Dan's eⁱ^pPage

Welcome to my eⁱ^ppage. On this page, I plan to post my investigation into what seems at first as a relatively easy equation. A lot of interesting math can be learned from investigating this interesting formula. I hope you enjoy it. Please direct all comments here.

First, let's look at the elements in the formula: e, p , and i. We are all familiar with p from elementary school. P is ratio of the circumference to the diameter of any circle. P was shown to be irrational by the Greeks. Irrational numbers are numbers that can not be represented as the ration of two integers. (Besides p , e and the square root of 2 are irrational.) Because of its irrationality, since the very beginnings of mathematics, mathematicians have been trying to produce p to more and more digits. The current world record (as of September 1997) is 1,000 billion places! You can see pi to 10 million decimal points, but beware, the file is very large. There are many ways to calculate p to arbitrary precision, though some algorithms are more efficient than others. There is a web page devoted to this topic that you can visit if you are interested. For the more mathematically inclined, one can visit this page.

P was also shown to be normal. Normality means that any of the 10 digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) in our number system are evenly distributed throughout the decimal expansion of p . In other words, knowing n digits of pi will be of no use in predicting the (n+1) digit of p because each of the 10 digits has an equal probability of being in each slot. For a more in-depth explanation, go here.

If you have taken Algebra II/Trigonometry in your high school studies, then you will be familiar with e. E is the natural logarithmic base. If one was to define a function as f(x)=log x, then f(x) would return as a result, y, such that 10^y=x. The log function can be generalized as follows:

f(x) = log _bx; bÎ R⁺

The variable b is referred to as the base of the logarithm. This time the result would be y, such that b^y=x. (Notice that in the first example, since the base was not specified, it was assumed to be 10.) Now we can easily define the natural logarithm as follows:

f(x) = log _e x; e » 2.718

Or more simply as f(x) = ln x

Notice that the base of the ln function is understood to be e. E is irrational like p , and was also proved to be transcendental. You can see 10 million digits of e on this page. For more information on e and a mathematical background, visit this page.

This leaves us with "i" still to discuss. What is the square root of -4? Your algebra teacher probably told you that there is no such creature. S/he was not totally correct. There is no number on the real number line that, when squared, will give you a negative result. Mathematicians, however, have discovered the imaginary number line, which is just like the real number line, but the elements on the line, when squared, give a negative result. The imaginary number line is made up of the real numbers, multiplied by a special number called i. Mathematicians defined i to be the square root of -1. With this background, we can get back to our original question, what is the square root of -4?

Ö (-4) = Ö (-1· 4) = Ö (-1) · Ö 4 = i · 2 = 2i

Note that all negative numbers can now have square roots, which are just the square roots of their absolute values times i. Despite their names, the imaginary numbers are far from imaginary. Imaginary numbers are just as real as their "real" counterparts. Historically, they were not considered as real, but recent discoveries since that time have shown them to be just as real as the reals. However, the name had stuck. Together, the real and imaginary numbers form the complex number plane. This plane is just like the coordinate plane studied in algebra, with the x-axis representing the real numbers, and the y-axis representing the imaginary numbers. For a more in-depth look at imaginary numbers, go here.

There are special rules for writing and calculating with the complex numbers. A complex number can be written in the form (x, y). In this form, the x is the real component and the y is the imaginary component. This can also be written as x + i· y. To add two complex numbers, you simply add the real and complex components, keeping them separate.

Example:

(1, 2) + (3, 4) = 1 + 2i + 3 + 4i = 1 + 3 + 2i + 4i = (1 + 3, 2 + 4) = (4, 6)

To multiply two complex numbers, one follows the laws for multiplying two binomials: (a, b)· (c, d) = (ac - bd, ad + bc)

Example:

(1, 2) · (3, 4) = (1 + 2i) · (3 + 4i) = 1 · 3 + 1 · 4i + 3 · 2i + 4i · 2i = 3 + 4i + 6i + 8i² = 3 + 8 · -1 + 10i = -5 + 10i = (-5, 10)

Note that i² = -1, i³ = -i, i⁴ = 1, and so on. Now that we have the preliminaries out of the way, we can move on to exploring eⁱ^p. This requires, as we have seen, raising e to a complex power. This is in fact the following:

eⁱ^p= (e^p)ⁱ = (eⁱ)^p

When one types that into a calculator that supports complex numbers (such as the TI-85), one gets (-1, 0), which of course is just -1. That may seem like a long way to go to come to such a simple solution! Well, it's the simplicity of the solution that makes one wonder! Why, when you raise two irrational numbers to each other, and then raise them to the square root of -1, do you get such a simple result? To understand that, one must first understand how one can raise a number to an irrational as well as an imaginary power.

Proceeding, we can rewrite our equation as follows:

eⁱ^p= -1

eⁱ^p+ 1 = 0

We can see why this equation has interested mathematicians for so many years. Five of the most important constants in mathematics appear linked and inseparable -- p , e, i, 1, and 0. No one ever imagined that they would ever be so wonderfully connected! (Yahoo has great links to pages about four of these constants.)

Before we can go to an explanation of this equation, we must familiarize ourselves with the cosine, sine, and tangent functions. The cosine and sine functions take as inputs an angle, either in degrees or radians. First, let us familiarize ourselves with these measurements. We all know that angles represent direction on a circle.

Any point on a circle can be represented by an (x, y) ordered pair, or by giving the number of degrees (or radians) to move counterclockwise from the x-axis (the angle, q ), and the distance to move (the radius, r). The Cartesian Coordinates (x, y) and the Polar Coordinates (r, q ) are related by the sine and cosine functions.

Definition:

x = r · cos q

y = r · sin q

sin q / cos q = tan q

These functions are very useful in understanding Euler's equation (as eⁱ^p=-1 is sometimes referred to). These functions can be represented as infinite sums.

cos q = 1 - q ²/2! + q ⁴/4! - q ⁶/6! + . . .

sin q = q - q ³/3! + q ⁵/5! - q ⁷/7! + . . .

E^x can also be represented as an infinite sum.

e^x = 1 + x + x²/2! + x³/3! + x⁴/4! + . . .

Note: If you are not familiar with the factorial function, it is defined as follows:

f(x) = x! = x · (x-1) · (x-2) . . . 3 · 2 · 1

Now, if we imagine substituting (ix) for x into the above equation, and remembering that i² = -1, i³ = -i, and i⁴ = 1, we can see that the equation is just the sum of cos x and i · sin A:

e^x = cos x + i · sin x

That beautiful relation is the key to the solution of eⁱ^p! Using the above as a jumping stone, one can derive the meaning of eⁱ^p+ 1:

eⁱ^p+ 1 = cos p + i · sin p + 1 = -1 + i · 0 + 1 = -1 + 1 = 0

And Euler's theorem bows before us as one of mathematics' most wonderful relations. Its simplicity and overall scope is overwhelming, and a sudden feeling of joy and accomplishment is felt after this simple statement is so eloquently proved. Of course, this has not been a very rigorous proof, for the infinite sum derivation of e, the sine, and the cosine functions were not proved or explored, but simply taken on faith. Their derivation is the subject of calculus, and I encourage you to take a book out on that subject and continue with the learning and investigation!

Of course, one can go on and discover many more interesting theorems that stem from the equation we just proved. Euler (and you and I) are able to derive the meaning of what at first seems like non-sense -- iⁱ:

eⁱ^p+ 1 = 0

eⁱ^p= -1

eⁱ^p= i² [by -1 = i²]

(eⁱ^p)ⁱ= (i²)ⁱ

(e^{i^2}^·^p) = (iⁱ)² [by (x^a)^b= (x^a^·^b) = (x^b)^a]

e^-^p= (iⁱ)² [by i²= -1]

Ö (e^-^p) = Ö (iⁱ)²

Ö (e^-^p) = iⁱ

iⁱ = Ö (1/e^p )
iⁱ = Ö (1/e^p )
iⁱ = Ö 1/Ö (e^p )

iⁱ = 1/Ö (e^p )

We end our discussion with a general derivation of xⁱ.

eⁱ^q= cos q = i · sin q

x = e^{ln x}

xⁱ^q= (e^{ln x})ⁱ^q= e^{ln x}^·ⁱ^·^q= eⁱ^·^{(ln x}^·^q⁾

xⁱ^q= cos (ln x · q ) + i · sin (ln x · q )

Example:

2ⁱ = cos (ln 2) + i · sin (ln 2)

2ⁱ » cos (0.693) + i · sin (0.693)

2ⁱ » 0.769 + i · 0.639

2ⁱ » 0.769 + 0.639i

This is not the end of all there is to know about Euler's equation, but simply the beginning. I encourage you to go on and investigate on your own into this and other wonderful math problems. I recommend you visit my Fractals Page, another interesting topic in the realm of mathematics. Another recommendation that I can make is Surreal Number Theory. Go to your local bookstore or library and pick up a copy on any one of these wonderful topics. Don't be afraid to email me regarding your findings.

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