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?
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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
eix=cos(x)+isin(x)
Number ecosine functionsine function
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:
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By looking at the formula cos(x)+isin(x) closely, I saw that it is a complex number of the form a+bi, so I realized that it could be modeled using the complex plane where: * cos(x) is the real x-coordinate (horizontal distance)
* isin(x) is the imaginary y-coordinate (vertical distance) Refer to the figure below for a refresher on how to interpret complex numbers using the complex plane: Illustration of the complex plane:
* Real part of the complex number is the x-coordinate.
* Imaginary part of the number is the y-coordinate.
* Four points are plotted so you can see the correspondence between x and y coordinates and the real and imaginary parts of the complex numbers.
Illustration of the complex plane:
* Real part of the complex number is the x-coordinate.
* Imaginary part of the number is the y-coordinate.
* Four points are plotted so you can see the correspondence between x and y coordinates and the real and imaginary parts of the complex numbers.
The analogy “complex numbers are 2-dimensional” helps us interpret a single complex number as a position on a circle. Let’s connect that to our Euler’s formula. If we associate the x- and y-axes with the real and imaginary part of the equation like before, that means that the real x-coordinate is the cosine of the angle x, and the imaginary y-coordinate is the sine of the angle x multiplied by the imaginary number, as shown in the graph below. Note: The angle marked by x in the diagram below, is the same as “x” in Euler’s formula. Properly, we need to write the angle x in radians, not degrees: One circle (360°) = 2π radians
∴ Half-circle (180°) = π radians
x
x
Since a half circle (180°) is equal to π radians, that means that when pi is substituted for x into the Euler’s formula, we’re traveling “pi” radians along the outside of the circle. Also, when x = 0, cos(0)+isin(0) = 1. So from the beginning point of 1, we will move π radians which 180°, half-way around the circle, putting us at -1, which is exactly what the Euler formula states since it says that cos(x)+isin(x)= -1. And we have now shown that using the complex plane. So now we know that the right side of Euler’s formula (cos(x) + i*sin(x)) describes circular motion with imaginary numbers and the trigonometry function of sine and cosine. Now let’s figure out how the left side of the formula, which is eiπ, equals to -1. θ
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θ
Explanation of eiπ= -1:
Instead of seeing the numbers on their own, you can think of -1 as something e had to “grow to” using exponentiation. Real numbers such as e, would have an fixed rate at which it would increase by, during exponentiation. In other words, interest rate is the rate at which the number e “collects” as it’s going along and increasing, growing continuously. Regular growth is simple — it keeps “pushing” a number in the same, real direction it was going. Imaginary growth is different — the “interest” we earn is in a different direction!
It’s like a jet engine that was strapped on sideways — instead of going forward, we start pushing at 90 degrees. The neat thing about a constant orthogonal (perpendicular) push is that it doesn’t speed you up or slow you down — it rotates you! Taking any number and multiplying by i will not change its magnitude, just the direction it points. * Regular exponential growth continuously increases ‘e’ by a set rate; imaginary exponential growth continuously rotates a number.
In imaginary growth, we apply i units of growth in infinitely small increments, each rotating us at a 90-degree angle. In real growth, we push growth in the same direction while compounding and continuously increasing. So while one pushes forward, the other rotates the evergrowing line of growth as shown the graph below. Also, the distance travelled around a circle is an angle in radians. We’ve found another way to describe circular motion just like using sine and cosine! So, Euler’s formula is saying that “exponential, imaginary growth traces out a circle”. And this path is the same as moving in a circle using sine and cosine in the complex plane.
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In conclusion, cos(x)+isin(x) and eix are two ways to move around a circle. While the first one uses cosine and sine, the second one uses number e and an imaginary exponent, and yet they both are equal to each other. This is part of the reason why this formula is so interesting. It defines the relationship between trigonometry and imaginary exponentiation in a very concise manner, and that is the true beauty of this equation. I would like to conclude my exploration of the Eulers formula using a mathematical joke which asks, “How many mathematicians does it take to change a light bulb?”, The answer to that is ” (which, of course, equals 1!!!).
