topics: |  math | young-adult |

a dream-fantasy in which a "number devil" appears in the dreams of robert, and initiates him into the pleasures of mathematics.

robert often has dreams in which a huge fish is gobbling him up, or he is falling down an endless slide, so when the number devil starts coming in his dream, it is a welcome relief.

 
 
    in robert's dreams, he is often swallowed by a 
    big ugly fish

in the first dream, the number devil introduces the 
infinitude of integrers, and introduces the 
multiplication series: 

			11 x 11 = 121
			111 x 111 = 12321
			1111 * 1111 = 1234321

but the series breaks down when there are more than 
ten one's, causing the number devil to explode 
and disappear.  

 
devil explodes when he realizes that 11 111 111 111
squared does not follow the pattern
(answer is 1234567 9012098 7654321)

the second dream deals with zero and its importance in representing
numbers, contrasting robert's birth year (1986) as written in roman
(MCMLXXXVI).  Squaring is called making numbers "hop"; and 1986 is explained
in terms of hops of ten - the positional system. 

	Still in a daze the next morning, Robert said to his mother, "Do you
	know the year I was born? It was 6x1 and 8x10 and 9x100 and 1x1000."

	"I don't know what's got into the boy lately," said Robert's
	mother... p. 46

 
devil breaks into a huge grin, revealing an endless series of teeth...

The third night introduces the prime numbers, and how finding prime
factors is a difficult task.   Fourth dream deals with irrational numbers
(called "unreasonable") and square roots (called "rutabagas").  the fifth
chapter (dream) introduces the triangle numbers 1, 3, 6, 10, ... and its
relations to the square numbers (4 = 1+3; 9 = 3+6 etc).  The sum of the first n
triangle numbers = n+1-th triangle number = (1 + n) + (2 + n-1) + ... giving
the relation n-1. n /2.   

subsequent dreams deal with Fibonacci numbers, pascal's triangle and its many
properties, combinations and permutations etc.  the inverted triangle
composites (evens, multiples of five) in the pascal triangle were quite
interesting for me also: 

 
inverted triangles emerge when looking at multiples of two. 
similarly, for multiples of five, etc. 

the relation between
combinations and pascal's triangle is developed through an interesting relationship: 

 
the diagonals are nC2, nC3 etc. even nC8. 

the tenth night introduces the golden section (sqrt(5)+1)/2; solution of the
recursive fraction x = 1 + 1/x; however this discussion is left half-finished
as the discourse moves to euler's relation f+v = e+1, and then to solids. 

the eleventh night is one of the more abstract, where the process of
mathematical proof is explained as a series of jumps across stones
on a mountain river - whether one can reach the other shore or not remains
uncertain.  

the final chapter introduces some historical figures (gustav klein is
translated as "dr. happy little".  the number devil is given a mayan name,
teplotaxl.  

on the whole, though there is no plot to speak of, but the
illustrations and the intrinsic interest of the matter keeps you going. 
the book's style and getup and its big fonts and well-executed
illustrations are quite appealing.  

certainly not for very young children, though.  

a historical quibble: the invention of zero is attributed to an ancient
chinese scholar. (12th night, p.245).