Thread: Cube Roots of -1 (complex numbers)

1. Cube Roots of -1 (complex numbers)

If anyone could explain how the following is done it would be greatly appreciated!

Find all cube roots of -1.

2. Originally Posted by faure72
If anyone could explain how the following is done it would be greatly appreciated!

Find all cube roots of -1.
x^3 = -1

x^3 + 1 = 0

Now factor:
x^3 + 1 = (x + 1)(x^2 - x + 1^2) = 0

Either
x + 1 = 0 ==> x = -1
or
x^2 - x + 1 = 0

So the other two solutions are:
x = [(-1) (+/-) sqrt{(-1)^2 - 4*1*1}]/(2*1)

x = [1 (+/-) sqrt{-3}]/2

-Dan

3. Thanks! Question though - doesn't i come into this at any point?

4. Originally Posted by faure72
Thanks! Question though - doesn't i come into this at any point?
sqrt{-3} = i*sqrt{3}

-Dan

5. Originally Posted by faure72
Find all cube roots of -1.
-1= cis(pi).
There are three cube roots of –1: cis(pi/3), cis(pi), and cis(-pi/3).

6. Originally Posted by Plato
-1= cis(pi).
There are three cube roots of –1: cis(pi/3), cis(pi), and cis(-pi/3).
cis?

cos(theta) + i*sin(theta)?

7. Originally Posted by ecMathGeek
cis?
cos(theta) + i*sin(theta)?
Yes that is the usual meaning.
I must tell you that I think that anyone working at this level who does not know that notation must not have paying close attention.

8. Originally Posted by Plato
Yes that is the usual meaning.
I must tell you that I think that anyone working at this level who does not know that notation must not have paying close attention.
Sure. I've never had a teacher represent it that way, so clearly I should know what it means.

9. Originally Posted by ecMathGeek
Sure. I've never had a teacher represent it that way, so clearly I should know what it means.
Have you had a course in complex analysis?
Or any course that considered the basic properties of complex numbers?

10. Originally Posted by Plato
Have you had a course in complex analysis?
Or any course that considered the basic properties of complex numbers?
No. Just about everything I know about complex numbers is (mostly) self-taught.

11. Originally Posted by ecMathGeek
No. Just about everything I know about complex numbers is (mostly) self-taught.
There are some very good textbooks you can use,
Look up Steven Krantz's book.

12. Originally Posted by ecMathGeek
No. Just about everything I know about complex numbers is (mostly) self-taught.
My dream this summer is to do the Erdos move and learn math for hours and hours a day. One thing I hope to do is complex analysis. I plan to use this.

I like Springer books.

13. Originally Posted by ThePerfectHacker
My dream this summer is to do the Erdos move and learn math for hours and hours a day. One thing I hope to do is complex analysis. I plan to use this.

I like Springer books.
I wish I could do that.

I intend to study Linear Algebra at the very least (I have not yet taken the class. I'm only in Differential equations right now .)

I should probably be in an accelerated math program because my class is too slow for me, but I have neither the time or money to do that.

P.S. Complex Analysis sounds like an awesome subject to learn.

14. Originally Posted by ecMathGeek
P.S. Complex Analysis sounds like an awesome subject to learn.
I think Complex Analysis is actually easy. Very similar to Calculus. And much easier than Advanced Calculus (Real Analysis).

Besides you can do some many wonderful things with them. Like evaluate unbelievable integrals and strange infinite summations.

And you can proof the Fundamental Theorem of Algebra in like 3 lines !!!

Somebody should write a book called "Elementary Complex Analysis" (get the pun )

15. Originally Posted by ThePerfectHacker
"Elementary Complex Analysis" (get the pun )
I think pun should be spelled "pow" (for at least 2 reasons).

The term "Elementary" is an oxymorn when used in mathematics. I've heard that "Elementary" courses tend to be some of the hardest!