Math Euler's Formula
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) eiπ=cos(π)+isin(π) eiπ= -1+i(0) eiπ= -1
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
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) eiπ=cos(π)+isin(π) eiπ= -1+i(0) eiπ= -1
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