How Does the Unit Circle Relate to Euler's Formula in Complex Numbers?

[MikeSv]

Hi everyone.

I was looking at complex numbers, eulers formula and the unit circle in the complex plane.

Unfortunately I can't figure out what the unit circle is used for.
As far as I have understood: All complex numbers with an absolut value of 1 are lying on the circle.

But what about numbers outside / inside of the circle and what does it have to do with Eulers formula?

Thanks in advance for any help,

Cheers,

Mike

 

[Paul Colby]

Every complex number has a polar form, $z = r e^{i \phi}$, where, $r$ and $\phi$ are real. The relation to the circle is $|r| > 1$ outside and $|r|<0$ inside the circle.

 

[MikeSv]

Hi and thanks for the reply!

But what exactly is the unit circle good for?

Thanks again,

Mike

 

[fresh_42]

Every complex number has a polar form, $z = r e^{i \phi}$, where, $r$ and $\phi$ are real. The relation to the circle is $|r| > 1$ outside and $|r|<0$ inside the circle.

... and $|r|<1$ inside the circle.

But what exactly is the unit circle good for?

What do you mean? Which kind of answer would satisfy you? Have you read already something about complex numbers?

The unit circle is one way to demonstrate $\cos$ and $\sin$ of an angle and both are related to Euler's identity of complex numbers. Thus it is an appropriate tool to deal with geometric properties of the complex plane, see e.g. https://en.wikipedia.org/wiki/Unit_circle
The unit circle plays also various roles in mathematics which are a little beyond B-level answers.

In general holds: "What is something good for?" depends completely on what is considered "good", which is a rather individual assessment.

 

[FactChecker]

Multiplication by a complex number on the unit circle causes a pure rotation in the complex plane of the number it multiplies. Because rotations and cyclic behavior are so important in geometry and physics, the complex numbers on the unit circle are very important. Many mathematicians would say that Euler's formula is the most important formula in mathematics. It always amazes me how well things work out when you use it. There are entire university courses in engineering and math where it is the central player.
(for instance, see
)

 

[Paul Colby]

Well, asking as a mathematician I wouldn't know. There is a rather extensive connection between circles and the complex plane. Most of this stuff was unearthed in the 1800's so it's not really cutting edge math. Look up modular groups. There is a wonderful two volume set "Theory of Functions" by Caratheodory published by Chelsea. One looks at 1-1 conformal mappings of the complex plane onto itself. This leads you to functions of the form,

$z' = \frac{az+b}{cz+d}$ where $ad-bc=1$

(could someone please tell me how to undo indents in this interface?)

​

Composing two of these mappings gives you another (called fractional linear maps) and yes, the new coefficients are found by viewing them as a matrix multiplication. My interest in this was sparked (quite a while ago) by the Lorentz group and $SL(2,C)$ connection and spinors and stuff.

 

[MikeSv]

So basically the unit circle is just used as graphical tool for complex numbers?
Can I represent all complex numbers on the circle?
And if the absolute value is > 1,is that just because r is used as a scaling factor?

/Mike

 

[fresh_42]

So basically the unit circle is just used as graphical tool for complex numbers?
Can I represent all complex numbers on the circle?

Of course not. All complex numbers $x+iy$ with $|x+iy|=\sqrt{x^2+y^2}=1\,$.

And if the absolute value is > 1,is that just because r is used as a scaling factor?

Yes. You can write all numbers (except $0$) as real multiple of a complex number on the unit circle. For zero there is no well defined correspondence.

 

[MikeSv]

Great!
And I guess the unit circle is useful when chansning from cartisian to polar form and back?

/Mike

 

[Mark44]

Great!
And I guess the unit circle is useful when chansning from cartisian to polar form and back?

Sort of.
The unit circle is important in the study of trigonometry inasmuch as all of the basic trig functions are defined in terms of the unit circle. For example, if (x, y) is a point on the unit circle (so that $x^2 + y^2 = 1$), then $\cos(\theta) = x, \sin(\theta) = y$, and so on. Here, $theta$ is the angle in radians, as measured counterclockwise from the positive x-axis.

 

[PeroK]

Great!
And I guess the unit circle is useful when chansning from cartisian to polar form and back?

/Mike

More fundamentally, there are two ways to think about complex numbers. The first is algebraically. But, things can often get messy very quickly.

The second is geometrically. This can often simplify things. For example, if you have an equation like:

$|z - 3 - 2i| = 5$

Then, geometrically, you can see immediately that $z$ describes a circle on the complex plane, centred at $3 + 2i$ with radius 5.

It's always a good idea to check whether geometry can help with a complex numbers problem and not just to plough ahead algebraically.