Euler’s Formula

Many scientists believe that the best way to understand our university, from the smallest subatomic particles to the largest galaxies, is through mathematics. Equations can help us explain patterns that are to complex, XXX or chaotic.

But mathematics is not just incredibly powerful to solve real life applications. Many mathematicians

As we have seen, many modern scientists believe that the natural world, from the smallest to the widest galaxy, can only be understood mathematically. They believe that mathematics is not only wildly useful; it enables us to see clearly the structure and beauty that lies beneath the surface of our world. It enables us to scrutinize and understand things that appear at first glance to be inscrutable: too vast, too tiny, or too chaotic.

Are there any equations that you think are particularly beautiful? Maybe you like the symmetry of Pythagoras' theorem or XXXX.

In this chapter we will learn about another equation that is not just XXXX, but has also inspired countless mathematics by its "beauty". In fact, the physicist and Nobel Laureate Richard Feynman called it “one of of the most remarkable, almost astounding formulas, in all of mathematics”! But first, we have to think about complex exponents.

Complex Exponents

In the previous chapter, we used the laws of Complex numbers to work out what it means to add, multiply or divide complex numbers. If we want to raise a complex number to a power, we can just multiply it repeatedly – for example, 2+3i3=2+3i·2+3i·2+3i.

The mathematician Leonard Euler asked what would happen if we instead put a complex number into the exponent – for example, what is 22+3i?

On the face of it, this question seems a bit far fetched. After all, we have been told since we were young that powers represent repeated multiplication. 2^3 means “multiply 2 by itself 3 times.” How can we possibly make sense of “multiply 2 by itself i times” or “multiply 2 by itself 1+2i times”? If you will remember, though, we have extended the definition of exponents before. Asking ourselves to “multiply 2 by itself -1 times” or to “multiply 2 by itself ½ of a time” does not really make sense, but we found a way to calculate expressions like 2^(-1), 2^(½), or even 2^(pi).

It turns out that we can extend our idea of exponents even further to include imaginary and complex exponents, but it requires thinking about exponents very differently than we are used to. To start, instead of using a base of 2,5, or even 10, we will use the natural base, e. In studying exponential functions, we learned that the function e^x can be defined in two ways:

If we take the second definition of e^x, we can consider what happens if, instead of substituting a real value for x, we substitute an imaginary number. For simplicity, we’ll start by making x=i. To visualize what is happening, we can take a vector with a horizontal (real) component of 1, and a vertical (imaginary) component of i/m, and we see what happens when we multiply that vector by itself m times. Use the slider below to see what happens as the value of m gets larger and larger.

Visual of the expression (1+x/m)^m with vectors representing a real part (1) and and imaginary part (dependent on x and m) that shows the product of this expression using vectors and triangles (similar to how we presented complex multiplication in the arithmetic chapter).

Add some more details to triangle approximation diagram (build-up in multiple steps, more explanations about what’s going on, so students don’t have to deduce everything by themselves)

There are several things to notice here. First, as the value of m becomes progressively larger, the value of the overall expression gets closer to a point on the unit circle. You might wonder why this expression gets closer to this particular point. To see why, play around with both of the sliders this time. What happens if we keep the value of m as large as possible, but change the value of x? What happens if we go to a different value of x, and vary the value of m? Record what you notice and wonder in the box below.

Text box: Record what they notice and wonder.

You may have noticed that as the value of x changes to different imaginary numbers, the value of the overall expression rotates to different points along the unit circle. You may also have noticed some interesting values of x. If you didn’t already, notice that the endpoint is taken halfway when the value of x is . This fact leads to a very interesting statement, that:

ei·π=

Or, alternatively . . .

ei·π+1=0

Known as Euler’s identity, this is one of the most famous statements in all of mathematics. According to Tobias Dantzig, some of Euler’s contemporaries regarded this identity as having a “mystic significance,” much like the Pythagoreans viewed Pythagoras’ theorem. In surveys of mathematicians and physicists, this identity comes up again and again as the most beautiful equation in all of mathematics. Studies have shown that when mathematicians look at this formula, it activates the same parts of the brain that are activated when you listen to a moving piece of music or take in a beautiful painting.

It may seem odd to find such great significance in a piece of mathematics. Euler’s identity is seen not only as interesting, but as beautiful, elegant, and profound. If this seems odd, consider how many unexpected and deep connections are at play in Euler’s identity. It connects nothingness (0), unity (1), the imaginary (i), and the infinite ( and e). It connects five of the most significant constants in the history of mathematics; all of which have long, fascinating histories, and all of which have, in some way, revolutionized the way humans understand and work with numbers.

It turns out that Euler’s identity is actually a special case of an even more far-reaching relationship. As you may have guessed, it is not a coincidence that when x=i, the point ends up rotating exactly halfway around the unit circle. As we know, an angle of radians corresponds to a rotation. If we look at other values of x, we see similar relationships. If you let the value of x=2i, the point rotates all three quarters one half one quarter of the way around the circle. To rotate the point a quarter of the way around the circle, the x-value needs to be , or 2i.In all of these cases, the coefficient of i tells us the radian angle through which the point rotates. This is why we usually write the expression not as eix, but rather as ei.

This relationship ends up being true for any value of x. If we let x=2π3·i in for x, the point will rotate exactly 2pi/3 radians, or degrees, along the unit circle. This means that e2π·i3= -1/2+sqrt(3)/2*ixxx. Even more generally, we can say that for any imaginary value i·θ, we can think of ei·θ as rotating a point along the unit circle an angle of theta radians. We can use what we know about trigonometry to write this idea in one compact statement:

We now have three distinct ways to express complex numbers: we already know the cartesian form a+bi and the polar form rcosθ+risinθ from the previous chapter. Euler's Formula can help us to convert the polar form into the new exponential form:

Where does the r prefix comes from (modulus), how to convert between cartesian and exponential, how e^(a + bi) can be converted to re^(bi)

Is the exponential form always the most useful? No: as we have seen, for adding and subtracting complex numbers the Cartesian form is extremely simple and intuitive geometrically. However, for multiplying and dividing complex numbers, the exponential form simplifies things considerably. To see how this works, consider the product the two complex numbers z=1+i and w=−1+3·i. As we have seen, z has a modulus of and an argument of . We can therefore write it in exponential form as z=2·ei·π4x. Similarly, we can write w as w=2ei·2π3. If we wanted to multiply z·w, we could write:

z=2·ei·π4 w=2ei·2π3 We can find the modulus and argument of both z and w and use that to write each in exponential form z·w=2·ei·π4·2ei·2π3 z·w=22·ei·π4+i·2π3 We can use exponent rules to multiply both complex numbers. z·w=22·eiπ4+2π3 z·w=22·ei·11π12 We can factor out an i and simplify our expression by adding both arguments.

Not only is this simpler than multiplying both numbers expressed in polar form; it has the advantage of matching exactly with how we know complex multiplication works. It shows that to find the product of two complex numbers, we only need to add subtract multiply divide their arguments and multiply add subtract divide their modulii. This form is similarly illuminating when we divide complex numbers.

Proving Euler's Formula

To start, this formula is very useful. When engineers use complex numbers to analyze electricity, they use Euler’s formula. When physicists use complex numbers to help answer big questions about the universe, they often rely on Euler’s formula as well. In general, this formula often offers the most simple and elegant way to perform computations with complex numbers.

There are many things we can do with Euler’s formula once we know it, but let’s explore a bit further exactly why Euler’s formula works and where it comes from.

Recall that we also can define e^(x) as the sum of the following infinite series:

1+x+x^2/2!+x^3/3!+x^4/4!+ . . .

This is known as the power series for e^(x). (To view a summary and explanation of the concept of power series, click here.) Normally we have thought of these power series having only real number inputs, but if we choose an imaginary number (i) and substitute it into the equation in place of x, something very interesting happens:

Math Commentary ei=1+i+(i)22!+(i)33!+(i)44!+(i)55!+(i)66!+...

ei=1+i-()22!-i()33!+()44!+i()55!-()66!+... We can use our understanding of powers of i to simplify this expression. ei=(1-()22!+()44!-()66+....) +(i()-i()33!+i()55!+...) Group all of the imaginary terms and all of the real terms. ei=(1-()22!+()44!-()66+....) +i(()-()33!+()55!+...) Factor i out of all of the imaginary terms. ei=cos()+i*sin() Notice that these new series are the power series for cos()and sin (), respectively!

We can therefore derive Euler’s formula directly from the power series for e^(x), sin (x), and cos (x)!

Circular Motion

As you can see, things can get out of control fairly quickly, and this is one way of thinking about how exponential growth works. We can also apply these ideas to other functions like x(t)=e^(2t) to see an even more rapid pace of growth, or x(t)=e^(-0.5t) to imagine exponential decay. But what happens if we introduce imaginary numbers into these exponential functions? To take perhaps the simplest example, how do these dynamics work if we make our function x(t)=e^(i*t)?

When studying calculus, we learned that if we differentiate the position function of an object, we should get its velocity position acceleration, is also e^(t). We can use the chain rule product rule quotient rule to find the derivate of xt above:

vt=ddteit=i·eitx

In terms of the particle’s motion, multiplying by i means that the velocity vector is the same length as the position vector, but it is rotated 90 counterclockwise. In other words, at any point in time, the velocity vector will be perpendicular parallel to the position vector. If we imagine our initial position vector, our velocity vector will appear as shown below:

Interactive starting with the initial position vector where the vectors pop up as soon as they enter in the blanks that explain how they should work.

We might be able to understand the motion of our particle even now, but to make things especially clear look at the acceleration. To find the acceleration, we can take the derivative of the velocity function:

at=ddti·ei·t=−ei·t

This means that the acceleration vector will also have the same magnitude as the velocity and position vectors, but it will point in the opposite same different direction as the original position vector. At any point in time, these vectors will stay in the same relationship: the acceleration vector pointing inward, the position vector pointing outward, and the velocity vector perpendicular to both.

Students of physics will recognize these conditions as the conditions for circular motion. If you haven’t taken physics, imagine a yo-yo spinning in a circle or a tetherball spinning on a pole. What keeps the yo-yo or tetherball in a circular path is a centripetal force that pulls the object towards the center of the circle: this is why the acceleration vector always points inward. Let’s press play on the diagram above and see how the particle actually moves.

As you can see, the particle traces a path around a circle of radius [1]]. What’s more: because the velocity and position vectors have the same magnitude, the particle is always moving around at a speed of unit/second. This means that when t=1, the particle will have moved a distance of 1 units around the circumference of the circle. When t=pi, the particle will have moved a distance of pi units around the circle, and so on. This serves to fully explain why Euler’s formula works the way it does. Because the speed of the particle is 1 unit/second, the input, which in this example represents the time passed, also represents the distance traveled by the particle. Because the circle has a radius of 1, the distance traveled by the particle is the same as the radian angle through which the particle has rotated, so we can really think of the input as representing that angle, multiplied by the imaginary unit i.

Fourier Series

Start by talking about the link between circular motion and waves.

Developed by the French mathematician and physicist Joseph Fourier, and originally invented to solve problems involving heat flow, the Fourier series has also extended its reach into our everyday lives and into almost every area of modern science. The general idea is that we can approximate or even represent almost any function as the sum of an infinite series of sine waves. It is similar to the idea of power series - that we can express certain functions as infinite sums of other functions - but it is also much stranger, because, again, Fourier claimed that an infinite series of sine waves could represent almost any function, no matter how complicated or weird.

To begin to see how this idea can be illuminating, not to mention extremely powerful, let’s look at an example. It is the same example that convinced the Pythagoreans that there were deep connections between mathematical structure, beauty, and the laws of nature: musical harmony. When you hear your favorite song on the radio, or listen to an orchestra perform a famous symphony, you are hearing an incredibly complicated mix of sounds. Each note played by, say, a violin, is made of what is called a fundamental frequency, which determines the actual pitch you hear, and a series of what are called overtones or harmonics. These overtones are multiples of the fundamental frequency, and the particular combination of softer and louder overtones gives the instruments you hear their timbre; the character or quality of their sound. Timbre is what makes a violin sound different than a piano, or an electric guitar sound different than an acoustic guitar.

What did Fourier series and Euler’s formula have to do with this? Well, the chord itself can be represented as a sum of different fundamental frequencies and overtones that are unique to the notes in the chord, as well as the particular instruments being played. These fundamental frequencies and overtones can be represented as sine waves, with a particular frequency (cycles per second) and intensity (amplitude). To see a simple example of how this might work, play a note on the piano below. Note both the fundamental frequency (the highest amplitude sine wave) and the various overtones that make up this one note.

Interactive: A piano (at least an octave) that students can click on to play notes or chords. The corresponding Fourier transform of the piano, both the overall sum and the series of the most important individual frequencies and overtones.

Now play a chord. For example, play the notes C, E, and G at the same time. This is a basic C major chord. Notice how we now have multiple fundamental frequencies as well as overtones, and how the wave representing the entire chord becomes much more complicated. The opening chord of “A Hard Day’s Night” is even more intricate; involving many more notes and 4 different instruments. But the principle is the same: We can break down the complicated, erratic graph that represents this chord into simple sine waves that represent the fundamental frequencies and overtones being played, and we can use this information to figure out what instruments played which notes. To see more about how this works mathematically, and how Euler’s formula fits into all of this, watch the following video by the popular math enthusiast Grant Sanderson:

More Applications

Waves are everywhere around us: you interact with electromagnetic waves whenever you listen to the radio, make a phone call, connect to WiFi, or use a microwave. You can see mechanical waves when you thrown a pebble into water, or feel the tremors of an earthquake. Light travels as waves, and even the most fundamental laws of nature, like gravity or subatomic particles, can be described using waves.

Euler's formula is a simple, elegant way to describe waves and circular motion mathematically – and that's why it should be no surprise that it has important applications everywhere in science, technology and engineering.

X-ray crystallography also allows scientists to investigate the structures of other molecules, like complex proteins and even some viruses. And many other medical imaging techniques, like Magnetic Resonance Imaging (MRI) use a similar principle: They probe chemical or physical structures that we cannot see using radiation, and then use the Fourier transform to interpret and reconstruct these structures from the data we receive.

Euler’s formula can help us describe alternating current, enabling engineers to create the electricity networks that power our homes and our cities. Structural engineers use both Euler’s formula to analyze the vibrations of tall buildings, bridges, and roads, to keep us safe when we fly on a plane or when an earthquake strikes. And Euler’s formula was even used to solve the Schrodinger equations, which form the backbone of quantum physics.

It is impossible to list all the different ways in which complex numbers have affected our lives, and all the scientific, technological and mathematical breakthroughs they have enabled.

Feynman called Euler’s formula “astounding” and “remarkable” not just because it is useful - but because it is beautiful, deep, and profound. Like the Pythagorean theorem, it represents a simple and elegant yet completely unexpected connection between different areas of math: between algebra and geometry, and between circular, wavelike motion and exponential change. Just as the Pythagoreans found an almost mystical significance in the simple ratios that helped them uncover the structure of a musical chord or the rotation of the planets, modern scientists and mathematicians find a similar sense of mystery, harmony, and beauty in Euler’s formula. Though we may have moved beyond the Pythagorean belief that “all is number,” Euler’s formula and the Fourier transform remind us of the power of numbers and of mathematics to uncover hidden structure and hidden beauty all around us: in music, in nature, and in the world at large.