Intuitive Understanding Of Euler's Formula

I hate you. I was planning to start a blog to better explain these concepts for engineering students but here you are doing that and doing that better than I could :frowning:

Great going brother, you are doing a great work. Keep them coming!

Is sin^2 (x) + cos^2 (x)=1 somehow related to Eulerā€™s theorem ?

Making some sense out of something that always seemed magical. Great work here. Thanks.

Wow, an article as beautiful as the mathematics itself!

@Adrian: Yep, great observation. Check out http://betterexplained.com/articles/intuitive-trigonometry/ for more details, but basically, Eulerā€™s theorem describes a circle with a rotation path (spinning around), and sin^2 (x) + cos^2 (x) = 1 describes the same circle via rectangular (grid) coordinates.

Here is also great explanation of why imaginary roots form a unit circle:

Just discovered your website through this particular article. Best thing Iā€™ve found anywhere on basic math. Just what people trying to learn math need! Thanks.

Thanks Charles, really glad you enjoyed it!

@Aditya Thanks a lot. Multiplication as you said should be considered as additions of new coordinates to our observation area.
ITS A NEW TOPIC NOW.
Have you noticed that the graphs of 2^(nx) and (nx)^2 {where n is an integer) intersect at 2 places. The graphs such as x^a and a^x(exceptions as mentioned earlier) intersect only once! Maybe this due to the fact that 2+2=4 and 22=4 and also 2^2=4. This symmetry is awesome and exists only for 0,1 and 2. (1=1=1 since the value of 1 is just 1 you just donā€™t get the chance to insert the operators.{I donā€™t think about 1+1 and 11 as using 1 two times would be a biased decision towards 2})
There is moreā€¦ about which I ask the question, but just as I get the answer the questions become even more intriguingā€¦

Complex growth:

Radius: How big of a circle do we need? Well, the magnitude is sqrt(6^2 + 8^2) = sqrt(100) = 10. Which means we need to grow for ln(10) = 2.3 seconds to reach that amount.

Why do we calculate ln(10)? Does this mean the time to reach that point through the radius is the same time to reach it by rotation?

I do not see the point?

Given that e^(i.pi) = -1, then 1/e^(i.pi) {or e^-(i.pi)} = 1/-1 = -1.
Another way I looked at this was e^-(i.pi) is e^-(i.x) which rotates clockwise to -1 at x=pi (instead of the counter-clockwise direction that e^(i.pi) took).
If I havenā€™t made a stupid mistake, then is it true to say that e^ negative(i.pi) + 1 = 0 and is similar to Eulerā€™s identity?

Hah, thanks Rohan ā€“ Iā€™m planning on adding some more community features to the site so youā€™ll still have a chance to explain things :).

Hi. Really amazing explanation.
I just want to ask you about the ā€œeā€, why this number? I mean, e is not a symbol, it represent the number 2,718ā€¦Does this number has a meaning for imaginary numbers representation? Is it used to calculate the circular path, or any other thing? I couldnā€™t find it out.

Inside the parentheses it should read: ā€œinstead of the counter-clockwise direction that e^(i.pi) tookā€ to reach -1 at x=pi".

Thanks Kalid. Great explanations, by the way. I am en route to a goal of understanding the electronic engineering maths (up to Fourier and Laplace) which I did over 40 years ago but only had a hazy grip of - and your explanations are making it much easier for me to visualise things.

Really appreciate the intuitive/analogy approach you use in all these articles. This concept in particular never really ā€˜clickedā€™ with me until now (20yrs after I first learned it!). Thanks and keep up the great work! Math is so important - youā€™re making a real difference in the world.

@Gabriel: Thanks, glad you liked it. e^x represents continuous growth (http://betterexplained.com/articles/an-intuitive-guide-to-exponential-functions-e/) and when combined with i in the exponent, represents continuous rotation.

@Don: Yep, youā€™re exactly right - having a negative sign means you are spinning the clockwise direction, and

$$e^{-i \cdot \pi} = -1$$

I will try to get it right this time.

Inside the parentheses it should read: ā€œ(instead of the counter-clockwise direction that e^(i.x) tookā€ to reach -1 at x=pi)ā€.

Hi again Kalid. I have had a quick look around and it seems to me that e^-i.pi = -1 is not normally expressed that way in the few sites I have visited on the subject. Would it not be more complete always to offer both exponentials ie: e^i.pi = e^-i.pi = -1 ?

Hi Don, Kalid,

Sorry to butt in on the conversation. I hope that you donā€™t mind.

Don, you ask if it would not be more complete always to offer both exponentials ie: e^i.pi = e^-i.pi = -1.

The reality is that both those numbers are actually the same number, just written in a different way. In fact there are an infinite number of way of writing -1 in exponential form - see these few for example:

e^i.pi, or e^i.3.pi, or e^i.5.pi, or e^i.7.pi ā€¦ etc.

And similarly, -1 can also be written using these examples:

e^-i.pi, or e^-i.3.pi, or e^-i.5.pi, or e^-i.7.pi ā€¦ etc.

All those numbers in exponential form are just alternative ways of writing the same number. They differ only by adding 2.pi (rotating anticlockwise) or subtracting 2.pi (rotating clockwise).

The reason is, of course, that 2.pi is the angle of rotation of a full circle - and turning a full circle always gets you back to where you started.

Remember this - each unique complex number is a unique point on the complex plane. If you use the rectangular form, there is only one way to write each point on the complex plane.

Butā€¦ the exponential form (probably the most useful form for writing complex numbers) has the interesting property that each unique point on the complex plane - each unique complex number - can be represented an infinite number of ways using the exponential form.

There is a standard method preferred when writing complex numbers in exponential form - call the ā€œprinciple argumentā€.

The principle argument for the exponential form means selecting the angle that is > -pi and <= +pi.

Therefore e^i.pi is preferred over e^-i.pi, and over all of the other infinite alternatives, when writing -1 in exponential form. Both those forms are correct, and equal, it's just that +pi is the preferred standard.

Along the same lines, the number 1, in exponential form, is e^0, but could also be correctly written as e^i.2.pi, or e^i.4.pi, or e^-i.2.pi etc. So, using the principle argument means that the preferred way of writing 1 in exponential form is e^0, because the angle of 0 lies between the range of the principle argument: between -pi and +pi.