Way out base: 2i

Donald Knuth author of the seminal multi-volume work, “The Art of Computer Programming” the father of the analysis of algorithms creator of the TeX computer typesetting system and many, many other contributions see http://en.wikipedia.org/wiki/Donald_Knuth#Humor

Quater-imaginary base “The quater-imaginary numeral system was first proposed by Donald Knuth in 1955, in a submission to a high-school science talent search.” – from http://en.wikipedia.org/wiki/Quater-imaginary_base base: 2i digits: { 0, 1, 2, 3 }

Quater-imaginary base Example: What is 1032i in base 10?

Quater-imaginary base k (2i)k 0 1 1 2i 2 -4 3 -8i 4 16 5 32i 6 -64 7 -128i 8 256 Example: What is 1032i in base 10? First, let’s build a table for 2i powers. 1 x (2i)2 + 0 x (2i)1 + 3 x (2i)0 -4 + 0 + 3 -1

Quater-imaginary base Example: 11012i is (-3-8i)10 1 x (2i)3 + 1 x (2i)2 + 0 x (2i)1 + 1 x (2i)0 -8i - 4 + 0 + 1 -8i - 3

Quater-imaginary base Example: 10300032i is -1310 1 x (2i)6 + 0 x (2i)5 + 3 x (2i)4 + 0 x (2i)3 + 0 x (2i)2 + 0 x (2i)1 + 3 x (2i)0 -64 + 48 + 0 + 0 + 0 + 3 -13

Quater-imaginary base So what is 10.22i in base 10? 1 x (2i)1 + 0 x (2i)0 + 2 x (2i)-1 = 2i + 0 + 2 x 1 / (2i) = 2i + 2 / (2i) = 2i + 1 / i = 2i – i = i Does 1 / i = -i? Proof: i x (1/i) = -i x i (multiply both sides by i) i / i = - (i x i) 1 = - (-1) 1 = 1

In researching this presentation, I noticed something very interesting (and somewhat disturbing) . . .

Donald Knuth or Dwight from “The Office”?????