- algorithm
- approximation
- bar
- binary
- binomial coefficients
- book
- cards
- carnival
- Carnival of Mathematics
- chocolate
- circle
- coins
- complex
- conjecture
- convolution
- counting
- decadic
- decimal
- diagrams
- Dirichlet
- Euclidean
- Euler
- factorization
- Fermat
- fibonacci
- fractal
- game
- games
- gcd
- graph
- groups
- Haskell
- integers
- irrational
- Ivan Niven
- Lagrange
- lehmer
- lucas
- MaBloWriMo
- making
- Mersenne
- moebius
- mu
- nim
- number
- numbers
- objects
- omega
- order
- pi
- powers
- prime
- primes
- primitive
- programming
- proof
- puzzle
- review
- roots
- sequence
- square
- strategy
- sum
- symmetry
- test
- theorem
- tree
- triangular
- two-player
- unit
- unity
- video
- visualization
- X
- zero-sum

### Blogroll

### Fun

### Reference

### Categories

- algebra (46)
- arithmetic (63)
- books (29)
- calculus (7)
- challenges (53)
- combinatorics (12)
- complex numbers (6)
- computation (46)
- convergence (9)
- counting (32)
- famous numbers (48)
- fibonacci (18)
- fractals (13)
- games (34)
- geometry (56)
- golden ratio (8)
- group theory (26)
- humor (6)
- induction (7)
- infinity (19)
- iteration (24)
- links (74)
- logic (6)
- meta (42)
- modular arithmetic (24)
- number theory (80)
- open problems (11)
- paradox (1)
- pascal's triangle (8)
- pattern (92)
- people (21)
- pictures (70)
- posts without words (25)
- primes (39)
- probability (6)
- programming (17)
- proof (71)
- puzzles (15)
- recursion (16)
- review (20)
- sequences (28)
- solutions (28)
- teaching (14)
- trig (3)
- Uncategorized (6)
- video (19)

### Archives

- November 2017 (2)
- October 2017 (1)
- September 2017 (1)
- July 2017 (4)
- June 2017 (4)
- May 2017 (9)
- April 2017 (7)
- March 2017 (5)
- February 2017 (4)
- January 2017 (3)
- December 2016 (4)
- November 2016 (6)
- October 2016 (6)
- September 2016 (2)
- August 2016 (5)
- July 2016 (2)
- June 2016 (4)
- May 2016 (4)
- April 2016 (2)
- March 2016 (3)
- February 2016 (9)
- January 2016 (8)
- December 2015 (5)
- November 2015 (29)
- August 2015 (3)
- June 2015 (2)
- April 2015 (1)
- May 2014 (1)
- December 2013 (1)
- October 2013 (1)
- July 2013 (1)
- June 2013 (1)
- May 2013 (1)
- April 2013 (3)
- March 2013 (3)
- February 2013 (2)
- January 2013 (5)
- December 2012 (3)
- November 2012 (4)
- October 2012 (5)
- September 2012 (1)
- August 2012 (4)
- July 2012 (1)
- June 2012 (6)
- May 2012 (2)
- April 2012 (3)
- March 2012 (1)
- February 2012 (4)
- January 2012 (5)
- December 2011 (1)
- November 2011 (7)
- October 2011 (4)
- September 2011 (6)
- July 2011 (2)
- June 2011 (4)
- May 2011 (5)
- April 2011 (2)
- March 2011 (4)
- February 2011 (1)
- January 2011 (1)
- December 2010 (1)
- November 2010 (4)
- October 2010 (2)
- September 2010 (1)
- August 2010 (1)
- July 2010 (1)
- June 2010 (2)
- May 2010 (3)
- April 2010 (1)
- February 2010 (6)
- January 2010 (3)
- December 2009 (8)
- November 2009 (7)
- October 2009 (3)
- September 2009 (3)
- August 2009 (1)
- June 2009 (4)
- May 2009 (5)
- April 2009 (4)
- March 2009 (2)
- February 2009 (1)
- January 2009 (7)
- December 2008 (1)
- October 2008 (2)
- September 2008 (7)
- August 2008 (1)
- July 2008 (1)
- June 2008 (1)
- April 2008 (5)
- February 2008 (4)
- January 2008 (4)
- December 2007 (3)
- November 2007 (12)
- October 2007 (2)
- September 2007 (4)
- August 2007 (3)
- July 2007 (1)
- June 2007 (3)
- May 2007 (1)
- April 2007 (4)
- March 2007 (3)
- February 2007 (7)
- January 2007 (1)
- December 2006 (2)
- October 2006 (2)
- September 2006 (6)
- July 2006 (4)
- June 2006 (2)
- May 2006 (6)
- April 2006 (3)
- March 2006 (6)

### Meta

# Category Archives: geometry

## Complex multiplication: proof

In my previous post, I claimed that when multiplying two complex numbers, their lengths multiply and their angles add, like this: In particular, this means that there are always different complex numbers whose th power is equal to : they … Continue reading

## Complex multiplication and roots of unity

If played around with the question from my previous post, you probably found something like the following: That is, as the powers of we get , , and with all possible sign combinations. Of course, since , if we continue … Continue reading

## Complexifying our dots

It’s time to up our game a bit. Previously we have considered some cool pictures with dots and bespoked circles, looking for patterns, without really considering what sort of mathematical objects these circles might represent. In fact, they turn out … Continue reading

## Totient sums

I took a bit of a break to travel to Japan for a conference, but I’m back now to continue the series I started with Post Without Words #10, a follow-up post, and Post Without Words #11. Recall that we … Continue reading

## A few words about PWW #10

If you still want to think more about the picture in my previous post, stop reading now! Here’s a simple way to think about how the picture is made, as noted by Fergal Daly. The th circle (starting with ) … Continue reading

Posted in geometry, pattern, pictures, posts without words
Tagged gcd, primitive, roots, unity
5 Comments