity: eix = cos(x) + i(sin(x))
The world of math today is one with endless possibilities. It expands into many different and interesting topics, often being incorporated into our everyday lives. Today, I will talk about one of these topics; the most mind-blowing and fascinating formula invented, called the “Euler’s formula”. This formula was created and introduced by mathematician Leonhard Euler. In essence, the formula establishes the deep relationship between trigonometric functions and the complex exponential function. Euler’s formula: eix=cos(x)+isin(x); x being any real number Wow -- we're relating an imaginary exponent to sine and cosine! What is even more interesting is that the formula has a special case: when π is substituted for x in the above equation, the result is an amazing identity called the Euler’s identity: eix=cos(x)+isin(x)
Euler’s identity: eiπ= -1
This formula is known to be a “perfect mathematical beauty”. The physicist Richard Feynman called it "one of the most remarkable, almost astounding, formulas in all of mathematics." This is because these three basic arithmetic operations occur exactly once each: addition, multiplication, and exponentiation. The identity also links five fundamental mathematical constants: the number 0, the number 1, the number π, the number e and the number i. But the question remains: how does plugging in pi for x give us -1? Why and how does the Euler’s formula work? So let’s get down to the details. When I saw this formula, I immediately started to think of analogies that could help me understand why eiπ gives us -1. My inquisitive curiosity on the formula led me to several resources that helped me formulate my explanation on why the equation is equal to -1. But before I dive into that, I will break up the formula and explicate some of its main components for a better understanding.
Exponent ix, with i being the imaginary number
The number e:
The number e, sometimes referred to as the “Euler’s number”, is a significantly important mathematical constant. Approximately, it is equal to 2.7182 when rounded, while the exact number extends to more than a trillion digits of accuracy! That is because e is an irrational number since it cannot be written as a simple fraction. The number e is the base of the natural logarithm. The logarithm of a number is the exponent by which another value (the base), must be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 10 to the power 3 is 1000. The natural logarithm is the logarithm to the base e. The natural logarithm of a number x is the power to which e would have to be raised to equal x. The imaginary exponent:
As we know, i= -1 or i2 = -1. The imaginary number helps in finding the square roots of many negative numbers, which is impossible to do otherwise. But Leonard Euler created the idea of the imaginary exponent, as shown in his Euler’s formula. He introduced a completely new concept. How the imaginary number works in this formula, will be later explained in my report. Sine and cosine functions are two of the prominent trigonometric functions, which you are already familiar with. Now that I have explained the math that makes up the Euler’s formula and given you a little background knowledge on it, I will now get down to the main question that I want to discuss: Why does the Euler’s formula work and why does eiπ equal to -1? My extensive research on this soon led me to an appropriate explanation: Euler's formula describes two equivalent ways to move in a circle. Think of Euler's formula as two formulas equal to each other; eix and cos(x)+isin(x) both of which explain how to move in a circle. Explanation of cosπ+isinπ= -1:
By looking at the formula cos(x)+isin(x) closely, I saw that it is a...
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