Why eulers formula

Is because they happen in nature. If you whistle the air pressure looks like a sine wave. If you ring a bell the bell moves in a sine wave.

In any kind of music if you look at the notes in music the sound they make the pressure waves look like sine waves. And circuits make sine waves, remember? We analyzed this circuit in great detail it was the LC circuit. We looked at the natural response of this and that was a sine wave. So, electric circuits make sine waves. All these things make sine waves. They occur in nature.

And we want to be able to analyze things that happen when sine waves are present. So, we have two things we love and we want to relate these two things. And these are going to be related through that. Euler's Formula. That's how we connect these two separate ideas.

Let me, let's go do that. So Euler's Formula says that e to the jx equals cosine X plus j times sine x. Sal has a really nice video where he actually proves that this is true. And he does it by taking the MacLaurin series expansions of e, and cosine, and sine and showing that this expression is true by comparing those series expansions. And I'm not going to repeat that here we're just going to state that as fact. And now we're going to look at this equation a little bit more.

So, this is the expression that relates exponentials that we love, to sines and cosines that we love. And part of the price of doing that is we introduce complex numbers into our world. Here's two complex numbers. This is where complex numbers come into electrical engineering. So we have to mention the other form of this formula which is e to the, I put a minus sign in here, e to the minus jx. And that equals cosine X minus J sine X.

There is another " Euler's Formula " about Geometry , this page is about the one used in Complex Numbers. But if you want to take an interesting trip through mathematics, you will discover how it comes about.

It was around , and mathematicians were interested in imaginary numbers. For any polyhedron that does not self-intersect, the number of faces, vertices, and edges are related in a particular way. Euler's formula for polyhedra tells us that the number of vertices and faces together is exactly two more than the number of edges. Euler's formula for a polyhedron can be written as:.

When we draw dots and lines alone, it becomes a graph. We obtain a planar graph when no lines or edges cross each other. We can represent a cube as a planar graph by projecting the vertices and edges onto a plane.

Euler's formula is proved using the utility problem: The three houses are to be connected to the 3 utility gas, water, and electricity.

They are to be connected in such a way that no pipe passes over the other pipe. To get a complete cycle with no intersection in any planar graph, we remove an edge to create a tree.

We repeat this process until the remaining graph is a tree. Consider our utility graph and apply Euler's formula graph theory. We find that there are 6 vertices and 9 edges. We need to verify Euler's formula and check for the number of faces.

We notice that we need 10 edges. However, the problem has only 9 edges. By this contradiction, we obtain Euler's formula proof. This two-dimensional planar graph when inflated into a solid becomes an octahedron.

The screencast was fun, and feedback is definitely welcome. I think it helps the ideas pop, and walking through the article helped me find gaps in my intuition. Learn Right, Not Rote. Home Articles Popular Calculus. Feedback Contact About Newsletter. Euler's identity seems baffling: It emerges from a more general formula: Yowza -- we're relating an imaginary exponent to sine and cosine! Not according to s mathematician Benjamin Peirce: It is absolutely paradoxical; we cannot understand it, and we don't know what it means, but we have proved it, and therefore we know it must be the truth.

Here's mine: Euler's formula describes two equivalent ways to move in a circle. If we examine circular motion using trig, and travel x radians: cos x is the x-coordinate horizontal distance sin x is the y-coordinate vertical distance The statement is a clever way to smush the x and y coordinates into a single number. What is Imaginary Growth? Let's step back a bit. But what does i as an exponent do? We're growing from 1 to 3 the base of the exponent. How do we change that growth rate?

We scale it by 4x the power of the exponent. The top part of the exponent modifies the implicit growth rate of the bottom part.

The Nitty Gritty Details Let's take a closer look. Some Examples You don't really believe me, do you? But remember, We want an initial growth of 3x at the end of the period, or an instantaneous rate of ln 3. Not today! Let's break down the transformations: We start with 1 and want to change it. That describes i as the base. How about the exponent? And it does: Tada! And now we modify that rate again by i : And now we have a negative rotation!

And, just for kicks, if we squared that crazy result: It's "just" twice the rotation: 2 is a regular number so doubles our rotation rate to a full degrees in a unit of time. Radius: How big of a circle do we need? Amount to rotate: What's the angle of that point? Why Is This Useful? Again, it's two ways to describe motion: Grid system: Go 3 units east and 4 units north Polar coordinates: Go 5 units at an angle of Happy math.

Appendix The screencast was fun, and feedback is definitely welcome. References: Brian Slesinsky has a neat presentation on Euler's formula Visual Complex Analysis has a great discussion on Euler's formula -- see p.

Join k Monthly Readers Enjoy the article? There's plenty more to help you build a lasting, intuitive understanding of math.